On the slow motion of a sphere in fluids with non-constant viscosities

International Journal of Engineering Science 48 (2010) 78–100

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On the slow motion of a sphere in fluids with non-constant viscosities Bong Jae Chung a, Ashwin Vaidya b,* a b

Department of Marine Sciences, University of North Carolina, Chapel Hill, NC 27599, USA Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA

a r t i c l e

i n f o

Article history: Received 30 April 2009 Received in revised form 16 June 2009 Accepted 26 June 2009 Available online 5 August 2009 Communicated by K.R. Rajagopal

Keywords: Drag reduction Power-law fluid Hot particle Pressure dependent viscosity

a b s t r a c t We study a slowly moving sphere in fluids where the viscosity depends upon factors such as shear-rate, temperature and pressure, with the flow field approximated by the Stokes flow past a sphere. We derive an expression for the stresses generated in the fluid due to these various factors. This gives us information about both, the force imposed by the fluid upon the sphere and also the reaction force due to the sphere upon the fluid, referred to as the stress density. The values of the force and stress density are numerically computed in each of the cases and analyzed for various values of the flow and material parameters. Our computations show interesting variations in the distribution of stress density in the fluid for the various cases and also give us valuable information about the effect of walls. Our calculations also indicate that particle heating or cooling can serve as a significant control parameter since the drag force upon the sphere increases dramatically for a cold particle and can be reduced considerably upon heating it. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we study the steady forces imposed upon a sedimenting rigid body in quiescent fluids where viscosity is not a constant but can vary due to various factors. The problem of resistance on a moving body due to a medium is classical and well studied problem. The development of the drag formula for the motion of a sphere in a viscous fluid, based on the derivation of the appropriate stream function, is well known and dates back to the work of Stokes [51]. Since a formal expansion of the velocity field in the Reynolds number is not permitted due to the Stokes and Whitehead’s paradox [61], a simple perturbation argument has been found to be insufficient at higher orders in Reynolds numbers. Oseen derived an approximation, also accounting for the inertial terms, which seems to yield substantially better results [59]. More recently, a promising technique has been proposed by Liao [30], namely the homotopy analysis method and claims to give the drag formula for a sphere for arbitrarily large values of Re [30]. With growing developments in the field of complex fluids, the question of drag on a body moving in complex fluids, particularly in non-Newtonian fluids, is a significant one. The contribution of the shear-thinning, shear-thickening and normal stress effects often found in complex fluids, to the drag and lift of immersed bodies is very much sought after information. The answers to these questions is of great interest for instance, in the transport of materials such as slurries and in the field of geophysics in the context of settling behavior of natural sediments (see [9,12,21,26] and references therein). Acharya et al. [1] provide a detailed account of theoretical and experimental work done until 1976 on the motion of a sphere in power-law and viscoelastic fluids. An approximate analytical solution using the stream function approach was also obtained for the

* Corresponding author. Current Address: Department of Mathematical Sciences, Montclair State University, Montclair, NJ 07043, USA. Tel.: +1 919 962 9622; fax: +1 919 962 2568. E-mail address: [email protected] (A. Vaidya).

0020-7225/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2009.06.010

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B.J. Chung, A. Vaidya / International Journal of Engineering Science 48 (2010) 78–100 Table 1 Some well known models for factors affecting viscosity. Factor

Model

References

1. Shear-ratea 2. Temperature

lðc_ Þ ¼ kjc_ jm (i) blðTÞ ¼ k1 eb1 T (ii) clðTÞ ¼ k1 2eb2 =ðRTÞ (i) lðpÞ ¼ a1 p a p (ii) lðpÞ ¼ ke 2 lð/Þ ¼ ð1  /=/m Þ2:5/m

[3,32] [54,32]

3. Pressure 4. Concentrationd a b c d

[3,32] [23] [29]

Various models available to describe this dependence such as Cross, Carreau, Ellis etc. Valid for T > 50K. Valid for a wide range of temperatures. Valid for suspensions.

