Scientia Iranica B (2018) 25(2), 790{798

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

Stability of thixotropic uids in pipe ow

M.H. Nahavandian, M. Pourjafar, and K. Sadeghy�

Center of Excellence in Design and Optimization of Energy Systems (CEDOES), School of Mechanical Engineering, College ofEngineering, University of Tehran, Tehran, P.O. Box 11155-4563, Iran.

Received 19 October 2016; received in revised form 26 December 2016; accepted 8 April 2017

KEYWORDSLinear stability;Thixotropic uid;Moore model;Spectral method;Pipe ow.

Abstract. Linear stability of a thixotropic uid obeying the Moore model is investigatedin pipe ow using a temporal stability analysis in which in�nitesimally small perturbations,represented by normal modes, are superimposed on the base ow and their evolution in timeis monitored in order to detect the onset of instability. An eigenvalue problem is obtained,which is solved numerically using the pseudo-spectral Chebyshev-based collocation method.The neutral instability curve is plotted as a function of the thixotropy number of the Mooremodel. Based on the results obtained in this work, it is concluded that the thixotropicbehavior of the Moore uid has a destabilizing e�ect on pipe ow.© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

Thixotropy appears to be a common e�ect amongindustrial uids such as drilling muds [1-3]. Waxycrude oils are also known to exhibit thixotropic be-havior at su�ciently low temperatures. The maincause of thixotropy in such complex uid systems arisesfrom an interaction between the uid elements and thesuspended particles or microstructures. In practice,these particles temporarily cross-link and form bonds,which are broken down by the action of shear. Si-multaneously, these bonds are reformed through theBrownian motion, but, because structure reform takesplace at a lower rate, in practice, one might witnessa time-dependent thixotropic e�ect. For instance, theviscosity of such uid systems may vary considerablywith time in addition to its being a function of theshear rate [1-3]. Such time-dependent behavior isexpected to a�ect the critical Reynolds number incon�ned ows [4,5]. In spite of its technologicalimportance, instability of thixotropic uids has rarely

*. Corresponding author. Tel/Fax: +98 21 82084011E-mail address: sadeghy@ut.ac.ir (K. Sadeghy)

doi: 10.24200/sci.2017.4498

been addressed in the past. One can notably mentionthe work carried out by Pearson and Tardy [6], andEbrahimi et al. [7], who have shown that thixotropy hasa stabilizing e�ect on the Sa�man-Taylor instability ina Hele-Shaw cell. On the other hand, Pourjafar et al. [8]have shown that in circular Coeutte ow, the e�ect ofthixotropy can be stabilizing or destabilizing dependingon the gap size and the severity of the uid's thixotropy.To the best of our knowledge, the e�ect of thixotropyon the stability of pipe ow has not been addressed inthe past.

The interest in the stability of pipe ow stemsfrom the fact that this particular geometry is widelyused for the transport of Newtonian and non-Newtonian uids [9-15]. In previous studies dealingwith non-Newtonian uids, the e�ect of shear-thinningand yield stress has already been investigated on thecritical Reynolds number in pipe ow. For example,we already know that shear-thinning has a stabiliz-ing e�ect on pipe ow [16]. Also, based on currentknowledge, it is well established that a non-zero yieldstress increases the critical Reynolds number in pipe ow [17,18]. As to thixotropic uids, it is also knownthat thixotropy can have a destabilizing e�ect on planePoiseuille ow [19]. Surprisingly, however, the e�ect ofthixotropy on the critical Reynolds number in circular

M.H. Nahavandian et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 790{798 791

Poiseuille ow still remains unexplored. In the presentwork, we intend to investigate the e�ect of thixotropyon the critical Reynolds number in pipe ow. Ourinterest in this particular ow arises from the fact thatpipelines are widely used for the transport of waxycrude oils, which are known to exhibit thixotropice�ects. Another motivation for the present study isthe fact that capillary rheometers are frequently usedfor the rheological characterization of non-Newtonian uids including thixotropic ones [20]. As a matterof fact, the viscometric data obtained using capillaryrheometers are commonly used for deciding on theirconstitutive equation. With these in mind, this paperaddresses the stability to small disturbances of circularPoiseuille ow for a typical thixotropic uid. To thatend, the paper is organized as follows. In the nextsection, the equations of motion are presented forthe thixotropic uid of interest (namely, the Mooremodel). The basic (i.e., unperturbed) solution is thendescribed for this particular uid. After that, weproceed with developing the linear stability analysisof the basic ow so-obtained. The numerical solutionto the stability problem is then given together withpresenting our understanding of its signi�cance. Thework is concluded by highlighting major �ndings.

