On the Validity of Boundary Layer Theory for Simulating

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Article

Nihon Reoroji Gakkaishi Vol.42, No.3, 161~167(Journal of the Society of Rheology, Japan)©2014 The Society of Rheology, Japan

1. INTRODUCTION

Exact solutions are rather rare in fluid mechanics, and this is particularly so for non-Newtonian fluids. This is perhaps why the field of non-Newtonian fluid mechanics relies so heavily on approximate theories such as boundary layer theory. Having said this, it should be conceded that while this theory has been very successful for Newtonian fluids, for non-Newtonian fluids its validity, in general, is in doubt. That is to say, there is always the danger that by dropping certain non-Newtonian terms from the governing equations, the non-Newtonian flavor of the flow is tarnished thereby affecting the physics of the problem. Also, it is by no means certain that the potential flow outside the boundary remains uninfluenced by the non-Newtonian behavior of the fluid―a point missed in virtually all boundary layer studies carried out in the past in relation to non-Newtonian fluids. Ironically, an exact solution is needed at the first place to assess the validity of boundary layer approximation for any given non-Newtonian fluid.

In a recent work Ahmadpour and Sadeghy1) have shown that for Bingham fluids, an exact solution can be found in von Karman flow (i.e., the swirling flow generated by a rotating disk in an otherwise quiescent fluid). This exact

solution provides us with a perfect tool to investigate the validity of boundary theory for Bingham fluids. With this in mind, it is the main objective of the present work to show that for Bingham fluids, the boundary layer theory is valid over a broad range of parameters. To achieve this goal, we will rely on the idea that a suitable similarity variable can be found [see Ref. 1] which transforms the governing partial differential equations into ordinary differential equations. (The idea, which was first introduced by von Karman2) while obtaining a self-similar exact solution for Newtonian fluids above a rotating disk [see, also, Refs. 3-5], has been shown to be valid for a variety of non-Newtonian fluids comprising shear-thinning fluid6), viscoelastic fluids7-10) and viscoplastic fluids.11))

The work is organized as follows: we start with presenting the governing equations in its most general form before simplifying them using the boundary layer approximation. The Bingham model will be introduced next as the rheological model of interest. We will then proceed with transforming the set of governing PDEs into ODEs using an appropriate similarity variable. The numerical method of solution used to solve the governing ODEs will be described next. Numerical results are presented showing the validity of boundary layer approximation in von Karman flow of Bingham fluids. The work is concluded by highlighting its major findings.

On the Validity of Boundary Layer Theory for Simulating von Karman Flows of Bingham Fluids

Ali AhmAdpour*, Mohsen GhAsemi*, Jalil JAmAli**, and Kayvan sAdeGhy*†

*Center of Excellence in the Design and Optimization of Energy Systems (CEDOES)School of Mechanical Engineering, College of Engineering, University of Tehran

P.O. Box: 11155-4563, Tehran, Iran **Islamic Azad University, Shoushtar Branch, Shoushtar, Iran

(Received : July 30, 2013)

The von Karman flow of an incompressible Bingham fluid is studied numerically under laminar and steady conditions. Boundary layer theory is used to simplify the equations of motion assuming that the flow is axisymmetric. A suitable similarity parameter is used to reduce the governing partial differential equations into a system of coupled ordinary differential equations. The equations so-obtained are then solved numerically using MATLAB software. The effect of yield stress is investigated on the validity of boundary layer approximation for this particular fluid when compared with known exact solution. It is shown that for Bingham fluids, the boundary layer theory is valid provided that the Reynolds number is sufficiently large and the Bingham number is sufficiently small.Key Words: Swirling flow / von Karman flow / Bingham fluid / Yield stress / Similarity solution / Boundary layer /

Waxy crude oil

† E-mail: [email protected]

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Nihon Reoroji Gakkaishi Vol.42 2014

2. MATHEMATICAL FORMULATION

The flow geometry consists of a disc of radius R located at z = 0 in a cylindrical coordinate system (r, θ, z), as shown in Fig. 1. The fluid of interest, which is initially at rest, is assumed to occupy the half-space (z > 0) only. The fluid is set into motion (due to the no-slip condition) as soon as the disk starts rotating around the z-axis with constant angular velocity Ω. It is this swirling motion which we would like to study in the present work―the so-called von Karman flow. Like von Karman2), we assume that the flow generated by the rotating diskd is laminar, steady, isothermal, and incompressible. For non-Newtonian fluids, the governing equations comprise the Cauchy equations of motion together with the continuity equation; that is,

(1a)

(1b)

(1c)