power-law fluid under the assumption that jn  1j << 1.1 Also, the high Reynolds number limit was analyzed using boundary layer theory. Chhabra and coworkers [6,7,13] also discuss the motion of spheres in fluids with elasticity and shear-thinning, using the Carreau model. Fosdick and Rajagopal [16] have studied the drag imposed on obstacles of arbitrary shape in second grade fluids. They are able to show that there is a reduction in drag due to the normal stress effects present in these fluids which is given by

F D ¼ F SD þ a1

 Z  1 jxj2  jAj2 n  k dA 4 @B

ð1Þ

where F SD is the Stokes drag, a1 is the first normal stress coefficient, x ¼ curl u, A is the symmetric part of the velocity gradient, n is the normal vector to the body and k is the direction of the flow. Some important recent, relevant experimental, theoretical and numerical investigations of this problem include [5,8,14,15,20,24,31,40–42,62]. Experimental evidence strongly suggests that the drag force can be reduced or enhanced in the presence of non-Newtonian components of the fluid when compared to a Newtonian fluid [14]. Experiments and numerical simulations seem to suggest that this variation in drag is highly dependent upon the particular polymeric fluid being employed in the experiment or the model being used in the numerical study. Theoretical work on this subject dates back to a few decades and is based on perturbation analysis. The theory does predict variations in drag with increasing Weissenberg and Reynolds numbers and is seen to be valid [15]. The correction to the drag force, denoted K, is given by the relation

K¼

FD 6pgaU 1

where F D is the drag force, g is the fluid viscosity, a is the radius of the sphere and U 1 is the terminal velocity of the sedimenting sphere. The correction factor, K is dependent upon the Reynolds and Weissenberg numbers which characterize Newtonian and non-Newtonian fluids and also the shape of the sedimenting body. The reader is referred to Happel and Brenner [22] for a list of values of K for oblate and prolate spheroids of varying eccentricities in Stokes flow. In this paper, we also treat the case of fluids where viscosity can vary due to factors other than shear-rate (see Table 1). It is well known [51,52] that viscosity has a strong dependence upon temperature and even pressure,

l ¼ lðh; pÞ

ð2Þ

where h refers to the temperature and p to the pressure. Although the dependence of viscosity on temperature is very well known and has been used in a various fluid flow problems, there is very sparse literature on its effect upon sedimenting bodies (see for instance [33,34]). The possibility of a dependence of viscosity on pressure for certain fluids was first hypothesized by Stokes [51]. Experiments strongly validate this dependence (see [23,53,60] and references therein). Recently Hron et al. [23] have proposed certain models for the viscosity function, namely

l ¼ apðxÞ; l ¼ l0 eapðxÞ

ð3Þ

where a is a constant. A recent series of papers by Rajagopal and co-workers [4,35,36,45–48] have looked at the mathematical and physical properties of such models. However, once again, the question of interaction of such fluids with sedimenting bodies remains unanswered. Taking into account all of the above factors, which can affect viscosity, we model the viscosity function as

lðc_ ; h; pÞ ¼ l0 f ðc_ ÞgðhÞhðpÞ

ð4Þ

where f ðc_ Þ, gðhÞ and hðpÞ are dimensionless functions of shear-rate, temperature and pressure, respectively. This model helps analyze the effect of each factor upon the drag as well as the combined effect of these factors. Eq. (4) is very relevant in geophysical contexts in the description of magma flows where shear-rate and temperature dependence is very strong. On the 1

And hence that jc_ jm is a constant, where m is the power-law index and c_ is the shear-rate.