2. Mathematical formulation

Figure 1 shows the ow geometry, which is a horizontalcircular rigid pipe of length L and radius R with itsaxis along the z-direction. The pipe is assumed tobe carrying an incompressible thixotropic uid underlaminar conditions. Neglecting gravity e�ects andassuming axial symmetry, the equations governing theisothermal ow of any uid are the Cauchy equationsof motion together with the continuity equation. Incylindrical co-ordinate systems, they are written as:

��@ur@t

+ ur@ur@r

+ uz@ur@z� u�2

r

�= �@p

@r

+1r@ (r�rr)@r

+@�zr@z� ���

r; (1)

��@u�@t

+ ur@u�@r

+ uz@u�@z� u�ur

r

�=

1r2@�r2�r�

�@r

+@�zr@z

; (2)

��@uz@t

+ ur@uz@r

+ uz@uz@z

�= �@p

@z+

1r@ (r�rz)@r

+@�zz@z

; (3)

1r@@r

(rur) +@uz@z

= 0; (4)

Figure 1. Schematics of the circular pipe ow.

where � is the uid's density; ur; u�; and uz are thevelocity components; and �ij are the components ofthe stress tensor.

The stress components in these equations need aconstitutive equation to be related to the velocity �eld.In this study, we assume that the thixotropic uid ofinterest obeys the \Moore model" [21,22]. For a uidobeying this simple thixotropic model, the viscosity isboth shear- and time-dependent, that is:

�( _ ; t) = �1 (1 + ��( _ ; t)) ; (5a)

where, _ is an e�ective shear rate de�ned by therelationship:

_ =p

2dijdij ; (5b)

where dij is the rate-of-deformation tensor de�ned by:

D = dij =24 @ur@r

12

�1r

�@ur@� �u�

�+ @u�

@r

� 12

�@ur@z + @uz

@r

�d21 =d12

1r

�@u�@� +ur

� 12

�@u�@z + 1

r@uz@�

�d31 =d13 d32 =d23

@uz@z

35 :(5c)

In Eq. (5), �( _ ; t) is the structural parameter, whichvaries between 1 to 0 with \one" denoting completestructure buildup and \zero" denoting complete struc-ture breakdown. The viscosities corresponding to thesetwo limiting cases are referred to by the zero-shear(�0) and in�nite-shear (�1) viscosities, respectively.It is worth mentioning that in the Moore model, theselimiting viscosities are related to each other through�, the simple relationship: � = �0��1

�1 . To repre-sent the time-dependency of the structural parameter,an appropriate kinetic equation is needed. In theMoore model, the kinetic equation takes the followingform [21,22]:

D�Dt

= �k1 _ �+ k2 (1� �) ; (6)

where k1 and k2 are model parameters controllingthe rates of structure breakdown and structure build-up, respectively. While k2 has the dimension of areciprocal time, k1 is dimensionless. Thus, the ratioK = k1=k2 can be regarded as the \characteristic

792 M.H. Nahavandian et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 790{798

time" of the Moore uid. In a sense, K controls thetime needed by the Moore to �nd its new con�gurationunder equilibrium conditions. Still, it should be notedthat for thixotropic e�ects to become important, Kshould be of the same order or even larger than the\characteristic time of the ow".

2.1. The basic owAs the �rst step in our instability analysis, we have to�nd out the velocity and viscosity �elds under steadyconditions{the so-called basic ow. Since the owis unidirectional, the equations of motion are simplyreduced to:

0 = �@p@z

+1r@@r

(r�rz); (7)

where p is the isotropic pressure, r is the radial distancefrom the pipe axis, z is the distance along the pipe,and �rz = Gr=2 is the shear stress (with G being thedriving pressure gradient @p=@z). On the other hand,under equilibrium conditions, the structural parameteris obtained as �ss = k2=(k1 _ ss + k2) so that theequilibrium shear stress becomes:

�ss(r) = �1�

1 + �k2

k1 _ ss + k2

�_ ss; (8)

where the subscript \ss" means steady state. Havingmultiplied this equation by k2=k1�1, after some simplerearrangement, we end up with the following quadraticequation for the steady shear rate:

_ 2ss +

�(1 + �)

k2

k1� �ss�1

�_ ss � k2

k1�1�ss = 0: (9)

In an attempt to solve this equation for the shear rate,we �rst try to make it dimensionless. To that end, wescale the radial position by R, the shear rate by U0=R,and the shear stress by GR, where U0 is the velocityscale (say, the centerline velocity). From Eq. (9), thephysically acceptable root is easily obtained as:

_ ss=12

24r2� (1+�)

Tx+

s�r2� (1+�)

Tx

�2

+2rTx

35 ;(10)

where Tx = k1GR=�1k2 is called the thixotropynumber. Figure 2 shows a plot of Eq. (10) as a functionof the thixotropic parameter, Tx, and the viscosityratio, �.