(1d)

where it has tacitly been assumed that the flow is axisymmetric (i.e., ∂/∂θ = 0). In these equations u, v, and w are the velocity components in the r, q, and z-directions, respectively (see Fig. 1). The last term in Eq. (1a) can be dropped if it is assumed that the fluid is inelastic. The symmetry of the stress tensor (which is true for simple fluids) means that the last term

in Eq. (1b) can also be dropped. The momentum equations then become

(2a)

(2b)

(2c)

As to the boundary conditions required to close the problem, we rely on the no-slip and no-penetration conditions at the wall; that is,

(3a)

Far from the disk, like von Karman2), we assume that the tangential and radial velocity components tend to zero; that is,

(3b)

To simplify Eqs. 2a-c, an order-of-magnitude analysis is used in the present work.12) At sufficiently high Reynolds numbers, the flow develops a new length scale, δ, in the z-direction. In the r-direction, however, the length scale is R. We start with the key boundary layer assumption that δ << r. With r = O( l ) and z = O(δ), we can proceed with the continuity equation (Eq. 2d) and show that u is by an order of magnitude larger than w. On the other hand, from Eqs. 2a it can be concluded that, for the inertia terms to be of the same order, the order of u must be the same as the order of v. On the other hand, based on Eqs. 2b and 2c, the order of velocity components can be determined as: u = O( l ), v = O( l ), and w = O(δ). Knowing the order of the three velocity components, we can now proceed with estimating the order of the inertia and stress terms in the equations of motion. The inertia terms in Eqs. 2a and 2b are of order one. On the other hand, the inertia terms in Eq. 2c are of order δ. Thus, based on the boundary layer approximation, we can drop the inertia terms from Eq. 2c. As to the order of the stress terms, we have

(4)

Based on the boundary layer approximation, tθz and trz are seen to be larger than the other stress terms. Therefore, in Eq. 2a the normal stress term, ∂trr / ∂r, can be neglected in comparison with the shear stress term, ∂trz / ∂z. Similarly, in Eq. 2b, the shear stress term (1/r2) (r2trθ) can be dropped in comparison with the normal stress term ∂tzθ / ∂z. Similarly, in Eq. 2c the normal stress term ∂tzz / ∂z can be neglected in

r∂∂

Fig. 1. Schematic showing the flow geometry and the coordinate system.

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AHMADPOUR • GHASEMI • JAMALI • SADEGHY : On the Validity of Boundary Layer Theory for Simulating von Karman Flows of Bingham Fluids

comparison with the shear stress term ∂trz / ∂r. The boundary layer equations (for inelastic fluids) then become:

(5a)

(5b)

(5c)

As to the stress terms appearing in the above equations, we assume that the fluid of interest is a viscoplastic fluid obeying Bingham rheological model. For Bingham fluids, the constitutive equation is of the following form13):

(6)

where ty is the yield stress, and µp is the plastic viscosity. Based on Eq. 6, it can be concluded that a Bingham fluid does not deform until the stress level reaches its yield stress, and when the yield stress is exceeded it flows with an apparent viscosity equal to η(γ4). It needs to be mentioned that, when a Bingham fluid (which is initially at rest) starts moving by a rotating disk it forms a two-layered structure consisting of a plug layer far from the disk and a sheared layer adjacent to the disk. It needs to be mentioned that in Eq. 6, eij is the rate-of-deformation tensor defined by

(7)

Also, the term |γ4 | in Eq. 6 is the magnitude of the deformation rate (or, more precisely, the second invariant of the rate-of-deformation tensor) defined by

(8)

where Einstein’s summation rule is adopted for the indices. On the other hand, the components of the rate-of-deformation tensor can be simplified as,

(9)

Therefore, across the yielded zone the stress terms become equal to

(10)

The apparent viscosity then becomes

(11)

Knowing the apparent viscosity, the boundary layer equations for the von Karman flow of Bingham fluids become:

(12a)

(12b)

(12c)

In an attempt to reduce the above set of coupled PDEs into ODEs, we follow von Karman’s idea for Newtonian fluids and assume that the tangential velocity has the following (separation-of-variable) form1):

(13a)

For Newtonian fluids, it can be shown that the other two velocity components should be of the following forms for compatibility reasons14):

(13b)

The last thing remained to be decided upon is the pressure gradient term, ∂p / ∂r in Eq. 12a. For Newtonian fluids it is easy to show that ∂p / ∂r is equal to zero.14) This notion has been found to be valid also for purely-viscous non-Newtonian fluids.11) Since Bingham fluids are inelastic like Ref. 11 we assume that ∂p / ∂r is negligible. With ∂p / ∂r also being equal to zero based on boundary layer approximation, one can conclude that the pressure, p, is constant everywhere. Knowing the form of the velocity profiles, we can then proceed with inserting these velocity components into the governing PDEs. But, before so-doing, we try to work with dimensionless parameters. To that end, we substitute:

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(14a)

Similarly, we define

(14b)

(The asterisk “*” above dimensionless variables will be dropped from subsequent equations for convenience.) We then proceed with approximating the deformation rate, γ4, as

(15)

where prime now denotes differentiation with respect to ζ. In dimensionless form, the terms appearing in the rate-of-deformation tensor become

(16)

Knowing the velocity components and stress terms above the disk, the governing equations (i.e., Eqs. 1a-d) become,

(17a)

(17b)

(17c)

where K1 = 1 + Bn/γ4; K2 = Bn/γ43; and K3 = r2 Re, where Bn = ty / µpΩ is the Bingham number and Re = ρR2Ω /µp is the Reynolds number.

The above equations are the equations governing boundary layer flow of a Bingham fluid above a rotating disk. It needs to be mentioned that Rashida11) was the first who tried to obtain these equations, but he made several errors in his derivation (which is perhaps why his work has not been published yet). It is easy to check that for a fluid lacking yield stress (i.e., for K1 = 1 and K2 = 0) the above system of differential equations reduce to the well-known equations for Newtonian fluids.2) In terms of the new variables, the boundary conditions become

(17d)

(17e)

It is interesting to note that, unlike Newtonian fluids, there is no self-similar solution for Bingham fluids; that is, the r-coordinate cannot be dropped from these equations. Still, these equations are very useful in that they can enable us to obtain the velocity components at a given r and z location and compare them with the exact solution reported in Ref. 1 to see if the boundary layer theory is valid for Bingham fluids. Since in Ref. 1 velocity profiles have been reported at dimensionless radius r = 1 (i.e., at the rim of the disk) we confine ourselves to obtaining boundary layer results at this location only (which happens to be the best location for the key boundary layer assumption that δ << r).

3. NUMERICAL METHOD

The fluid mechanics problem as posed by Eqs. 17a-e constitutes a two-point boundary value problem (BVP) with no analytical solution in close sight. Therefore, we have decided to rely on numerical method for obtaining a converged solution. To that end, we first recast the governing equations in the following form:

(18a)

(18b)

(18c)

The above set of equations can then be solved using ‘bvp4c’ package of the commercial software MATLAB. It is an efficient solver based on the collocation method which produces continuous solution on a mesh xi (i = 1,2,…, Nmax). The approximate solution satisfies the set of ODEs at both ends and also at the midpoint of each interval [xi, xi+1] using a fourth-order accurate Lobatto IIIA formula. In the course of the computations, the residual at the boundary condition is controlled to be uniformly small. The nonlinear system of algebraic equations so-obtained is then solved iteratively using MATLAB linear solver. The solution is initialized by

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Newtonian values for the similarity functions and the absolute value of the error for terminating the process is set at 10-6. The far field similarity variable, ξ, is initially set at 11 and then it is gradually increased while the value of Y1 and Y3 are monitored at the far field for it to fall below a certain threshold of 10-6. In all cases considered here, ξ = 35 was found to be adequate to meet the criterion mentioned above. For more information about ‘bvp4c’ scheme and its performance in solving boundary value problems, the reader is referred to Ref. 15.

4. RESULTS AND DISCUSSION

Before proceeding with presenting our numerical results for the boundary layer flow of Bingham fluids above a rotating disk, the code developed in this work had to be verified first. To that end, we have decided to rely on the data published in textbooks for Newtonian fluids. In Ref. 16, it has been reported that for Newtonian fluids we have: F'(0) = 0.5102, G'(0) = -0.6159, H(∞) = -0.8830. These data agree remarkably well with the numerical (boundary layer) results obtained in the present work for Newtonian fluids, i.e., F'(0) = 0.5100, G'(0) = -0.6159, H(∞) = -0.8845. Having verified the code, we are now ready to present our boundary layer results for swirling flow of Bingham fluids. Our main objective is to find the range of applicability of boundary layer theory for Bingham fluids. Figure 2 shows a comparison the tangential velocity profiles, G, obtained using our boundary layer analysis with the exact solution reported in Ref. 1 for three Reynolds numbers of Re = 1, 10, and 100 at a typical Bingham number of Bn = 10. This figure shows that the boundary layer theory is applicable for swirling flow of Bingham fluids provided that the Reynolds number is sufficiently large. Interestingly, even at a Reynolds number as small as Re = 10, the boundary layer theory is predicted to do a nice job (see, Fig. 2b).