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other hand, one can find that the flows of inhomogeneous materials such as suspensions in fluids to be an example where pressure dependence on viscosity can be meaningful. We consider the case of a slow, steady translational motion of a rigid body in a fluid medium. We are able to write an expression for the net forces and torques imposed upon a sedimenting body of any shape in Newtonian and non-Newtonian fluids. Specifically, we can show that the net force (F) and torque (M) in a non-Newtonian fluid can be decomposed into

FðuÞ ¼FS ðuÞ þ ReFI ðuÞ þ kFV ðuÞ  V ðuÞ MðuÞ ¼MS ðuÞ þ ReMI ðuÞ þ kM

ð5Þ ð6Þ

where the subscripts S, I and V represent the Stokesian, inertial and viscoelastic contribution to the force and torque and Re and  k represent the Reynolds number and non-Newtonian parameter or those coming from the particle heating or pressuredependent viscosity. Eq. (6), in particular has been dealt with in some detail earlier in a variety of fluids [18,56–58], Newtonian, power-law and also second order viscoelastic fluids. The Eqs. (5) and (6) permit us to analyze the contribution of the different aspects of the fluid to the forces. In order to estimate these force components, we write down the velocity field past the moving body as

u ¼ us þ w

ð7Þ

where us refers to the Stokes flow around the body which is known analytically for the case of a sphere and w is the remnant field which contributes to the nonlinearity of the problem. The decomposition is based on the idea employed earlier by Galdi and co-workers (see [19] and references cited within). We replace Eq. (7) into Eq. (5) and consider only the part of the force which is evaluated at us . We realize that the assumption on velocity in this paper is approximate and merely a first attempt at understanding the effects of non-constant viscosities. While the creeping flow for a Newtonian fluid is the Stokes flow, the same cannot be said for creeping flows of non-Newtonian fluids. The reason for our decomposition is that only the creeping flow of a viscous fluid past a sphere is known analytically while the same for more complex fluids is not. The Stokes and Whitehead’s paradox [61] prevent us from writing out any sort of perturbative form of u in the Reynolds number and hence we only have access to the creeping flow approximation of the velocity field in the limits of vanishingly small Re and k. A more rigorous approach would involve obtaining a perturbation solution of u in terms of the parameter  k. Additionally, it must be pointed out that the task of computing appropriate creeping flow velocities for the non-constant viscosity flows is very complex, both analytically and numerically. Our problem is further complexified in that we treat the effect of multiple factors, namely, shear-rate, temperature and pressure upon viscosity all of which cannot be treated collectively through perturbation analysis. The advantage of our formulation is its convenience and relative ease in calculating forces upon bodies of different shapes in a variety of fluids. Despite our assumption, our computations reveal interesting results regarding drag on a sphere. In Section 2, we derive the expression for the net force imposed upon the fluid upon the sedimenting body in a power-law fluid. In the subsequent Section 3, we treat the case of sedimentation of a heated (and cold) sphere in a viscous and powerlaw fluid. The novelty of our argument lies in modeling the dependence of viscosity on temperature implicitly i.e. we write l ¼ lðxÞ by choosing an appropriate function such that the effect of temperature on viscosity is to reduce viscosity in the immediate vicinity of the body. The objective of our analysis is to see if heating or cooling the surface of the particle induces substantial drag variations. Section 4 is devoted to the case of fluids with pressure-dependent viscosity and in Section 5, we discuss the numerical scheme employed to compute the resulting integrals which are presented in the final Section 6. 2. Forces in a power-law fluid 2.1. The governing equations We assume that the body B is free-falling in an unbounded fluid under the influence of gravity g with a translational velocity n. Though the experiments on particle sedimentation are performed in confined tanks, it is observed that when the aspect ratio of particle to tank diameter is sufficiently small, the effects of the walls can be ignored. The problem will be studied in a frame which is attached to the body (see Fig. 1). The governing equations for motion of the fluid, are given by

Re u  gradðuÞ ¼ div T div ðuÞ ¼ 0 u ¼ 0 at @ X

ð8Þ

lim ðuðxÞ þ nÞ ¼ 0

jxj!1

The governing equations for the body, on the other hand, are given in terms of the forces and torques acting upon it, namely

R R@ X @X

T  n ¼ me g yTn¼0

where me ¼ ðqb  qf ÞjBj is the effective mass.