Having found the shear rate pro�les at equilib-rium, we can obtain the basic velocity pro�les bysimply integrating the equation: _ s = duz=dr. Indimensionless form, the result is:

uz(r) =(1� r2)

8+r � 1

2

�1 + �Tx

�� 1

2

�r2

+1� �Tx

��sr2

4+ r

�1� �Tx

�+�

1 + �Tx

�2

+ 2�ln�

2� 2�Tx

+ r

+ 2

sr2

4+ r

�1� �Tx

�+�

1 + �Tx

�2��:� ��Tx2

�� 2�ln�

2� 2�Tx

+ 1

+ 2

s14

+1� �Tx

+�

1 + �Tx

�2��:� ��Tx2

�+

12

�12

+1� �Tx

��s

14

+1� �Tx

+�

1 + �Tx

�2

:(11)

As expected, this equation reduces to the well-knownparabolic velocity pro�le for Newtonian uids by sim-

Figure 2. Shear rate pro�les as a function of the thixotropic number, Tx, and the viscosity ratio, �.

M.H. Nahavandian et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 790{798 793

ply setting � = 0. The importance of this semi-analytical solution is that it can easily be used toinvestigate the e�ect of Tx and � on the velocitypro�les. As such, it can be regarded as an exactsolution for the pipe ow of Moore uids. And,as we know, exact solutions are quite rare in non-Newtonian uid dynamics-mainly because of the non-linearity of their constitutive behavior. This is perhapswhy the �eld of non-Newtonian uid mechanics reliesso much on the computational techniques. However,computer codes need to be veri�ed �rst and this shouldpreferably be done by comparing their output with anexact solution. To this should be added the fact thatexact solutions can provide us with a better insightas to the physics of any given uid mechanics problem.Obviously, exact solutions are indispensable tools in the�eld of uid mechanics, and this by itself highlights theimportance of Eq. (11) by itself. It is worth mentioningthat our semi-analytical solution has enabled us to �ndthe velocity pro�les without any need to add di�usionterms to the kinetic equation as previously used byBillingham and Ferguson [23]{see also [19] who resorted

to the same trick in order to obtain converged resultsin their uid mechanics problem.

Figures 3 and 4 show typical basic- ow velocitypro�les using Eq. (11). As can be seen in Figure 3,by an increase in �, the velocity pro�les deviate moreand more from the parabolic pro�le of Newtonian uids. This is because at su�ciently large �, thelogarithmic term in Eq. (11) becomes progressivelymore important. It is also interesting to note thatthe centerline velocity decreases by an increase in theviscosity ratio. Figure 4 shows the e�ect of Tx numberon the axial velocity pro�le for a given �. As can beseen in this �gure, the centerline velocity increases byan increase in Tx. Obviously, the two parameters inthe Moore model (i.e., � and Tx) have opposite e�ectson the velocity pro�les.

3. Linear stability analysis

Having found the basic solution, we can now proceedwith their linear stability analysis. Using the idea ofnormal modes, we substitute:

Figure 3. E�ect of the viscosity ratio on the axial basic ow velocity pro�les, u, obtained at (a) Tx = 10 and (b)Tx = 100.

Figure 4. E�ect of the viscosity ratio on the axial basic ow velocity pro�les u in various Tx numbers: (a) � = 10 and (b)� = 100.

794 M.H. Nahavandian et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 790{798

uz (r; z; t) = �uz + uz = �uz + w(r)eqt+ikz; (12a)

u� (r; z; t) = 0 + u� = f(r)eqt+ikz; (12b)

ur (r; z; t) = 0 + ur = u(r)eqt+ikz; (12c)

� (r; z; t) = ��+ "(r)eqt+ikz; (12d)