To obtain a better insight about the range of validity of the boundary layer theory for swirling flow of Bingham fluids, Fig. 3 shows the L∞ error norm as a function of the Reynolds number for different values of the Bingham number. (The error norm is defined as L∞ = ||vE - vB||∞ = max (|vE - vB|i,j) where vE is the tangential velocity obtained from the exact solution, as reported in Ref. 1, and vB is the result obtained based on our boundary layer analysis.) As can be seen in Fig. 3, the boundary layer approximation becomes progressively more accurate by an increase in the Reynolds number. As a matter of fact, if we take 0.01 as the maximum allowable error norm for the boundary layer approximation to be regarded as

0<i,j<Nmax

Re

Nor

mof

Erro

r

100 101 102 103

10-3

10-2

10-1

Bn = 1Bn = 5Bn = 10Bn = 15Bn = 20

Fig. 3. The effect of the Reynolds number on the norm of error at different Bingham numbers.

ζ

G

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1

Reference [1]Present work

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G

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0

0.2

0.4

0.6

0.8

1


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G

0 5 10 15 20 25 30

0

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0.8

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(a) (b) (c)

(a)

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G

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0.8

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G

0 5 10 15 20 25 30

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G

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(a) (b) (c)

(b)

ζ

G

0 5 10 15 20 25 30

0

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G

0 5 10 15 20 25 30

0

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1


ζ

G

0 5 10 15 20 25 30

0

0.2

0.4

0.6

0.8

1


(a) (b) (c)

(c)

Fig. 2. The effect of the Reynolds number on the tangential velocity profile, G, obtained at Bn = 10:

a) Re = 1, b) Re = 10, c) Re = 100.

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acceptable, based on Fig. 3 the threshold Reynolds number, Retb (i.e., the Reynolds number above which the boundary layer approximation is valid) increases by an increase in the Bingham number. For example, while at Bn = 1 the threshold Reynolds number is roughly equal to Re = 18, for Bn = 10 it is roughly equal to Re = 75. One can therefore conclude that by an increase in the yield stress, the boundary layer approximation becomes less valid. This notion can better be seen in Fig. 4 which shows a plot of the threshold Reynolds number, Retb, as a function of the Bingham number.

Figure 5 shows the effect of the Bingham number on the boundary layer thickness obtained at Re = 120. (This Reynolds number is deemed to be large enough for the boundary layer theory to be valid.) For Newtonian fluids, the thickness of boundary layer is defined as the point at which we have G = 0.01. For Bingham fluids, however, we have realized that it is more appropriate to define the boundary layer thickness as the distance above the disk at which point the shear rate drops to γ4 = 0.01 [see Ref. 1]. As can be seen in Fig. 5, the boundary layer becomes thicker by an increase in the Bingham number. This results is consistent with the exact solution reported in Ref. 1. (It is also quite straightforward to show that like the exact solution our boundary layer analysis is able to show that the torque needed to rotate the disk increases by an increase in the fluid’s yield stress.) The fact that the bulk predictions made on the basis of our boundary layer analysis is consistent with the exact solution reported in Ref. 1 confirms that having dropped so many terms from the equations of motion (based on the boundary layer theory) does not affect the physics of the problem for Bingham fluids. That is to say that the boundary layer approximation is capable of capturing all facets of the swirling flow for Bingham fluids.

5. CONCLUDING REMARKS

A comparison between the boundary layer results obtained in the present work for swirling flow of Bingham fluids with the exact results reported in Ref. 1 suggests that for Bingham fluids the boundary layer approximation progressively loses its validity by an increase in the yield stress. Interestingly, however, the Reynolds number above which the boundary layer theory can safely be used in von Karman flow of Bingham fluid is of the order 100 (even for Bingham numbers as large as 20). Based on the results obtained in the present work, it can be concluded that for swirling flow of Bingham fluids the boundary layer approximation can safely be used provided that the Reynolds number is sufficiently large and the Bingham number is sufficiently small.

ACKNOWLEDGEMENT

The authors wish to express their gratitude to Iran National Science Foundation (INSF) for supporting this work under contract number 92011995. Special thanks are also due to the respectful reviewer for his/her constructive comments.

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Bn

Re tb

0 5 10 15 200

20

40

60

80

100

120

Fig. 4. The effect of the Bingham number on the threshold Reynolds number, Retb.

Bn

δ

0 2 4 6 8 10 12 14 16 18 208

8.5

9

9.5

10

10.5

11

11.5

Fig. 5. The effect of the Bingham number on the dimensionless boundary layer thickness (Re = 120).

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On the Validity of Boundary Layer Theory for Simulating