ð9Þ

B.J. Chung, A. Vaidya / International Journal of Engineering Science 48 (2010) 78–100

81

Fig. 1. The creeping flow field past a sphere with streamlines. The reference frame is attached to the center of the sphere which is moving in the e1 direction.

For the fluid model, we select the generalized Newtonian or the power-law model, where the stress tensor is given by

T ¼ pI þ lð1 þ PÞm A

ð10Þ

where T is the Cauchy stress tensor, l is the coefficient of viscosity. This particular model is chosen in order to avoid any possible singularities in the shear-thinning case which could emerge from regions of vanishing velocity gradients. The kinematical tensor A ¼ 2D where

1 ðL þ LT Þ 2 L ¼ grad u D¼

ð11Þ ð12Þ

In this model, P accounts for the shear-thinning or shear-thickening aspect of the fluid and is given by

P ¼ jjA : Aj

ð13Þ

where j is the consistency index. When m < 0, the fluid is shear-thinning and if m > 0 the fluid is shear-thickening. When m ¼ 0, Eq. (10) reduces to the standard Newtonian fluid. We introduce non-dimensional variables

x ¼

x ; X0

¼ u

u ; U0

n ¼ n ; U0

p¼

p P0

ð14Þ

and define the dimensionless parameters

Re ¼

qU 0 X 0 jU 2 ; k ¼ 20 l0 X0

ð15Þ

where Re represents the Reynolds number and k is a parameter which accounts for the non-Newtonian-ness of the fluid. When k ¼ 0, the shear-dependent function vanishes and we go back to the case of a Newtonian fluid. Increasing the value of this parameter has the same effect as increasing the concentration of the polymer. As a result of the non-dimensionalization, ignoring the bars on the non-dimensional variables, our governing equations can be written in the following form:

  Re u gradðuÞ ¼ div pI þ ð1 þ kjA : AjÞm A div ðuÞ ¼ 0 u ¼ 0 at @ X lim ðuðxÞ þ nÞ ¼ 0

jxj!1

ð16Þ

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2.2. Expression for force Our eventual objective in this study is to understand the contribution to the drag force imposed upon a sedimenting body by the surrounding fluid medium due to different components of the fluid. Based on the argument provided in Appendix A, we can write the equation for the force imposed by the fluid upon the sedimenting body as

F ¼ Re

Z

ðiÞ

u  gradðuÞ  h þ

X

k 2

Z

ðiÞ

TE ðuÞ : Dðh Þ

ð17Þ

X

We define the inertial contribution to the force as

FI ¼

Z

ðiÞ

u  grad u  h

ð18Þ

X

and

FPL ¼

Z

1 2

ðiÞ

ð1 þ k½A : AÞm A : Aðh Þ

ð19Þ

X

is the power-law effect of the force. When k ¼ 0, then we obtain the Stokesian force expression. As a result the net force can be written in the simplified form

F ¼ FPL þ ReFI

ð20Þ

In order to simplify the argument, motivated by the fact that the motions of the particle and liquid are in the creeping flow regime, we write field, u ¼ us þ wðRe; k; mÞ where us is the Stokes velocity field. Hence, the net force can be written as

FðuÞ ¼ F0;PL þ ReF0;I þ Nðus ; wÞ

ð21Þ

where the superscript ‘0’ indicates the evaluation of the force at the zeroth-order. The term Nðus ; wÞ represents the remnant term.2 However, in the absence of any simple approximation of w, the best we can do for this nonlinear problem is to compute the forces imposed by the power law model upon a sphere at u ¼ us . The question of immediate concern is the drag force and its dependence upon the parameters k and m. We may further simplify this expression for the torque by writing the Stokes velocity P ðiÞ field, us ¼ 3i¼1 ni h . This representation is used to decouple the motion of the fluid from that of the body and will prove to be very convenient in our calculations that follow. We choose, without loss of generality, our reference frame attached to the body ð1Þ in such a way as to make n ¼ ðn1 ; 0; 0Þ. As a result, with us ¼ n1 h , the expression for the net force may be expressed very simply in terms of the auxiliary field. 2.3. Sedimentation in a power-law fluid We first treat the case of the sphere moving in a Newtonian fluid, i.e. we set k ¼ 0 in Eq. (17). As a result the expression of force becomes

