Phys. Fluids 32, 024105 (2020); https://doi.org/10.1063/1.5129220 32, 024105

© 2020 Author(s).

On the stability of a heated rotating-disk boundary layer in a temperature-dependent viscosity fluidCite as: Phys. Fluids 32, 024105 (2020); https://doi.org/10.1063/1.5129220Submitted: 27 September 2019 . Accepted: 16 January 2020 . Published Online: 04 February 2020

R. Miller, P. T. Griffiths, Z. Hussain , and S. J. Garrett

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On the stability of a heated rotating-diskboundary layer in a temperature-dependentviscosity fluid

Cite as: Phys. Fluids 32, 024105 (2020); doi: 10.1063/1.5129220Submitted: 27 September 2019 • Accepted: 16 January 2020 •Published Online: 4 February 2020

R. Miller,1 P. T. Griffiths,2 Z. Hussain,3 and S. J. Garrett4,a)

AFFILIATIONS1School of Engineering, University of Leicester, Leicester LE1 7RH, United Kingdom2Centre for Fluid and Complex Systems, Coventry University, Coventry CV1 5FB, United Kingdom3Centre for Mathematical Modelling and Flow Analysis, Manchester Metropolitan University, Manchester M1 5GD,United Kingdom

4School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, United Kingdom

a)Author to whom correspondence should be addressed: sjg50@leicester.ac.uk

ABSTRACTThe paper presents a linear stability analysis of the temperature-dependent boundary-layer flow over a rotating disk. Gas- and liquid-typeresponses of the viscosity to temperature are considered, and the disk rotates in both a quiescent and an incident axial flow. Temperature-dependent-viscosity flows are typically found to be less stable than the temperature independent cases, with temperature dependences thatproduce high wall viscosities yielding the least stable flows. Conversely, increasing the incident axial flow strength produces greater flow stabil-ity. Transitional Reynolds numbers for these flows are then approximated through an eN -type analysis and are found to vary in approximateconcordance with the critical Reynolds number. Examination of the component energy contributions shows that flow stability is affectedexclusively through changes to the mean flow. The results are discussed in the context of chemical vapor deposition reactors.

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I. INTRODUCTION

A convectively unstable flow is the one featuring the manifesta-tion of small disturbances in an otherwise laminar state. These con-vective instabilities develop and grow downstream, eventually gain-ing sufficient amplitude to trigger non-linear behavior that causesthe flow to transition from the laminar to turbulent regime. As such,there is a long history of interest in the mechanisms and controlof these instabilities. The effects of small disturbances on the lam-inar flow over a flat plate were first investigated by Tollmien1 andSchlichting,2 who considered the amplification of linear, wave-likedisturbances (aptly named Tollmien–Schlichting waves) as the pri-mary mechanism for flow instability. Later, Gregory et al.3 inves-tigated the formation of co-rotating stationary vortices (crossflowinstabilities) on the rotating disk.

The physics of a rotating disk situated in a stationary fluidcan be described as a centrifugal pump or fan. As the disk rotates,

viscous effects cause the surrounding fluid to rotate with it. Thefluid is dragged down toward the disk surface before flowing radi-ally outwards to be cast off at the disk edge. The seminal workof von Kármán4 found exact similarity solutions to the Navier–Stokes equations for the infinite-plane rotating disk problem, allow-ing them to be reduced to a set of ordinary differential equations thatcharacterize the mean flow.

Much of the early motivation for the study of the stability ofthe rotating-disk flow is drawn from aerospace engineering; boththe rotating disk and the swept wing exhibit inflectional mean-flowprofiles susceptible to crossflow instabilities. These arise from invis-cid effects and manifest as spiral vortices as first observed exper-imentally by Gray5 for a swept wing and later by Gregory et al.3

for the rotating disk. The rotating disk, however, proved advanta-geous for the study of crossflow instabilities due to its axisymmetricgeometry and exact similarity solutions of the Navier–Stokes equa-tions allowing for significantly simplified mathematical investigation

Phys. Fluids 32, 024105 (2020); doi: 10.1063/1.5129220 32, 024105-1

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when compared to the swept wing. The study of rotating-disk flowshas subsequently developed into its own field, with direct indus-trial applications as well as remaining a model flow of fundamentalinterest.

One of the key differences between the swept wing and therotating disk boundary layers is that the latter is subject to Coriolisforces. Malik6 discovered an additional instability mechanism aris-ing from this by performing a numerical analysis to compute theneutral curves for stationary instability modes in the rotating-diskboundary layer. This work was the first departure from the previ-ous fourth-order Orr–Sommerfeld type analyses in favor of a sixth-order system that includes rotational Coriolis and viscous terms. Thework, also verified asymptotically by Hall,7 found that two branchesexist: the previously known upper branch associated with crossflowinstability (type I) and a lower branch associated with streamlinecurvature (SC) and the balancing of Coriolis and viscous forces (typeII). The type II mode has been found in all subsequent studies andis known to be particularly important when traveling modes arepermitted; see, for example, the work of Hussain et al.8 Observingthe type II disturbance experimentally has, however, proved diffi-cult as the type I mode is almost always dominant for all practicalflows where unavoidable roughness acts to select modes that rotatewith the disk surface. The work presented by Fedorov et al.9 is oftenpointed to as a rare observation of the type II mode, and more recentexperimental work10,11 has shown the potential for traveling modesto be selected.

This paper considers the linear convective stability behaviorof flows over the rotating disk under a temperature-dependent vis-cosity model and is closely related to the sister publication Milleret al.12 concerning the Blasius/Falkner–Skan boundary layer stabil-ity under the same viscosity model. There are, however, a num-ber of key physical differences between the stability of the vonKármán and Blasius/Falkner–Skan boundary layers. Notably, whilerotation of the disk is sufficient for a potentially unstable bound-ary layer due to the inviscid nature of the crossflow instability,Tollmien–Schlichting waves are generated through viscous effectsand as such external forcing is required to generate a boundary layer(and, therefore, instability) over the flat plate. The two-dimensionalnature of the plate problem also renders all instabilities “travel-ing,” with no direct analog to the stationary vortices formed onthe disk. Nevertheless, the methodology presented here is muchthe same as that presented in the work of Miller et al.12 and theinterested reader is referred to there for an in-depth discussionregarding two-dimensional temperature dependent boundary layerflows.

While normal-mode linear stability analyses typically aim tomodel the early stages of transitional flows, numerous studies havehighlighted the importance of nonlinearity and non-normality inthe prediction of transition to turbulence, as well as their role inthe formation of primary instabilities. The eN analysis developedby van Ingen13 (reviewed extensively in van Ingen14 and utilizedin this paper) is often used to consolidate nonlinearity within thelinear framework, utilizing the linear amplification rates to findempirical agreement with transitional Reynolds numbers observedin experiments. Bertolotti et al.15 utilize parabolic stability equationsto examine both non-parallel and nonlinear effects on the Blasiusboundary layer stability, concluding that while both are destabiliz-ing with reference to the linear results, neither is significant enough

to account for the discrepancies between neutral curve data obtainedfrom theory and experiments.

Non-normal studies have allowed for the investigation ofglobal instabilities in boundary layers and their role in transi-tion. In response to the assertion of Huerre and Monkewitz16

that a linear analysis is sufficient for investigating global behavior,Chomaz17 identifies that instability behavior becomes rapidly non-linear beyond a global threshold. Lingwoodan18 provided the bench-mark for the absolute instability of the rotating disk boundary layerflow, finding theoretical and experimental agreement of the flowbecoming absolutely unstable at a Reynolds number of 510 and tran-sition to turbulence at 513. From this, it was concluded that absoluteinstability acts as a linear threshold, beyond which non-linear behav-ior in the flow begins to dominate and develops into fully turbulentflow. Briggs’ pinching criterion states that the absolutely unstableparameters must lie within the range of those that are convectivelyunstable (that is, the absolute neutral stability curve is containedwithin the convective neutral curve). To this end, Healey19 showsthat the absolute neutral stability curve associated with the rotat-ing disk flow also exhibits the two branch structure of its convectiveequivalent. Utilizing front propagation theory and direct numericalsimulation (DNS), Cossu et al.20 examined the nonlinear behaviorof disturbances beyond the threshold for global instability, findingthat for large-amplitude impulses, the globally unstable behavior ofthe Blasius boundary layer is decisively nonlinear, yet the wave frontspeed is identical to that of its linear counterpart. The findings ofHealey19 and Cossu et al.20 lead to the conclusion that conduct-ing local linear analyses of unstable flows over both the disk andthe plate is still of importance when extending to a global stabil-ity analysis to inform the nonlinear behavior. While nonlinearity,non-normality, and absolute instability are beyond the scope of thework presented here, it is acknowledged that they form an active areaof research that is greatly informed by the behavior of convectivelyunstable flows. As such, the work presented here and in the work ofMiller et al.12 can be considered as a first step toward understand-ing the global criteria for instability and transition in modified vonKármán and Blasius/Falkner–Skan flows. The interested reader isdirected to Ref. 17 for an extensive review of non-normal, nonlinear,and global studies and their role in boundary layer stability. In addi-tion, the review presented by Lingwood and Alfredsson21 includesrecent developments regarding absolute instability over rotating diskboundary layers.