� (r; z; t) = �� + y(r)eqt+ikz; (12e)

p (r; z; t) = �p+ �(r)eqt+ikz; (12f)

where �uz; �p; ��; and �� represent the basic solution asrepresented by Eqs. (10) and (11), and � is thekinematic viscosity. It needs to be mentioned that,based on the temporal instability analysis, in theseequations, the axial wavenumber (k) is a real numberwhereas the growth rate (q) is a complex number.Now, as the next step, we insert these equations intothe time-dependent set of the governing equations.After neglecting the nonlinear terms based on the\linear stability analysis," we end up with the followingequations for the perturbations amplitude functions:

uq + ik�uzu =�1�d�dr

+2r

��dudr

+ 2d��drdudr

+ 2��d2udr2

+ ikyd�uzdr� k2��u; (13a)

f q + ikf �uz =���fr2 � f

rd��dr

+��rdfdr

+d��drdfdr

+ ��2 d2fdr2 � k2��f ; (13b)

qw + fd�uzdr

+ ikw�uz =�ik��

+1r

(d��dr

dwdr + �� d

2wdr2 + dy

drd�uzdr +

y d2 �uzdr2 + ikud��

dr + ik�� dudr

)� 2k2w��;

(13c)

ur

+dudr

+ ikw = 0: (13d)

To calculate the viscosity function in these equations,the e�ective shear rate should be calculated �rst from

Eq. (5a). To that end, we �rst write down the rate ofdeformation tensor as shown in Box I (see Eq. (5b)),where we have dropped all @=@� terms thanks tothe ow being axisymmetric in cylindrical coordinatesystem. The e�ective shear rate can then be obtainedas:

_ =

vuuuuuuuuuuuuuuuuuuuut

2

(�@ur@r

�2

+�urr

�2

+�@��uz + uz

�@z

�2)

+�@u�@z

�2

+�@ur@z

�2

+�@ (�uz + uz)

@r

�2

+ 2@ur@z

@ (�uz + uz)@r

+u2�r2 +

�@u�@r

�2

� 2@u�@r

u�r:

(14a)

This equation can be simpli�ed as:

_ �s�

@�uz@r

�2

+ 2@uz@r

@�uz@r

+ 2@ur@z

@�uz@r

: (14b)

The above equation can be recast into the followingform:

_ =

vuut�@�uz@r

�2"

1 + 2@uz/@r@�uz/@r

+ 2@ur/@z@�uz/@r

#

=����@�uz@r

����vuut1 + 2

@uz/@r@�uz/@r

+@ur/@z@�uz/@r

!: (14c)

Having dropped all nonlinear terms and using thebinomial, we obtain:

_ � S [1 + (@uz/@r + @ur/@z)] ; (14d)

where, \S" refers to the sign of d�uzdr . Knowing the shear

rate for the perturbed form of the Moore model, thekinetic equation (see Eq. (6)) takes the following form:

q"+ ud��dr

+ ik�uz " = �k1S�

���dwdr

+ iku�

+ "�

� k2": (14e)

D =

2664 @ur@r

12

� 1r

�@ur@� � u�

�+ @u�

@r

� 12

�@ur@z + @(uz+�uz)

@r

�d21 = d12

1r

�@u�@� + ur

� 12

�@u�@z + 1

r@(uz+�uz)

@�

�d31 = d13 d32 = d23

@(uz+�uz)@z

3775 :Box I

M.H. Nahavandian et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 790{798 795

It should also be noted that there is a relationshipbetween the disturbed amplitude of the kinematicviscosity and the structural parameter itself. To obtainthis relationship, Eq. (5) is re-written as:

� = �1h1 + �

���+ �

�i= �1

�1 + ���

�+ ��1�

= �� + ��1�: (14f)

On the other hand, knowing that � = ��+ �, we obtain:

" = (1/��1) y: (14g)

Hence, Eq. (14e) can be re-written as:

qy + ud��dr

+ ik�uz y = �k1S�

(�� � �1)�dwdr

+ iku�

+ y�� k2y: (14h)

As the next step, we substitute: a = Rk, � = R2q=�1,u� = u=U0, f� = f=U0, �� = ��=�1, and y� = y=�1.We then combine Eqs. (13a) to (13c) with Eq. (14e)in order to obtain the following r- and �-momentumequations. We �nally end up with following equationsfor the dimensionless (still unknown) functions \u�"and \f�":