F ¼ F0;V þ ReF0;I

ð22Þ

Hence it follows that a rigid particle, B experiences a net force, which is given by

F0;V ðus Þ ¼

1 2

Z

ð1Þ

Aðus Þ : Aðh Þ ¼

X

3 X

n1 /10 e1

ð23Þ

i¼1

where

/10 ¼

1 2

Z

ð1Þ

ð1Þ

Aðh Þ : Aðh Þ

ð24Þ

X

and Eq. (23) represents the Stokes’ drag. Note that since we assume that the sphere is only moving in the e1 direction, only this part of the force is relevant. Furthermore, using Eq. (65), in Appendix B, we see that FI ðus Þ ¼ 0. Therefore in the creeping flow approximation, the inertial contribution to the force cannot be captured. As a second case, let us consider the case of a sphere moving in a power-law fluid where the fluid is modeled by Eq. (10). As a result, the expression for the power-law contribution to the net force can be specifically written as

F0;PL ðuS Þ ¼

1 2

Z

ð1Þ

ð1 þ kjAðuS Þ : AðuS ÞjÞm AðuS Þ : Aðh Þ

ð25Þ

X

2 In our earlier work [18,19], when evaluating torques imposed by fluids upon bodies, we are able to provide an upper estimate for the remnant term N. However, in the case similar techniques fail and we have no way of finding any apriori estimates for N and therefore a formal justification for neglecting it is still forthcoming.

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B.J. Chung, A. Vaidya / International Journal of Engineering Science 48 (2010) 78–100

It seems that shear-thinning or shear-thickening effects of the fluid can play a role in modifying the viscous drag even for very slow flows. In order to get a feeling for the magnitude of variation in the drag forces, we compute it for certain specific cases of integer m which represent shear-thickening and shear-thinning fluids. Example 1. m ¼ 1 (shear-thickening).

F0;PL ðuS Þ ¼

1 2

Z

ð1Þ

X

ð1 þ kjAðuS Þ : AðuS ÞjÞAðuS Þ : Aðh Þ ¼ ð/10 n1 þ /11 n31 Þe1

ð26Þ

where /10 is given in Eq. (24) and

/11 ¼

Z

1 2

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ðkjAðh Þ : Aðh ÞjÞAðh Þ : Aðh Þ

X

Example 2. m ¼ 1 (shear-thinning).

F0;PL ðuS Þ ¼

1 2

Z

ð1Þ

ð1 þ kjAðuS Þ : AðuS ÞjÞ1 AðuS Þ : Aðh Þ ¼

X

Z X

n1

ð1Þ

ð1Þ

ð1 þ kn21 jAðh Þj2 Þ

ð1Þ

Aðh Þ : Aðh Þ:

ð27Þ

3. Sedimentation of a hot particle A second problem that we deal with in this paper concerns the sedimentation of a hot particle in a fluid medium and the relevant forces acting upon it. The fundamental equations governing the motion of such a hot particle are similar in general form to the ones dealt with in the earlier section. We must in addition to the fluid equations, also consider the steady state thermal equation in non-dimensional form, namely

u  rT ¼

1 DT Pe

ð28Þ

where T ¼ Tðr; hÞ represents the heat distribution in the fluid, Pe is the Peclet number and u is the fluid velocity. We further assume that the flow is very slow and Pe ! 0. As a result, the Eq. (28), reduces to