The significance of viscous effects on flow instability has ledto interest in variable viscosity flows. Strong temperature gradientscan cause the viscosity of a fluid to vary immensely and, as such,there is a great deal of literature studying temperature dependentviscosity flows. Ling and Dybbs22 assert that an inverse-linear rela-tionship between temperature and viscosity accurately models thebehavior of a range of real fluids. Kafoussias and Williams23 utilizedthis relationship in their study of the free-forced vertical flow withtemperature dependent viscosity over a flat plate. There, it is foundthat fluids that become more viscous with increasing temperaturecreate both a narrower temperature profile and a velocity profile,while the effect is reversed for fluid viscosities that decrease withtemperature. Wall and Wilson24 considered the effects of two dif-ferent exponential-type viscosity temperature relationships on Bla-sius flow stability. In both models, they noted that for an isother-mally heated plate, an exponentially decaying viscosity produces a

Phys. Fluids 32, 024105 (2020); doi: 10.1063/1.5129220 32, 024105-2

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destabilized flow, while an exponentially increasing viscosity is morestable. An inverse linear temperature dependent viscosity law is alsoconsidered by Jasmine and Gajjar25 in their study of flow stabilityover a rotating disk, and similar qualitative effects as cited by Kafous-sias and Williams23 are observed, as well as finding that a viscos-ity that decreases more rapidly with increasing temperature yieldsa less stable flow. Again, this broadly agrees with the findings ofWall and Wilson.24 The work presented in this paper considers theinverse-linear relationship between viscosity and temperature uti-lized by Ling and Dybbs,22 Kafoussias and Williams,23 and Jasmineand Gajjar,25 amongst others.

There is a wealth of literature concerning flows over rotatingmedia, most of which set the fluid as otherwise stationary. Morerecent studies examined the effects of an oncoming flow over rotat-ing media to better understand transitional flows in industrially rel-evant configurations. Chen and Mortazavi26 produce a model of aChemical Vapor Deposition (CVD) reactor featuring a forced flowparameter to represent the injection of gas into the system. Gar-rett et al.27 present a linear stability analysis (both asymptotic andnumerical) of the enforced axial flow over a rotating cone, where it isfound that a stronger axial flow results in delaying the onset of tran-sition in both modes of instability. This is extended to the flow overa rotating disk in the work of Hussain et al.,8 where a similar result isfound.

The work presented in this paper has particular relevance tovertical CVD systems where it is important that laminar flow ismaintained over a heated substrate rotating in a forced flow. Thisis necessary to promote regular and cohesive film growth. Longitu-dinal roll instabilities are well established in buoyancy-driven flowswith a free convection element28 and have been observed in CVDreactors at Rayleigh numbers (the Rayleigh number is the productof the Grashof and Prandtl numbers, i.e., the ratio of buoyancy andviscous forces multiplied by the ratio of momentum and thermaldiffusivities) beyond 1708.29 While there is a great deal of literatureregarding solutions to the laminar flow in a CVD reactor, there is (tothe authors’ knowledge) no such study on the existence of unstableor transitional flows due to forced convection. Convective instabil-ities arising from forced flow are perhaps not considered due to adichotomic view of the flow being laminar or turbulent. Typically,the operating Reynolds number of a vertical CVD reactor is in therange of 1–200,29,30 which most would consider to be firmly in thelaminar regime. However, modifications to the stability solutionsof flows over both the disk and the plate have shown that destabi-lizing effects are possible, as demonstrated by Wall and Wilson24

where a critical Reynolds number of approximately 220 is achievedat one extreme of their viscosity temperature dependence param-eter. Hence, it is possible that certain destabilizing criteria may besufficiently influential to produce primary instabilities within theoperational parameters of a CVD reactor. It is widely accepted thatthe steep temperature gradients in a CVD reactor cause variationin all physical properties of the fluid,29–31 supporting the need fora temperature dependent viscosity model, and as such this studyserves as the first step toward a larger stability analysis of flows inCVD reactors.

In particular, the work presented here considers the convectiveinstability of the boundary layer resulting from the forced flows offluids with temperature dependent viscosity incident on a heatedrotating disk. It proceeds as follows: In Sec. II, we formulate and

solve the equations representing the steady, mean flow of the fluidover the disk. The linear convective stability of these flows is thenassessed in Sec. III, via the formulation of neutral stability curvesand eN and integral energy analyses. Finally, our conclusions arediscussed in Sec. IV.

II. PROBLEM FORMULATION AND MEAN-FLOWANALYSISA. Derivation of the mean-flow equations

Consider a heated disk of infinite radius, rotating about its cen-ter of axis z∗ with angular velocity Ω∗. The disk is situated perpen-dicularly downstream from an incompressible Newtonian fluid. Asthe disk rotates, fluid is entrained toward its surface and moves withvelocity U∗ = (u∗, v∗, w∗), representing boundary-layer velocitiesin the radial, azimuthal, and axial directions (r∗, θ, and z∗), respec-tively. The disk is heated to a temperature T∗w , while the tempera-ture of the fluid in the free stream is T∗∞; the fluid in the boundarylayer has temperature T∗. Note that in all that follows, an asteriskindicates a dimensional quantity.

In the rotating frame of reference, the flow can be described bythe Navier–Stokes equations in cylindrical polar coordinates,

∇∗ ⋅U* = 0, (1a)

( ∂

∂t∗+ U∗ ⋅ ∇∗)U∗ + 2Ω∗ ×U∗

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶R1

+ Ω∗ ×Ω∗ × r∗´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

R2

= − 1ρ∗∇∗p∗ +

1ρ∗∇∗ ⋅ τ∗, (1b)

( ∂

∂t∗+ U∗ ⋅ ∇∗)T∗ = k∗

ρ∗C∗p∇∗2T∗, (1c)

where t∗ is the time, Ω∗ = (0, 0, Ω∗) is the angular velocity vector,r∗ = (r, 0, 0) is the radial position vector, ρ∗ is the density, p∗ is thepressure, k∗ is the thermal diffusion coefficient, and C∗p is the specificheat of the fluid. The viscous stress tensor is τ∗ = μ∗γ∗, where μ∗is the viscosity and γ∗ = [∇∗U∗ + ∇∗(U∗)T] is the rate-of-straintensor. The terms labeled R1 and R2 are rotational terms arising fromCoriolis and centrifugal acceleration forces, respectively. The heatcontinuity Eq. (1c) is coupled to Eqs. (1a) and (1b) via the imposedtemperature dependence of the fluid viscosity

μ∗ = μ∗∞1 + ε∗(T∗ − T∗∞)

.

Here, μ∗∞ is the fluid viscosity in the free stream and ε∗ is the temper-ature dependence constant in inverse temperature units. Note thatε∗ is an intrinsic property of the fluid and changing ε∗ would repre-sent a fundamentally different fluid. For an isothermally heated disk(as considered here), ε∗ > 0 represents the viscothermal behavior ofa liquid (a fluid that becomes less viscous with increasing tempera-ture), while ε∗ < 0 represents that of a gas (a fluid that becomes moreviscous with increasing temperature, with appropriate considerationof parameters to ensure positive viscosity). Applying ε = 0 returns aconstant viscosity and uncouples Eq. (1c) from Eqs. (1a) and (1b).

Phys. Fluids 32, 024105 (2020); doi: 10.1063/1.5129220 32, 024105-3

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Ling and Dybbs22 cite this expression for μ∗ as an accurate repre-sentation of the relationship between viscosity and temperature fora range of real fluids such as water and crude oil; this motivates itsuse here.

The motion of the fluid is influenced both by the rotation of thedisk and the axially enforced element toward the disk surface. Theenforced axial flow component is derived from the radial pressurebalance at the boundary layer edge

U∗e∂U∗e∂r∗

= − 1ρ∗

∂p∗

∂r∗, (2)

where U∗e = C∗r∗ is the flow velocity at the boundary-layer edge.The constant C∗ represents the strength of the enforced flow ininverse time units.

We non–dimensionalized (1) by the scaling variables

U∗ = (u, v, δw)U∗∞, (r∗, z∗) = (r, δz)L∗, t∗ = tL∗

U∗∞,

p∗ = p (U∗∞)2ρ∗, T∗ − T∗∞ = T (T∗w − T∗∞), ε∗ = εT∗w − T∗∞

,

where U∗∞ = L∗Ω∗, with L∗ defined as a generic length scale. Theboundary-layer scaling constant is δ. We note that setting ε as greateror less than 0 remains representative of liquid- or gas-type viscositybehavior, respectively, consistent with the definition of ε∗.

The laminar flow is steady and axisymmetric, meaning thatderivatives with respect to t and θ are neglected. We define theReynolds number as Re = U∗∞L∗ρ∗/μ∗∞ and, with Re≫ 1, determinethe governing equations at leading order to be

1r∂(ru)∂r

+∂w

∂z= 0, (3a)

u∂u∂r

+ w∂u∂z− (v + r)2

r= rT2

s +1

δ2Re(μ∂

2u∂z2 +

∂μ∂z

∂u∂z), (3b)

u∂v

∂r+ w

∂v

∂z+

uvr

+ 2u = 1δ2Re

(μ∂2v

∂z2 +∂μ∂z

∂v

∂z), (3c)

u∂w

∂r+ w

∂w

∂z= − 1

δ2∂p∂z

+1

δ2Re(μ∂

2w

∂z2 + 2∂μ∂z

∂w

∂z), (3d)

u∂T∂r

+ w∂T∂z= 1δ2PrRe

∂2T∂z2 , (3e)

where it is concluded that δ = O(Re−1/2) and the boundary-layerthickness is δ∗ = δL∗ =

√μ∗∞/ρ∗Ω∗. Note that Pr = C∗p ρ∗/k∗ is the

Prandtl number that is henceforth set to Pr = 0.72. This is a suitablevalue for various fluids including air.