D2����D2u�a2r�2 +

Du�a2r�2 �

u�R2a2r�4 + u�

�+D���

�(

1a2r�2 �

R2

a2 )D3u�

+ 2�

1a2r2 � 1

Ra2r3

�D2u�

+�

1R2a2r4 � 2

Ra2r3 � 1Rr

�Du�

��

2Rr

+4

R3a2r5 � 2Rr3a2

�u��

+ �����R2D4u�

a2 +D3u�a2r�2 �D

2u��

3� 1Rr�

��Du�

�� 4Rr� +

1R2r�2

�� u

�6

R2r�4a2 � 2R2r�2 �

a2

R2

��+ �

��D2u�

a2 � Du�Rr�a2 +

u�R2r�2a2 +

u�R2

�+ i�u�U0a�u�z�1R

� y�RD3�u�za

� 2RDy�D2�u�za

� 1aD�u�zRD2y� � 1

aD�u�zRy�

+a

�1RU0

2u��u�z�

+iRea

�D�u�zDf� � �u�zD2u� �Du�D�u�z

+ f�D2�u�z �D�u�zDu��

= 0: (15a)

For the �-momentum equation, we obtain:

D����� f�Rr� +Df�

�+ ���

��f�a2

R2 � f�R2r�2

+1Rr�Df

��

+ ���2�1D2f� � iaReR2 �u�zf�

� f� �R2 = 0; (15b)

where D = ddr . As to the z-momentum equation, we

have:

y���1�R2 + i

aR

�u�z + Txk2S + k2GR�

= �2iU0Txk2S

Ra

�D2u� � u�

R2r�2 + 1Rr�Du

��+ aRu�U0

!(��� + 1) ; (15c)

where y� is de�ned as:

y� =� (�� + 1)

�1�R2 + i aR �uz + Txk2

�S + k2�GR: (15d)

Also, � is de�ned as:

� =� 2iU0Txk2�S�Ra

(D2u� � u�R2r�2 +

1Rr�Du

�)

+aRu�U0

�: (15f)

The above set of di�erential equations (governing theperturbations) needs six boundary conditions to beamenable to a numerical solution. Based on the no-slipcondition, the three velocity perturbations and theiramplitudes must vanish at the wall. On the other hand,from Eq. (13d) the boundary condition on w can betranslated into du

dr + ur = 0 at the wall as well as at

the centerline. To capture the �rst mode of instability,the dimensionless ampli�cation factor, �, should be setequal to zero.

796 M.H. Nahavandian et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 790{798

4. Method of solution

Since our main target is to �nd out the criticalReynolds number, it su�ces to obtain the neutralstability curves by �nding the critical wave number atwhich we have: max[real(�cr(Recr; �cr))] = 0. In theMATLAB code, we used the \polyig" function to dealwith the eigenvalue problem resulting from Eqs. (15a)and (15b). In addition, for solving the stability equa-tions, we have relied on the pseudo-spectral based onChebyshev polynomials and using the Gauss-Lobattocollocation points [24]. The fundamental considerationof spectral method is to assume unknown functions ofu� and f� as the base functions, that is:

u� (x)=NXn=0

anTn (x) ; f� (x)=NXn=0

bnTn (x); (16)

where Tn(x) = cos(n cos�1(x)) is the nth Chebyshevpolynomials. Since the polynomials are de�ned in[�1; 1], we use the transformation x = 2r � 1 to mapthe interval of [0; 1] (from pipe centerline to its wall)to [�1; 1]. The collocation points used in this work arethe Gauss-Lobatto points de�ned by:

xj = � cos�j�N

�; j = 0; 1; 2; :::; N: (17)

In the next section, we report instability results ob-tained in this way for the critical Reynolds number asa function of � and K (or, Tx number). We havechecked the e�ect of the number of base (or, trial)functions on the accuracy of our numerical results andreached the conclusion that using N = 60 can ensuregrid-independent results. To validate the code, we canrely on published Newtonian results for pipe ow ofNewtonian uids. In [25], the critical Reynolds numberfor Newtonian uids has been reported to be equal to2000. Our spectral code renders a value of roughly2004, which is quite close to that reported in thisreference (see Figure 5).

5. Results and discussions

Having veri�ed the code, we are now ready to presentour new results addressing the e�ect of a uid'sthixotropy, as represented by the thixotropy number(Tx), on the instability picture in pipe ow. For com-pleteness, we also address the e�ect of viscosity ratio onthe critical Reynolds number. Figure 6 shows the e�ectof the thixotropy ratio on the critical Reynolds numberas well as on the critical wave number for several viscos-ity ratios, �. As can be seen in this �gure, for any given�, by an increase in Tx, the critical Reynolds number isdecreased monotonically, meaning that the thixotropyratio in the Moore model has a destabilizing e�ect onthe pipe ow. Interestingly, longer wave numbers (or,