Imposing the similarity solutions,

u = rU(z), v = rV(z), w =W(z), p = P(z), T = Θ(z),

(4)

leads to a modified form of the von Kármán equations

2U + W′ = 0, (5a)

U2 + U′W − (V + 1)2 − (μU′)′ − T2s = 0, (5b)

2U(V + 1) + V′W − (μV′)′ = 0, (5c)

WW′ + P′ − μW′′ − 2μ′W′ = 0, (5d)

PrWΘ′ −Θ′′ = 0, (5e)

where μ = 1/(1+εΘ) and primes indicate differentiation with respectto z. We note that (5d) is obtained from a leading order expansionof P(z) in terms of δ. At leading order, one finds that P(z) = P(z = 0)= P0. The solution at the next order is then determined from (5d).The system is subject to the boundary conditions

U(0) = V(0) =W(0) = P(0) = Θ(0) − 1 = 0, (6a)

U(z →∞)→ Ts, V(z →∞)→ −1, Θ(z →∞)→ 0, (6b)

which reflect the no-slip criteria and Coriolis force balance condi-tions in the rotating frame. Note that the condition for the outerradial velocity is modified according to the pressure balance con-dition at the boundary-layer edge due to the enforced axial flow.The conditions for Θ(0) and Θ(z → ∞) are indicative of a heateddisk.

The familiar temperature-independent problem is returnedwhen ε = 0, and the enforced axial flow is eliminated when Ts = 0.Setting ε = Ts = 0 returns the standard von Kármán equations.

With the exception of (5d), the system (5) is solved utilizinga Newton–Raphson searching routine to find a suitable conditionfor the unknowns U′(0) and V′(0), alongside a double precision,fourth-order Runge–Kutta integration scheme to solve the meanflow functions through the boundary layer. Convergence is reachedat zmax = 20 to a tolerance of 10−7, which can be considered represen-tative of the free stream although plots will be truncated to z = 10 tobetter visualize the effects of varying the free parameters. Note that(5d) is independently solvable for the temperature independent case,i.e., ε = 0,

P − P0 = −(2U +W2

2),

with the stipulation that P(z = 0) = P0. However, the introduction ofa temperature dependent viscosity leads to an additional term owingto the viscous stress tensor, 2μ′W′, and an incomplete solution thatrequires additional numerical integration is obtained

P = −(2μU +W2

2) + ∫ μ′W′ dz,

in this case, the P0 term is absent because, as noted above, it isobtained in the leading order expansion of P(z).

B. Mean-flow profiles1. Temperature-dependent viscosity (no axial flow)

Figure 1 shows the resulting mean-flow profiles for a range ofε values when Ts = 0. We see that μ is uniformly 1 for the tempera-ture independent case of ε = 0. For ε ≠ 0, the viscosity converges onμ(z → ∞) = 1; this is due to the uniform temperature in the

Phys. Fluids 32, 024105 (2020); doi: 10.1063/1.5129220 32, 024105-4

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FIG. 1. Mean-flow profiles for −0.75 ≤ ε ≤ 0.75 in 0.25 increments. The black lines indicate the temperature independent (ε = 0) solution.

free stream. Prior to this convergence, the viscosity is everywheregreater or less than one, i.e., the viscosity is increased or decreasedthroughout the boundary layer, for ε < 0 and ε > 0, respectively.

Increasing the value of ε is found to move the radial velocityprofile, U(z), closer to the disk surface. This can be interpreted as anarrowing of the fluid boundary layer. Increasing ε also reduces theviscosity at the wall, reducing the wall shear stress and the viscousinteraction between the disk and the fluid. Hence, the ability of thedisk to accelerate the fluid is diminished, resulting in a reduced U(z)profile jet.

The azimuthal profile, V(z), reaches convergence at a reducedvalue of z as ε is increased and is further evidence of the narrowingeffect. This is consistent with the absolute value of V′(z) at the wallincreasing with ε, showing that V(z) grows more significantly withan increasing distance from the disk surface. An interesting detailof the V′(z) profile is that it becomes inflectional for ε < 0, meaningthat there is a small region for which the azimuthal velocity increaseslinearly through the boundary layer, sharply increasing close to thedisk surface and more steadily at a greater distance. This is of partic-ular relevance to the results presented in Sec. III E and are discussedin further detail there.

Figure 1 also demonstrates that increases in ε reduces the ver-tical acceleration of W(z) in the region close to the disk, as wellas the converging outflow velocity closer to the free stream. Thecontinuity equation suggests that this axial response is related tothe effects of ε on the radial profile. We interpret this physically

as a result of the reduced viscosity inhibiting the centrifugal pumpeffect.

In contrast to the behavior of the velocity profiles, the meantemperature profile, Θ(z), moves away from the disk surface with ε.Mathematically, this is a consequence of the W(z)-dependent termin (5e): as ε is increased, the term is decreased due to the reductionin W(z). In turn, Θ′(z) is reduced, resulting in a shallower gradientin the Θ(z) profile. As such, while the momentum boundary layer isnarrowed, increasing ε results in a broadening of the thermal bound-ary layer, meaning the reduction in outflow must be favorable to thetransferral of heat from the disk to the fluid.

Where values of ε match, the solutions presented here are inexact agreement with those of Jasmine and Gajjar.25

2. Enforced axial flow (no temperature dependence)Figure 2 shows mean-flow solutions for a range of Ts val-

ues for ε = 0. The prominent feature of the radial velocity pro-file, U(z), is the free stream convergence to the applied value ofTs according to the boundary condition outlined in (6). The max-imum value of U(z) increases with increasing axial flow strengthalthough the characteristic inflection is suppressed for higheraxial flow strengths. This maximum also occurs at a moderatelyincreased distance from the disk surface as the axial flow strength isincreased. The profiles then converge to the boundary condition ata reduced distance, which is interpreted as a gentle narrowing of the

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FIG. 2. Mean-flow profiles for the range 0 ≤ Ts ≤ 0.3 in 0.05 increments. The black lines indicate the zero axial enforcement (Ts = 0) solution.

boundary layer (consistent with the observed effects of temperaturedependence).

The azimuthal velocity profile, V(z), moves closer to the disksurface with increasing axial flow strength. Recall that the axial flowparameter Ts represents the ratio of forced flow to the rotationalspeed of the disk. As such, increasing Ts inhibits the influence ofthe disk’s rotation on the flow, meaning that the fluid converges tozero rotational velocity [V(z) → 0] at a reduced distance from thedisk surface.

The axial flow profiles, W(z), exhibit a change in shape forTs ≠ 0 and become unbounded with increasing the distance fromthe disk. While acceleration in the axial direction is to be expectedwith enforced axial flow, the outflow velocity should still, in reality,converge at a certain distance from the disk surface. This unphysicalunbounded acceleration is instead a result of the boundary conditionimposed at U(z →∞). However, the physical element of increasedoutflow does support the narrowing of the boundary layer as fluid isejected from the disk surface more rapidly. This is discussed furtherby Hussain et al.8 and, where parameter values match, the solu-tions presented here are in exact agreement with those presentedthere.

3. Combined effects

Figure 3 shows the combined effects of a temperature depen-dent viscosity and enforced axial flow on the mean flow over therotating disk.

The interaction of the two effects is most clearly observed in theviscosity profiles. As the axial flow strength is increased, the viscosityprofiles for the cases when ε ≠ 0 move closer to the disk surface.The change in viscous behavior then becomes localized to the regionvery close to the disk surface, and the viscosity converges to that ofthe free stream over a reduced distance. This is consistent with thenarrowing of the boundary layer seen from increasing Ts in Fig. 2(where ε = 0).

Increasing the values of ε and Ts individually, both result in theradial velocity profile, U(z), converging to the free stream bound-ary condition at a reduced distance from the disk surface. WhenTs ≠ 0, we see that increasing ε results in similar movement of theradial profile toward the disk surface. Likewise, when ε ≠ 0, the effectof increasing Ts results in the profile maximum moving away fromthe wall, but the overall convergence of the profile occurs closer tothe disk surface.

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FIG. 3. Mean-flow profiles for a range of temperature dependencies and axial flow strengths. Solid lines (–) Ts = 0; dashed lines (−−) Ts = 0.15; dashed–dotted lines (−⋅)Ts = 0.3. Line colors represent ε = −0.75 (blue), 0 (red), and 0.75 (yellow).

Similarly, the azimuthal profiles reveal that increasing both Tsand ε produce effects consistent with those observed in the indi-vidual cases, i.e., increasing both parameters shifts the V(z) pro-file toward the disk surface. In summary, the individually observedeffects of increasing ε and Ts combine to result in further narrowingof the boundary layer.

Interestingly, the effect of varying the temperature dependenceon the axial and temperature profiles appears to be impacted by thestrength of enforced axial flow. When Ts = 0.15, varying ε has a neg-ligible impact on W(z), while at Ts = 0.3, the effect of increasing ε isthe opposite of that seen when Ts = 0, i.e., the outflow accelerationis increased. Similarly, Θ(z) has a negligible variation with ε whenTs = 0.15, while in the Ts = 0.3 case, the Θ(z) profile narrows withincreasing ε as opposed to the broadening effect seen when Ts = 0.Recalling the continuity equation, the effects on both W(z) and Θ(z)can be attributed to the boundary condition for U(z) when Ts ≠ 0.

We see that the radial profile converges to zero whenTs = 0, while increasing ε moves the radial profile toward the disksurface. As such, the area under the U(z) curve decreases withincreasing ε, which decreases the magnitude of W(z) throughoutthe boundary layer. However, when Ts ≠ 0, the radial profile con-verges to Ts, which drastically increases the area under U(z) andresults in the unbound acceleration observed in the W(z) profile.For sufficient axial flow, the narrowing effect of increasing ε furtherincreases the area under the U(z) curve. Hence, the axial outflow

velocity is increased throughout the boundary layer as can beobserved when Ts = 0.3. Due to its dependence on W(z), the Θ(z)profile is moved toward the disk surface when Ts = 0.3, narrowingthe thermal boundary layer (as opposed to the thickening effect seenwhen varying ε at Ts = 0).

Summary data for the mean-flow profiles for a range of flowparameters are shown in Table I.

III. LINEAR CONVECTIVE INSTABILITY ANALYSISIn this section, we derive the perturbation equations for the

linear instability of the flow and proceed to present a full stabilityanalysis under various parameter regimes.

A. Derivation of the perturbation equationsThe mean-flow solutions are now subject to small perturbation

quantities leading to

u(r, θ, z, t) = rR

U(z) + u(r, θ, z, t), (7a)

v(r, θ, z, t) = rR

V(z) + v(r, θ, z, t), (7b)

w(r, θ, z, t) = 1R

W(z) + w(r, θ, z, t), (7c)

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TABLE I. Mean-flow data for a range of flow parameters.

ε Umax U′(0) V′(0) Θ′(0) W(z→∞)

Ts = 0

−0.75 0.1884 0.2282 −0.2216 −0.3279 −1.0531−0.50 0.1855 0.3520 −0.3869 −0.3334 −0.9951−0.25 0.1828 0.4397 −0.5136 −0.3322 −0.93770.00 0.1808 0.5102 −0.6159 −0.3286 −0.88450.25 0.1794 0.5710 −0.7022 −0.3239 −0.83580.50 0.1784 0.6254 −0.7773 −0.3188 −0.79150.75 0.1779 0.6753 −0.8444 −0.3135 −0.7511

Ts = 0.15

−0.75 0.2153 0.2365 −0.2340 −0.3588 . . .

−0.50 0.2127 0.3684 −0.4117 −0.3717 . . .

−0.25 0.2103 0.4625 −0.5499 −0.3775 . . .

0.00 0.2083 0.5381 −0.6626 −0.3806 . . .

0.25 0.2069 0.6029 −0.7583 −0.3824 . . .

0.50 0.2058 0.6607 −0.8419 −0.3836 . . .

0.75 0.2051 0.7136 −0.9168 −0.3844 . . .

Ts = 0.3

−0.75 0.3005 0.2640 −0.2545 −0.3991 . . .

−0.50 0.3003 0.4187 −0.4518 −0.4192 . . .

−0.25 0.3001 0.5311 −0.6076 −0.4309 . . .

0.00 0.3000 0.6219 −0.7358 −0.4389 . . .

0.25 0.3000 0.6997 −0.8451 −0.4449 . . .

0.50 0.3000 0.7689 −0.9409 −0.4496 . . .

0.75 0.3000 0.8318 −1.0265 −0.4536 . . .

Ts Umax U′(0) V ′(0) Θ′(0)

ε = −0.75

0.00 0.1884 0.2282 −0.2216 −0.32790.05 0.1927 0.2288 −0.2245 −0.33660.10 0.2015 0.2316 −0.2287 −0.34710.15 0.2153 0.2365 −0.2340 −0.35880.20 0.2352 0.2436 −0.2402 −0.37160.25 0.2626 0.2528 −0.2471 −0.38510.30 0.3005 0.2640 −0.2545 −0.3991

ε = 0

0.00 0.1808 0.5102 −0.6159 −0.32860.05 0.1847 0.5133 −0.6269 −0.34470.10 0.1937 0.5225 −0.6429 −0.36210.15 0.2083 0.5381 −0.6626 −0.38060.20 0.2296 0.5599 −0.6851 −0.39970.25 0.2592 0.5879 −0.7097 −0.41920.30 0.3000 0.6219 −0.7358 −0.4389

TABLE I. (Continued.)

Ts Umax U′(0) V ′(0) Θ′(0)

ε = 0.75

0.00 0.1779 0.6753 −0.8444 −0.31350.05 0.1813 0.6791 −0.8619 −0.33770.10 0.1902 0.6918 −0.8867 −0.36110.15 0.2051 0.7136 −0.9168 −0.38440.20 0.2270 0.7443 −0.9508 −0.40760.25 0.2575 0.7838 −0.9878 −0.43070.30 0.3000 0.8318 −1.0265 −0.4536

p(r, θ, z, t) = 1R2 P(z) + p(r, θ, z, t), (7d)

T(r, θ, z, t) = Θ(z) + Θ(r, θ, z, t), (7e)

with the coefficients arising from the non-dimensionalization pro-cess. The governing equations are now non-dimensionalized undernew length, velocity, time, and pressure scales

U∗ = (u, v,w) r∗s Ω∗, (r∗, z∗) = (r, z) δ∗, t∗ = tL∗

r∗s Ω∗,

p∗ = p (r∗s Ω∗)2ρ∗, T∗ − T∗∞ = T (T∗w − T∗∞), ε∗ = εT∗w − T∗∞

.

where r∗s is the local radius of the disk at which instability occurs.The local Reynolds number is, therefore, defined as

R = r∗s Ω∗δ∗ρ∗

μ∗∞= r∗sδ∗= rs, (8)

which is interpreted as a non-dimensional radial distance along thedisk.

Upon substituting (7) into the newly non-dimensionalizedequations, the viscosity function undergoes a Taylor expansion tofirst order, allowing the mean flow form to be retained,

μ = 11 + ε(Θ + Θ)

≈ 11 + εΘ

(1 − εΘ1 + εΘ

) = μ + μ,

where μ = −εΘ/(1 + εΘ)2. The resulting stability equations are thenlinearized with respect to perturbation quantities, giving

∂u∂r

+ur

+1r∂v

∂θ+∂w

∂z= 0, (9a)

Lu +rU′w

R+

UuR− 2(V + 1)v

R

= −∂p∂r

+μR(∇2u − 2

r2∂v

∂θ− u

r2 ) +rU′′μ

R2 +2UR2

∂μ∂r

+μ′

R(∂u∂z

+∂w

∂r) +

rU′

R2∂μ∂z

, (9b)

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Lv +rV′w

R+

Uv

R+

2(V + 1)uR

= −1r∂p∂θ

+μR(∇2v +

2r2

∂u∂θ− v

r2 ) +rV′′μ

R2 +2UrR2

∂μ∂θ

+μ′

R(∂v∂z

+1r∂w

∂θ) +

rV′

R2∂μ∂z

, (9c)

Lw +W′w

R= −∂p

∂z+μR∇2w +

W′′μR

+rU′

R2∂μ∂r

+V′

R2∂μ∂θ

+2R(μ′ ∂w

∂z+ W′ ∂μ

∂z), (9d)

LΘ + Θ′w = 1RPr∇2Θ, (9e)

where

L = ∂

∂t+

rUR

∂

∂r+

VR

∂

∂θ+

WR

∂

∂z,

∇2 = ∂2

∂r2 +1r∂

∂r+

1r2

∂2

∂θ2 +∂2

∂z2 .

This system is now subject to a parallel-flow approxima-tion, wherein all r coefficients are replaced by R. This approx-imation is representative of a constant boundary-layer thicknessacross the disk surface. Physically, the rotating-disk boundary-layer does not exhibit radial growth and the terminology is bor-rowed from the approximation’s use in growing boundary layers,such as Blasius or Falkner–Skan flows. Following this substitu-tion, the stability equations become separable in r, θ, and t, allow-ing the perturbation quantities to be expressed in normal modeform,

(u, v, w, p, Θ) = (u, v, w, p, Θ)ei(αr+nθ−ωt),

where the variables marked with a tilde are the perturbation eigen-functions dependent on z. Here, α = αr + iαi is the complexradial wavenumber such that αi < 0 denotes convectively unsta-ble disturbances that grow radially. The azimuthal wavenumber,n = βR, is an integer quantity representing the number of vorticespresent on the disk, and ω is the frequency of the disturbance.Disturbances that rotate with the disk surface are represented byω = 0 in this frame of reference and are henceforth referred to asstationary.

The perturbation quantities in (9) are now expressed in thenormal mode form, and all O(R−2) terms are considered negligible,

iαu +uR

+ iβv + w′ = 0, (10a)

Du +Wu′

R+ U′w +

UuR− 2(V + 1)v

R

= −iαp +μ(−α2u − β2u + u′′)

R+

U′′μR

+μ′(u′ + iαw)

R+

U′μ′

R,

(10b)

Dv +Wv′

R+ V′w +

Uv

R+

2(V + 1)uR

= −iβp +μ(−α2v − β2v + v′′)

R+

V′′μR

+μ′(v′ + iβw)

R+

V′μ′

R,

(10c)

Dw +Ww′

R+

W′wR= −p′ +

μ(−α2w − β2w + w′′)R

+W′′μ

R

+iμ(αU′ + βV′)

R+

2(μ′w′ + W′μ′)R

, (10d)

− iωΘ+ iαUΘ+ iβVΘ+WΘ′

R+Θ′w = (−α

2Θ − β2Θ + Θ′′)RPr

, (10e)

where D = i(−ω + αU + βV) and μ = −εΘ/(1 + εΘ)2. The sys-tem (10) states the governing perturbation equations as a quadraticeigenvalue problem of the form A2α2 + A1α + A0 = Q, whereQ = (u, v, w, p, Θ)T is the vector of eigenfunctions, and the quanti-ties Aj are matrices containing the coefficients of the O(αj) terms.The eigenfunctions are then computed according to the bound-ary conditions that constrain the perturbations to reside within theboundary layer

u(y = 0) = v(y = 0) = w(y = 0) = w′(y = 0) = Θ(y = 0) = 0,

(11a)

u(y →∞)→ v(y →∞)→ w(y →∞)→ p(y →∞)

→ Θ(y →∞)→ 0. (11b)

The neutral temporal and spatial stability solutions areobtained via a Chebyshev polynomial discretization method. Anexponential map is used to transform the Gauss–Lobatto colloca-tion points into the physical domain. The stability equations are thensolved as primitive variables over 100 collocation points distributedbetween the upper and lower boundaries, with the exception of theconditions described in (11) which are imposed at z = 0 and z = zmax,where suitable mean-flow convergence is again found at zmax = 20.Further details of the numerical method employed here can be foundin the work of Alveroglu,32 and the code used here is based on thatdeveloped by Alveroglu during that related work.