Figure 5. Critical Reynolds number for Newtonian uids.

equivalently, shorter wavelengths) are excited when Txis increased (see Figure 6(b), right). Unfortunately, a uid obeying Moore model is shear-thinning in additionto being thixotropic. One might therefore wonder ifthe e�ect shown in Figure 6 is a true re ection of thethixotropy or not. Based on available data (see, for ex-ample, Ref. [16]), we already know that shear-thinninghas a stabilizing e�ect on pipe ow. Therefore, theresults presented in Figure 6 show that the destabilizinge�ect of thixotropy is so strong that it has eclipsed thestabilizing e�ect of shear-thinning, so much so that thenet e�ect is destabilizing (see Figure 6).

Figure 7 shows the e�ect of the viscosity ratio onthe critical Reynolds number as well as on the criticalwave number for several thixotropy ratios, Tx. Ascan be seen in this �gure, for any given Tx, by anincrease in �, the critical Reynolds number is increasedmonotonically, meaning that the viscosity ratio in theMoore model has a stabilizing e�ect on the pipe ow.Interestingly, shorter wave numbers (or, equivalently,longer wavelengths) are excited when � is increased(see Figure 7(b)). As mentioned above, a uid obeyingMoore model is shear-thinning and, for a given �1,its shear-thinning behavior is intensi�ed when � isincreased. On the other hand, the e�ect of shear-thinning on pipe ow is already known to be stabilizing.Thus, the destabilizing e�ect of the viscosity ratio is notsurprising (see Figure 7).

The strong e�ects of the thixotropy number andthe viscosity ratio in the Moore model on the instabilitypicture of pipe ow can best be seen by plotting theneutral instability curves in Figure 8. In Figure 8(a),we have shown the e�ect of Tx for a typical � = 10and in the right plot, we have shown the e�ect of �for a typical Tx = 10. The destabilizing e�ect of Txis evident in the left plot (cf. results at Tx = 10 withresults at Tx = 100). Similarly, the stabilizing e�ect of

M.H. Nahavandian et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 790{798 797

Figure 6. E�ect of the thixotropy number, Tx, on the critical Reynolds number.

Figure 7. E�ect of the viscosity ratio, �, on the critical Reynolds number.

Figure 8. E�ect of the thixotropy ratio at constant viscosity ratio: (a) � = 10 and viscosity ratio at constant Tx numberand (b) Tx = 10 on the critical Reynolds number.

� is clear in the right plot (cf. results at � = 0:5 withresults at � = 1).

6. Concluding remarks

In the present work, we have relied on the Moore modelto investigate the e�ect of a uid's thixotropy on thecritical Reynolds number in pipe ow using a linear,temporal, normal-mode, and stability analysis. Basedon the results obtained in the present work, it can be

concluded that thixotropic uids obeying Moore modelare less stable than their corresponding Newtonian uids. That is to say that, the thixotropy number(i.e., the breakdown-to-rebuild ratio) has a destabi-lizing e�ect on the pipe ow. Since a uid obeyingMoore model is both shear-thinning and thixotropic,interpretation of the results appear to be a tricky taskat �rst sight. However, Fortunately, because shear-thinning is already known to have a stabilizing e�ect onpipe ow, the destabilizing e�ect found for the Moore

798 M.H. Nahavandian et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 790{798

model can be ascribed to its thixotropic behavior,which is so strong that it eclipses the stabilizing e�ectof shear-thinning.

References

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Biographies

Mohammad-Hosein Nahavandian is a PhD stu-dent of Mechanical Engineering at Amirkabir Uni-versity of Technology, Tehran, Iran. He receivedMSc degree from University of Tehran and BSc de-gree from Iran University of Science and Technology,Tehran, Iran. His research interests include rheology,non-Newtonian uid mechanics, instability, multiphase ows, CFD, Nano- uids, and porous media.

Mohammad Pourjafar is a PhD student of Mechan-ical Engineering at University of Tehran. He receivedMSc degree from University of Tehran and BSc degreefrom University of Guilan, Tehran, Iran. His researchinterests are rheology, non-Newtonian uid mechanics,poro-elastic instability, and peristaltic ows.

Kayvan Sadeghy is a Professor of Mechanical En-gineering at University of Tehran. He received hisPhD degree from University of Toronto, Toronto,Canada. He obtained his BSc and MSc degrees fromUniversity of Tehran, Tehran, Iran. He conducts ex-perimental/numerical/analytical research in the �eldsof complex uids, non-Newtonian uid mechanics, andinstability, among others.