B. Neutral stability curvesWe now proceed to compute the neutral stability curves for

various parameter combinations of the mean flow. In all cases, thestability of the flow will be examined by plotting neutral points(αi = 0) in the (R, n)- and (R, ψ)-planes, where ψ = tan−1(β/αr) isthe orientation angle of spiral vortices relative to a circle concentricto the disk. Both n and ψ are physical, measurable quantities. Whilen must be interpreted as an integer, it is permitted to take any valueduring the mathematical analysis.

All neutral curves obtained are found to have a two-lobed struc-ture: the upper representing the type I crossflow instability and thelower representing the type II viscous instability. This is consis-tent with all previous work on the rotating disk boundary layer, asdiscussed in Sec. I. Our discussions focus on the critical Reynolds

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FIG. 4. Vortex-number and -angle neutral stability curves for the range −0.75 ≤ ε ≤ 0.75 in increments of 0.25.

number, Rc, which is the lowest Reynolds number that permits anunstable mode.

1. Temperature-dependent viscosity (no axial flow)Figure 4 shows neutral stability curves for stationary distur-

bances (ω = 0) for a range of temperature dependencies. We seethat ε = −0.75 is the most unstable case for both modes by a signifi-cant margin. Examining the type I mode, we find that increasing thetemperature dependence from ε = −0.75 results in a sharp increasein the critical Reynolds number. This stabilizing behavior contin-ues through the gas-type regime, to the temperature independentcase, and into the liquid-type regime to ε = 0.25. However, a fur-ther increase in the value of ε results in a reduction in the criticalReynolds number, such that the case when ε = 0.75 is in fact lessstable than the temperature independent case. It can, therefore, beinferred that a turning point in the stabilizing behavior of the type Imode exists in the range 0 ≤ ε ≤ 0.5.

The type II mode exhibits similar behavior although it appearsthat the maximum of the type II critical Reynolds number occurs

when ε = 0. It is possible that any temperature dependence leads todestabilization of the type II mode. The type II mode is also signifi-cantly less stable for cases when ε < 0, such that when ε = −0.5, thetwo modes cannot be visually distinguished, while for the case whenε = −0.75, the type II mode is dominant.

In spite of its non-monotonic relationship with the criticalReynolds number, increasing ε appears to consistently increase thecritical number of vortices nc, as well as the range of unstable nvalues. We see that the critical wave angles for both modes followthe same trends with changing ε; with the ε = −0.75 neutral curveencompassing the largest range of unstable wave angles.

2. Axial flow (no temperature dependence)Figure 5 shows the stationary mode neutral stability curves for

a range of axial flow strengths for the case when ε = 0. As Ts isincreased, the critical Reynolds number is significantly increased,stabilizing the type I mode. The value of nc also increases sig-nificantly, as well as the range of unstable values of n enclosedby the successive curves. Conversely, the type II mode is initially

FIG. 5. Vortex-number and -angle neutral stability curves for the range 0 ≤ Ts ≤ 0.3 in increments of 0.05.

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FIG. 6. Vortex-number and -angle neu-tral stability curves for a range of axialflow strengths and temperature depen-dencies. Curves show −0.75 ≤ ε ≤ 0.75in increments of 0.25 for the indicated Ts

value or 0 ≤ Ts ≤ 0.3 in increments of0.05 for the indicated ε value. The blackcurves are provided as a reference tothe temperature independent, zero axialenforcement (ε = Ts = 0) case.

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destabilized for small values of Ts, before steadily stabilizing as Tsis increased further. From this, it can be inferred that there is aminimum stability of the type II mode in the region 0 < Ts < 0.1.

For larger values of Ts, although both modes are shifted alongthe R-axis, the type II mode becomes more prominent as the type Imode undergoes greater stabilization. It is likely that further increas-ing Ts will lead to the type II mode becoming the dominant form ofinstability.

As might be expected, the behavior of the critical wave anglefollows that of nc. The range of unstable wave angles is decreased asTs is increased although the unstable angles enclosed by the curvesare significantly increased.

3. Combined effectsFigure 6 depicts the neutral curves for a range of temper-

ature dependences at a constant, non-zero enforced axial flowand vice versa. We see that the effect of increasing ε is con-sistent with that seen in Fig. 4, where increasing ε results inincreased flow stability limited to some positive value of ε, beyondwhich flows are less stable. At Ts = 0.15, the turning point forstabilizing behavior of increasing ε appears in the range 0 ≤ ε≤ 0.5, which is consistent with the unenforced case (i.e., Ts = 0).When Ts = 0.3, the stabilizing effect is extended and the pointof maximum stability occurs in the range 0.25 ≤ ε ≤ 0.75. Thewave angles enclosed by the neutral curves increase significantlyfor all ε values when Ts = 0.15, increasing further still whenTs = 0.3.

Examining the type II mode behavior, the lower branch criti-cal Reynolds number increases as ε is increased; an effect consistentwith that seen in Fig. 4 for the range ε ≤ 0. Examining the individualcurves in this range shows that a flow with ε =−0.5 is type I dominantwhen Ts = 0.15 and Ts = 0.3, unlike for the case when Ts = 0 wherethere is no obvious distinction between the branches. However, forthe cases when ε > 0, there does not appear to be an immediately dis-cernible relationship between the type II critical Reynolds numberand the value of ε.

Increasing Ts yields effects consistent with that seen in Fig. 5, inthat both modes are stabilized (the type I mode more significantly),the range of vortex values enclosed by the neutral curves increases,and the enclosed range of wave angles decreases.

When ε = −0.75, we find that the curves are significantly lessstable than those in Fig. 5 and that the type II mode is dominantfor all Ts values. As Ts is increased, the type I mode is apprecia-tively stabilized. This further enforces the dominance of the type IImode caused by the temperature dependence (in spite of it also beingstabilized).

When ε = 0.75, we observe neutral curves very similar to thoseof the temperature-independent case presented in Fig. 5. The curvesfor values of Ts ≥ 0.1 are slightly stabilized for this value of tem-perature dependence, while the cases when Ts = 0 and Ts = 0.05,both yield a reduced critical Reynolds number when compared tothe temperature-independent problem.

Figure 7 plots a range of ε values against their associated type Icritical Reynolds numbers at various Ts values; each curve, therefore,represents a different fixed axial flow strength. Note that the rangeof ε values reported on is extended beyond that used for the neutralcurves. In each case, an optimal value of ε > 0 (with respect to the

FIG. 7. Plot of the type I critical Reynolds number arising from each Ts = 0, 0.15,and 0.3. The dotted lines indicate a reference for the critical Reynolds numberassociated with the temperature independent case (i.e., ε = 0) at each Ts.

maximum value of the critical Reynolds number) is determined suchthat further increases beyond this point result in flows that are pro-gressively less stable. We find that the most stable flow is producedfor an increasingly greater value of ε as Ts is increased. Further-more, the range of ε values that results in flows that are more stablethan the temperature independent case is increased with Ts. Criti-cal data for the range of ε and Ts values considered are presented inTable II.

C. Growth rates and e N analysisUsing an eN type analysis, as outlined by van Ingen,14 we are

able to investigate the evolution of the complex radial wave numberfor a range of axial flow strengths and temperature dependencies.First, the amplitude of a disturbance at radial positions r and r + dris considered,

a(r) = ∣(u, v, w, p, Θ)∣ = ∣(u, v, w, p, Θ)∣e−αir ,

a(r + dr) = ∣(u, v, w, p, Θ)∣e−αi(r+dr).

This leads to the ratio of amplitudes

a(r + dr)a(r) = e−αidr .

Recalling the definition of the Reynolds number (8), we replace rwith R and integrate from some initial position R0 (interpreted as thesmallest Reynolds number, at a fixed value of n, for which instabilityis observed) to the location under consideration, R. This leads to theamplitude ratio

N = ln( aa0) = −∫

R

R0

αi dR.

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TABLE II. Critical data for a range of ε values at fixed Ts values and vice versa. TypeII critical data are indicated by bracketed headings. Values marked by . . . are omitteddue to the indefinite division between each branch.

ε Rc nc αr ,c ψc (Rc) (nc) (αr ,c) (ψc)

Ts = 0

−0.75 . . . . . . . . . . . . 164 10 0.2096 15.55−0.5 214 15 0.3450 11.02 . . . . . . . . . . . .

−0.25 260 19 0.3624 11.11 . . . . . . . . . . . .

0 287 23 0.3842 11.41 462 22 0.1325 19.200.25 293 25 0.4114 11.60 441 22 0.1390 19.660.5 290 27 0.4404 11.72 434 23 0.1462 19.760.75 282 28 0.4718 11.78 433 24 0.1526 19.85

Ts = 0.15

−0.75 . . . . . . . . . . . . 186 18 0.2617 19.77−0.5 284 36 0.4402 16.06 . . . . . . . . . . . .

−0.25 345 46 0.4624 16.02 469 33 0.1772 21.610 379 53 0.4793 16.16 459 34 0.1739 22.560.25 390 60 0.5267 16.16 440 34 0.1830 22.680.5 389 61 0.5327 16.28 442 36 0.1905 22.690.75 383 66 0.5850 16.21 454 38 0.1970 22.69

Ts = 0.3

−0.75 . . . . . . . . . . . . 229 33 0.2991 25.15−0.5 391 76 0.4692 22.45 . . . . . . . . . . . .

−0.25 485 103 0.5186 22.23 580 59 0.2011 26.790 536 122 0.5557 22.23 560 60 0.2060 27.200.25 554 135 0.5950 22.23 541 61 0.2165 27.220.5 557 145 0.6365 22.21 549 64 0.2241 27.200.75 552 153 0.6800 22.18 574 69 0.2309 27.18

Ts Rc nc αr ,c ψc (Rc) (nc) (αr ,c) (ψc)

ε = −0.75

0 . . . . . . . . . . . . 164 10 0.2096 15.520.05 . . . . . . . . . . . . 167 12 0.2275 16.770.1 . . . . . . . . . . . . 175 14 0.2409 18.310.15 . . . . . . . . . . . . 186 18 0.2617 19.770.2 . . . . . . . . . . . . 199 22 0.2721 21.570.25 . . . . . . . . . . . . 214 27 0.2854 23.340.3 . . . . . . . . . . . . 229 33 0.2991 25.15

ε = 0

0 287 23 0.3842 11.41 462 22 0.1325 19.200.05 308 29 0.4175 12.71 441 24 0.1470 20.170.1 339 39 0.4490 14.34 443 28 0.1609 21.280.15 379 53 0.4793 16.16 459 34 0.1739 22.560.2 427 71 0.5073 18.13 485 41 0.1858 23.990.25 479 94 0.5326 20.16 519 49 0.1965 25.550.3 536 122 0.5557 22.23 560 60 0.2060 27.20

TABLE II. (Continued.)

Ts Rc nc αr ,c ψc (Rc) (nc) (αr ,c) (ψc)

ε = 0.75

0 282 28 0.4718 11.78 433 24 0.1526 19.850.05 305 36 0.5093 12.92 424 27 0.1683 20.520.1 340 48 0.5475 14.46 433 32 0.1829 21.510.15 383 66 0.5850 16.21 454 38 0.1970 22.690.2 433 88 0.6194 18.12 487 46 0.2092 24.070.25 490 117 0.6512 20.12 527 56 0.2207 25.570.3 552 153 0.6800 22.18 574 69 0.2309 27.18

This quantity can be computed for a range of fixed vortex numbers nfrom which an enveloping curve can be fitted. The enveloping curvecan then be used to approximate a transition region by comparisonto empirical transition data. Cooper and Carpenter33 utilized thismethod for transition prediction of the flow over a rotating disk witha compliant surface, and the in-depth review by van Ingen14 high-lights a number of comparisons between eN predictions and stabilityexperiments. Malik et al.34 determined that N = 10.7 according toa transitional Reynolds number of 513, later updating this to be inthe region 9 ≤ N ≤ 10.35 It is noted here that modifications to theflow may result in a variation of the transitional N-factor althoughfor the purposes of this qualitative study, N = 9 will be utilized as thereference amplitude ratio required for the prediction of transitionalflows.

Figure 8 displays the enveloping N-curve formed from arange of amplitude ratios from the temperature independent, zeroenforced flow case (i.e., ε = Ts = 0). Here, the curve crosses at N = 9

FIG. 8. Amplitude ratio curve for ε = Ts = 0. The dashed curves (−−) representamplitude ratio curves at fixed n; the solid curve is the enveloping N-factor curveused to determine a transitional Reynolds number.

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when R ≈ 521, showing good agreement with the results presentedby Wilkinson and Malik.35 Figure 9 shows the N-curves for a rangeof temperature dependencies and axial flow strengths. It can be seenthat by varying ε for fixed Ts, the temperature independent case hasthe largest transitional Reynolds number. It is likely that this behav-ior mimics that of the neutral curves, where a maximum transitionalReynolds number exists. Interestingly, for the case when Ts = 0.3,the transitional Reynolds number for flows with ε = 0.75 is lowerthan the temperature-independent case, in spite of the latter hav-ing a larger Rc than the former. This indicates that the relationshipbetween the critical and transitional Reynolds numbers is not strictlymonotonic.

We note that the gradient of the N-curve is reduced withincreasing Ts. This suggests that a stronger axial flow reduces the dis-turbance growth rate and delays the onset of transition. This effectis consistent for all ε values considered and is most likely the rea-son for the previously discussed increase in the transitional Reynoldsnumber in spite of a reduced critical Reynolds number. AdditionalN-curve data are presented in Table III.

D. EigenfunctionsIt is useful to examine the type I eigenfunctions which will

be used to conduct an energy balance analysis in Sec. III E. Here,all eigenfunctions are assessed at R = Rc + 200 (i.e., well intothe unstable regime), and the value of α is chosen such that themost amplified disturbance, at this particular Reynolds number, isselected.

The plots depicted in Fig. 10 are the magnitudes of the pertur-bation eigenfunctions for ε in the range −0.75 ≤ ε ≤ 2. We see that,as ε is increased, the eigenfunction profiles are narrowed. This effectcan largely be attributed to a mirroring of the mean-flow profiles.

FIG. 9. Amplitude ratio curves for a range of ε and Ts values. Solid lines (–) rep-resent Ts = 0; dashed lines (−−) represent Ts = 0.15; dashed–dotted lines (−⋅)represent Ts = 0.3. Line colors blue, red, and yellow represent ε = −0.75, 0, and0.75, respectively. The black, dotted line (⋯) indicates N = 9, i.e., the transitionalN-factor.

TABLE III. Transitional Reynolds numbers, R(N = 9), for a range of temperaturedependencies and axial flow strengths.

Ts ε R(N = 9)

0 −0.75 3290 0 5210 0.75 5030.15 −0.75 5000.15 0 7510.15 0.75 7580.3 −0.75 7320.3 0 10830.3 0.75 1053

A notable exception is seen where the ∣Θ(z)∣ profile narrows as εis increased, as opposed to the broadening of the mean-flow pro-file depicted in Fig. 1. We further notice that for positive values ofε, the two maxima of the |u(z)| profile become more prominent asε is increased and occur closer to the disk surface. Increasing ε inthe range −0.75 ≤ ε ≤ −0.5 decreases the maximum of the |v(z)| pro-file. Increasing ε beyond this leads to an increase in the maximumof the |v(z)| profile. Similarly, increasing ε initially also decreases themaximum of the ∣w(z)∣ profile. The limit of this behavior occursin the range 0 ≤ ε ≤ 0.5, where an increase in the temperaturedependence parameter thereafter results in the profile maximumincreasing.

Figure 11 depicts the magnitudes of the perturbation eigen-functions for a range of enforced axial flow strengths. We see thatthe velocity perturbation profiles narrow as Ts is increased, decay-ing closer to the disk surface. Again, this can be seen as a mir-roring of the mean-flow profiles in Fig. 2. For small values ofTs, the two maxima of the radial eigenfunction |u(z)| are bothreduced. The maximum located further from the disk surface con-tinues to decrease for greater values of Ts, eventually becomingentirely suppressed in the case when Ts = 0.3. It is inferred thatthis is a direct response to the suppression of the mean radialvelocity inflection seen in Fig. 2. We also observe that the maxi-mum of the azimuthal eigenfunction is increased as the axial flowstrength is increased, whereas the maximum of the ∣w(z)∣ profile isreduced.

Although not shown here, the plots for the combined effectsof temperature dependence and enforced axial flow on the per-turbation eigenfunctions are such that increasing ε produces anarrowing effect consistent with that seen in Fig. 10 and thiseffect is strengthened by the presence of enforced axial flow.Enforced axial flow also acts to amplify the primary maximumof the |u(z)| profile, as well as the maxima of both the |v(z)|and ∣Θ(z)∣ profiles. Conversely, the maximum of the axial pro-file is significantly reduced for all ε values considered here. Othereffects such as the variation in profile maxima with increasing εare largely consistent with the Ts = 0 case. Similarly, the effectof increasing Ts is largely consistent with the ε = 0 case. Theprimary maximum of the |u(z)| profile and the maxima of the|v(z)|, ∣w(z)∣, and ∣Θ(z)∣ profiles are all increased for all Ts valuesconsidered.

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FIG. 10. Type I eigenfunctions for −0.75 ≤ ε ≤ 2 with 0.25 increments. The black curve on each plot is provided as reference to the case of ε = Ts = 0.

E. Energy balance analysis

It is possible to conduct an analysis of the energy balancewithin the system to better understand the underlying mechanismsof the flow instability. Following Cooper and Carpenter,33 an energy

balance equation is now derived from the eigenfunctions to exam-ine the energetic input and output of a disturbance to the meanflow.

To begin, the momentum stability Eqs. (9b)–(9d) are multipliedby their corresponding velocity component and summed

u∂u∂t

+ v∂v

∂t+ w

∂w

∂t+ U(u

∂u∂r

+ v∂v

∂r+ w

∂w

∂r) +

VR(u

∂u∂θ

+ v∂v

∂θ+ w

∂w

∂θ)

+WR(u

∂u∂z

+ v∂v

∂z+ w

∂w

∂z) +

U(u2 + v2)R

+ U′uw + V′vw +W′w2

R

= −(u∂p∂r

+ v∂p∂θ

+ w∂p∂z) +

μR(uj

∂σij

∂xi) +

μ′

R[u(∂w

∂r+∂u∂z) + v( 1

R∂w

∂θ+∂v

∂z) + 2w

∂w

∂z]

+1R[u∂(U

′μ)∂z

+ v∂(V′μ)

∂z+ w

∂(W′μ)∂z

] +w

R(U′

∂μ∂r

+V′

R∂μ∂θ

+ W′ ∂μ∂z). (12)

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FIG. 11. Type I eigenfunctions for 0 ≤ Ts ≤ 0.3 with 0.05 increments. The black curve on each plot is provided as reference to the case of ε = Ts = 0.

Here, σij represents the viscous stress terms

σ11 =∂u∂r

σ12 =1R∂u∂θ

σ13 =∂u∂z

,

σ21 =∂v

∂rσ22 =

1R∂v

∂θσ23 =

∂v

∂z,

σ31 =∂w

∂rσ32 =

1R∂w

∂θσ33 =

∂w

∂z.

Introducing a new kinetic energy variable, e = (u2 + v2 + w2)/2, leads to (12) being re-written in the following form:

∂e∂t

+∂(Ue)∂r

+1R∂(Ve)∂θ

+1R∂(We)∂z

+U(u2 + v2)

R+ U′uw + V′vw +

W′(w2 − e)R

= −[∂(up)∂r

+1R∂(vp)∂θ

+∂(wp)∂z

] − upR

+1R[∂(μujσij)

∂xi− μσij

∂uj

∂xi]

+1R[∂(μ

′uw)∂r

+1R∂(μ′vw)

∂θ+∂(μ′w2)

∂z− μ′′w2] +

1R[∂(μ

′e)∂z

− μ′′e]

+1R[∂(U

′μu)∂z

+∂(V′μv)

∂z+∂(W′μw)

∂z−U′μ

∂u∂z− V′μ

∂v

∂z−W′μ

∂w

∂z]

+1R[∂(U

′μw)∂r

+1R∂(V′μw)

∂θ+∂(W′μw)

∂z−U′μ

∂w

∂r− V′μ

R∂w

∂θ−W′μ

∂w

∂z−W′′μw], (13)

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which is further averaged over a single time period and azimuthalmode (removing all derivatives with respect to t and θ) and inte-grated across the boundary layer,

dEdr= − ∫

∞

0U′⟨uw⟩ + V′⟨vw⟩ +

W′

R⟨w2⟩dz

EPRS

− 1R∫

∞

0μ⟨σij

∂uj

∂xi⟩dz

EDV

− 1R∫

∞

0⟨up⟩dz

PW

− 1R∫

∞

0U(⟨u2⟩ + ⟨v2⟩) −W′⟨e⟩dz

SC

− 1R∫

∞

0μ′′⟨w2⟩ + W′′⟨μw⟩ + U′(⟨μσ31⟩ + ⟨μσ13⟩)

+ V′⟨μσ32⟩ + W′⟨μσ33⟩dzAVV

, (14)

where

E = ∫∞

0U⟨e⟩ + ⟨up⟩

− μ(⟨uσ11⟩ + ⟨vσ12⟩ + ⟨wσ13 + ⟨uw⟩⟩) + U′⟨μw⟩R

dz.

The terms labeled EPRS represent Energy Production terms due toReynolds Stresses; EDV represent Energy Dissipation terms due toViscosity; PW represent Pressure Work terms; SC are StreamlineCurvature terms arising from the disk’s rotation; and AVV are Addi-tional terms arising from the temperature dependent (Variable) Vis-cosity. Note that many terms from (13) disappear upon integrationdue to the mean flow and perturbation boundary conditions. Here,the time-averaged quantities have the form ⟨xy⟩ = x⋆y + xy⋆, wherex⋆ indicates the complex conjugate of x. The perturbations retainthe normal mode form; hence, r and z derivatives can be expressedas iαu and u′, respectively.

The energy contributions of each term in (14) are determinedvia numerical integration of the eigenfunctions and are shown inFig. 12 for the temperature independent, zero axial flow case (ε = Ts= 0). We see that many of the integrated terms are negligible and asimplified energy balance equation can then be expressed to give theTotal Mechanical Energy (TME) as

− 2αi

²TME

≈ ∫∞

0V′⟨vw⟩dz

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶EPRS

− 1R ∫

∞

0μ⟨σij

∂uj

∂xi⟩dz

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶EDV

, (15)

where the terms have been normalized by E. The above returnsαi < 0 (the criteria for an unstable mode) when EPRS > EDV. That is,a disturbance is amplified when the energy produced by the distur-bance outweighs energy dissipated in response. Note that the AVVterms have not been considered here since they are equal to zero forthe temperature independent (i.e., ε = 0) case. The relevance of theAVV terms to (15) is discussed overleaf.

Figure 13(a) shows the energy integral values split into thephysical components for a range of temperature dependencies. TheTME curve shows that both positive and negative values of ε can

FIG. 12. Results of the energy balance integral showing the energy contributionof the individual components for ε = Ts = 0. Note that EPRS = P1 + P2 + P3 andSC = SC1 + SC2 + SC3.

increase the total mechanical energy of the disturbance. Interest-ingly, the minimum value of TME does not occur at ε = 0, butrather at ε ≈ 0.25, although the TME does not change significantlyin the range 0 ≤ ε ≤ 0.5. The location of this minimum aligns withthe ε value responsible for the maximum critical Reynolds num-ber on the Ts = 0 curve in Fig. 7. Further comparison betweenFigs. 7 and 13(a) shows an inverse proportionality between Rc(ε)and the TME. That is, a reduced critical Reynolds number alsoresults in greater disturbance amplification. It can also be inferredthat the flow stability criteria at Rc is consistent with that atRc + 200 for all values of ε considered here and as such the dis-turbance growth rate over this range of Reynolds numbers is unaf-fected by ε. The EDV curve shows that increasing ε from ε = −0.5increases the viscous dissipation associated with a disturbance. How-ever, the gradient of the EDV curve is significantly shallower thanthat of the EPRS curve, leading to an overall increase in the dis-turbance TME. The minimum dissipation occurs when ε = −0.5,below this value a far greater viscous dissipation is observed. Refer-ring back to Fig. 4, we note that the ε = −0.5 case lacks an inflectionbetween the type I and II branches, leading to a type II dominantinstability for the cases when ε < −0.5. As a result, the most unsta-ble vortex number (at which the eigenfunction is assessed) occurson the type II branch, which may explain the departure from thetrend in the otherwise type I dominant flows. We see that the addi-tional terms that arise when ε ≠ 0 are mostly negligible for therange of ε values considered here, only adding a small amount ofenergy to the system for the cases when ε < −0.5, demonstrat-ing that the normalized relationship in (15) is viable without theinclusion of the variable viscosity terms. The lack of influence ofthese AVV terms leads to the conclusion that the variations in theenergy profiles, and hence the overall flow stability, are affectedby changes to the mean flow profiles, rather than the temperatureperturbations.

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FIG. 13. (a) Component energy contributions and (b) total and component energy dissipation due to viscosity for a range of temperature dependencies. The vertical andhorizontal black, dashed lines are provided as reference to ε = 0 and to zero energy contribution, respectively.

In order to better understand the role of temperature depen-dent viscosity in the flow energetics, the dissipation term is manip-ulated to reflect a constant viscosity component (μ ≡ 1) and atemperature dependent component,

− 1R ∫

∞

0μ⟨σij

∂uj

∂xi⟩dz

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶EDV

= − 1R ∫

∞

0⟨σij

∂uj

∂xi⟩dz

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ND

− 1R ∫

∞

0(μ − 1)⟨σij

∂uj

∂xi⟩dz

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶TDD

. (16)

Here, the term labeled ND represents Newtonian Dissipation, andthe term labeled TDD represents Temperature Dependent Dissi-pation. The terms established in (16) are plotted in Fig. 13(b). Asε is increased, the Newtonian dissipation briefly reduces, beforeincreasing significantly. Conversely, the dissipation associated withthe temperature dependent component is seen to decrease quasi-linearly as ε increases, where TDD becomes positive when ε > 0,acting as an energy production term and reducing the overall viscousdissipation.

Figure 14 shows the energy integral values for a range ofaxial flow strengths. Unlike the temperature dependent viscosity,the formulation of enforced axial flow used here does not changethe stability equations from the standard von Kármán formula-tion. Hence, it is known that any variation in the stability of theflow is a result of variations of the mean-flow profiles. We seethat as Ts is increased, each of the curves moves toward zero; theenergy produced by the disturbance is decreased and the viscousdissipation decreases in turn, reducing the total mechanical energy.The results seen here are consistent with the stabilizing effect ofincreasing Ts demonstrated in Fig. 5, where higher Reynolds num-bers are required to amplify a disturbance sufficiently to causeinstability.

Figure 15 shows the energy contribution of each componentof the energy balance equation for a range of temperature depen-dencies at three fixed axial flow strengths. We see that the gen-eral trend resulting from increasing ε remains unchanged whenenforced axial flow is applied. In the case when Ts = 0.15, eachcurve is notably shallower, which is consistent with the dampeningeffect of enforced axial flow seen in Fig. 14. The Ts = 0.3 curvesare shallower still, where overall variation appears to be largelyinsignificant. However, recalling Fig. 6, we observe that the criticalReynolds number does in fact vary significantly for a range of ε val-ues when Ts ≠ 0. As such, enforced axial flow cannot fully suppressthe destabilization associated with temperature dependent viscos-ity flows; it is, however, helpful in dampening the growth of thedisturbances.

FIG. 14. Component energy contributions for a range of axial flow strengths.

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FIG. 15. Component energy contributions and total and component energy dissipation due to viscosity for a range of temperature dependencies. Solid lines (–) representTs = 0; dashed lines (−−) represent Ts = 0.15; dashed–dotted lines (−⋅) represent Ts = 0.3.

IV. DISCUSSION AND CONCLUSIONS

In Sec. II, the formulation and solution of the mean-flow equa-tions were presented for a range of temperature dependent viscosi-ties and enforced axial flow strengths. We found that negative valuesof the temperature dependence value ε result in a steady conver-gence of the velocity and temperature profiles to their respectiveboundary conditions as z → ∞, indicating a broader boundary-layer thickness. Conversely, positive values of ε result in conver-gence of the profiles at a reduced distance from the disk sur-face and, therefore, a narrower boundary layer. The physical rela-tion between the temperature dependency and the boundary layerthickness can be described through the viscosity at the wall andits relation to the wall shear, where changes to the viscous forceshere affect the impact of the disk’s rotation on the surroundingfluid.

Increasing the axial flow strength results in a reduced conver-gence distance of the mean flow profiles with the exception of theaxial velocity profile, which—as to be expected for forced flow inthis direction–exhibits acceleration throughout the boundary layer.An enforced flow results in a gentle narrowing of the boundary layeras the influence of the disk’s rotation is diminished.

When combining the two parameters, the effects they exhibitindividually upon the mean flow are generally consistent whenapplied simultaneously. Notable exceptions are the effects uponthe axial velocity and temperature profiles [W(z) and Θ(z), respec-tively]. Beyond a certain axial flow strength, the effect of increasingthe temperature dependence is reversed. More specifically, whenTs = 0.3, the axial velocity is increased with increasing ε, and thetemperature profile converges at a reduced distance (i.e., the thermalboundary layer is narrowed). This is due to the non-zero conver-gence of the radial velocity profile when Ts ≠ 0, which then inhibitsthe characteristic profile inflection.

Having detailed the physics behind the mean-flow behaviorand its variation with temperature dependence and enforced axialflow, it is pertinent to apply the same to the observed effects withregard to flow stability. In Sec. III, we observed that solutionsto the stability equations can be plotted in the form of neutral

stability curves, where the stability of the flows can be assessedprimarily through the location of the critical Reynolds number. Itwas found that negative values of ε lead to significantly destabilizedflows when compared to the temperature independent case. As εis increased, the type I critical Reynolds number increases and theresulting flows are more stable. When comparing this behavior tothat of the mean flow, one might conclude that the narrowing of theboundary layer is favorable to flow stability. However, increasing εinto the positive regime continues this stabilizing trend, up until apoint where beyond which the critical Reynolds number reduces,eventually to a value less than that of the temperature independentcase (ε ≈ 0.6 as detailed in Fig. 7). The type II mode appears tobe destabilized for all non-zero values of ε considered, suggestingthat any variation in the viscosity through the boundary layer willpromote this instability.

The stability response to variations in axial flow strength hasalso been investigated. We find that the type I mode is stabilizedsignificantly as Ts is increased. Again, referring to the mean flowsuggests that the narrowing of the boundary layer caused by increas-ing Ts is favorable to the stability of the type I mode. Referring backto Fig. 2 shows that the characteristic inflection of the radial pro-file is inhibited as Ts is increased due to the boundary conditionon U(z). This inflection is physically responsible for the crossflowinstability; hence, if the converging velocity beyond the inflectionis closer to the profile maximum, the instabilities arising from thevelocity gradient should be less significant. It is also noted in Fig. 2that, when Ts = 0.3, the inflection is completely suppressed. How-ever, Hussain et al.8 noted that an inflection exists in the resolvedvelocity profile (the resolved velocity Q = U cosϕ + V sinϕ, where ϕis the resolution angle from the radial profile in the direction of rota-tion) for all values of Ts, and hence a crossflow instability will alwaysarise.

The type II behavior is more difficult to characterize physically.The eventual stabilizing effect of increasing Ts can be explained asthe forced flow suppressing the influence of the disk’s rotation (fromwhich the type II instability arises). However, the small range of Tsvalues where increasing Ts is destabilizing to the type II mode fallsoutside of this explanation. Recall that the type II mode is viscous

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in nature; perhaps, this weakly enforced axial flow encourages theviscous interaction between the fluid and the disk.

We have also assessed the radial growth rate of disturbancesfor a range of temperature dependencies and axial flow strengths,utilizing them to find disturbance amplitude ratios and plottingN-factor curves that approximate the location of a transitionalReynolds number. We find that the lowest transitional Reynoldsnumbers correspond to the ε values that produce the least sta-ble flows, i.e., ε = −0.75. Increasing the enforced axial flow heresignificantly increases the transitional Reynolds number, as wellas suppressing the growth rate of the N-factor curve. The over-all behavior largely imitates that of the neutral curves. As such,the physical mechanisms that allow instabilities to form at lowerReynolds numbers are likely also responsible for the developmentof these instabilities into fully turbulent flow over a reduced dis-tance. A notable result is that enforced axial flow may still delaytransition in spite of the destabilizing effect of increasing ε beyondits most stable value; the reduced gradient of the N-factor curvesinduced by enforced axial flow means that the development of dis-turbances from instability to turbulence is delayed, and thereforea reduced critical Reynolds number is not necessarily indicativeof an earlier onset of transition. Experimental work is required toverify this, but the results presented here imply that, for certaincases, the radial position where crossflow vortices form could occurcloser to the disk center, while the turbulent region develops furtherdownstream.

In Sec. III E, the magnitudes of the perturbation eigenfunctionswere plotted and an energy balance equation was derived from thestability equations. The total energy of the system follows the trendestablished between the critical Reynolds number and ε, where theminimum disturbance energy and the maximum critical Reynoldsnumber approximately coincide when ε ≈ 0.25. A particularly inter-esting result is that the additional disturbance terms arising fromthe inclusion of a temperature dependent viscosity are found to benegligible. We, therefore, conclude that the linear stability charac-teristics of such flows are predominantly governed by variations inthe basic states. It is established in (15) that the energy productionterm (EPRS) is dependent on V′(z) and the viscous dissipation term(EDV) is dependent on μ(z;Θ(z)). With this in mind, recall themean flow solutions from Sec. II. As ε is increased, the gradient ofthe azimuthal profile in Fig. 1 increases as the profile is narrowed,leading to a significant change in the shape of the V′(z) profile. Thischange in the V′(z) profile is clearly favorable as ε is increased in therange −0.75 ≤ ε ≤ 0.25 (where the upper limit is the point of max-imum stability), reducing the energy production term and therebycreating a more stable flow. Perhaps beyond this, the steep veloc-ity gradients through a narrow boundary layer render the flow moresusceptible to instability.

It has already been suggested that certain mean flowbehaviors—such as the suppression of the radial profile inflection—are in part responsible for the stabilizing effect. It is seen in Fig. 14that increasing Ts leads to each profile moving toward zero. Exam-ining the V′(z) profile behavior with increasing Ts in Fig. 2 showsthat although the absolute value of V′(0) is increasing (see alsoTable I), the profile is also narrowed such that the area enclosedby the V′(z) curve is reduced. As such, the energy produced bythe disturbance is significantly reduced. Figure 15 shows that thiseffect is consistent for all temperature dependencies considered here.

Enforced axial flow evidently acts as a dampener to the amplificationof disturbances.

Where values match, the neutral curve data presented herefor varying enforced axial flow are in excellent agreement with thedata presented in the work of Hussain et al.8 An alternative forcedflow formulation is presented in the work of Chen and Mortazavi,26

where the mean flow of a chemical vapor deposition reactor is mod-eled. The forced flow parameters are expressed as modifications tothe mean flow variables U(z) and W(z), where two constants controlthe ratio of rotational to axial influence on the flow. The formula-tion utilized by Chen and Mortazavi26 allows for all flows betweenthe standard, rotation-only von Kármán flow and forced flow-onlystagnation flow. The effects observed on the mean flow are consis-tent with those seen here, where the radial inflection is suppresseddue to the imposed boundary condition. As noted by Hussain et al.,8

the advantage of the formulation utilized in this work is that Ts doesnot appear in the stability equations, while the formulation used byChen and Mortazavi26 would modify the stability equations. How-ever, the formulation utilized here is only suitable for enforced axialflow strengths where rotation is dominant. As such, to extend thisanalysis to account for weak rotational influence on the forced flowover a disk (i.e., Ts > 1), a formulation similar to that of Chen andMortazavi26 would be more suitable.

We have limited the work presented here to stationary modes ofinstability that rotate with the disk surface. The most obvious exten-sion of this work would be to investigate traveling modes. Typically,traveling modes yield higher critical Reynolds numbers with reducedamplification rates; hence, stationary modes are often the primarysource of linear convective instability. This was first detailed by thevisualization experiments of Gregory et al.,3 where the existence oftraveling modes was verified theoretically, but stationary vorticeswere observed on china clay disks. However, experimental work11

has since shown highlighted the importance of traveling modes overa smooth and clean disk surface.

The behavior of traveling modes in the formulation presentedhere can be speculated from the results of previous studies. For tem-perature dependent viscosity flows (without axial forcing), Jasmineand Gajjar25 found that for all values of ε considered, increasingthe frequency of the disturbance is mildly stabilizing to the type Imodes, while highly destabilizing with respect to the type II mode.Similarly, for flows with axial forcing (without a temperature depen-dent viscosity), Hussain et al.8 found that increasing the disturbancefrequency is mildly stabilizing to the type I mode and heavily desta-bilizing to the type II mode. They also investigate some negative fre-quencies, which are indicative of a disturbance traveling slower thanthe disk: these modes are found to be the most amplified although itis suggested that less-amplified near-stationary frequencies are morelikely to be selected due to a lower critical Reynolds number. Itmay, therefore, be expected that traveling modes in this formulationwould most likely appear when ε ≪ 0 and Ts ≈ 0 as the stationarymodes at these parameter values are the most amplified according tothe amplitude ratio curves and integral energy analysis (Figs. 9 and15, respectively).

ACKNOWLEDGMENTSR.M. would like to thank the EPSRC for funding this research

(Award No. 1658206).

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Phys. Fluids 32, 024105 (2020); doi: 10.1063/1.5129220 32, 024105-21

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