DT ¼ 0

ð29Þ

With the advection term now ignored, we may assume symmetry in the azimuthal direction, hence T ¼ TðrÞ. Putting this in Eq. (29) and solving for T, we have

TðrÞ ¼

aðT 0  T 1 Þ þ T1 r

ð30Þ

where a is the radius of the sphere and we employ boundary conditions

Tðr ¼ aÞ ¼ T 0 ;

Tðr ! 1Þ ¼ T 1

We can likewise treat the case of a hot particle (T 0 > T 1 ) or a cold particle (T 1 > T 0 ). A full modeling of this problem would involve working with the coupled system of fluid and the convection equations. However, a simple way to treat this problem would be to consider the viscosity of this fluid to vary with spatial coordinates by employing the Reynolds viscosity model

lðTÞ ¼ eaTðrÞ  is the rate of growth or decay, combined with Eq. (30) to get where a

lðrÞ ¼ l1 e

 aad r

T 1 a

ð31Þ

 d for sake of convenience. Our where l1 ¼ e and d ¼ T 0  T 1 . We further define a non-dimensional parameter, b ¼ a upcoming numerical analysis will consider the result of variations in b in the forces affecting the sphere. Since the frame of reference is placed upon the body, as the hot particle moves, it immediately interacts with the fluid in its immediate vicinity. As a result, in the case of a hot particle, the viscosity of the fluid close to the body would be lower than that far away from it, i.e. l0 < l1 while for the case of a cold particle, the inequality is reversed, so l0 > l1 . A point to be noted regarding the choice of the viscosity function given by Eq. (4) is that the model is physically valid in cases where (a) the fluid influence, due to viscosity variation, upon the particle is negligible and hence thermal diffusion in the liquid occurs much faster that the translational speed of the particle and (b) density variations with temperature are negligible when compared to viscosity variations. We make an additional assumption regarding the density variation of liquids with temperature. Based upon the literature on various liquids [27,28,49,63,25] we assume that though viscosity depends strongly upon temperature, the same is not true for density. Table 2 provides data on the changes in density and viscosity of different types of liquids (water, sulfolane, ethers) with temperature. Experiments indicate that for several liquids, for a temperature variation of about the density variation over about 30 °C, the viscosity can change by upto 50% whereas variations in density are only around 3%.

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Table 2 Some examples of density and viscosity variations in a few different fluids with temperature. Liquid

Temperature (°C)

Density (g/cc)

Viscosity (Pa-s)

Watera

20 40 60 80 25.15 30.15 35.15 40.15 45.15 50.15 20 40 60 80

0.9982 0.9922 0.9831 0.9716 1.2665 1.2620 1.2575 1.2531 1.2487 1.2442 1.0211 1.0059 0.9907 0.9731

1.003 0.653 0.467 0.355 0.0115 0.01005 0.00883 0.00778 0.00691 0.00616 0.00419 0.00274 0.00186 0.00119

Sulfolaneb

Diethylene glycol monomethyl etherc

a b c

Ref. [62]. Ref. [49]. Ref. [27].

The non-dimensional form of the governing equations is obtained using the same scheme adopted in Eqs. (14) and (15) with characteristic viscosity l1 used for purpose of non dimensionalization. Hence we have

h i ba Re u gradðuÞ ¼ div pI þ e r A div ðuÞ ¼ 0

ð32Þ

u ¼ 0 at @ X lim ðuðxÞ þ nÞ ¼ 0

jxj!1

where the ‘bars’ on the non-dimensional quantities have been again dropped for convenience. The expression for the resulting force acting upon such a body is obtained from a similar manipulation as performed in Section 2.2. We therefore have the equation

F ¼ FHP þ ReFI

ð33Þ

I

HP

where F is as given in Eq. (18) and F

FHP

Z

1 ¼ 2

ba r

(‘HP’ referring to hot particle), is given by

ðiÞ

e A : Aðh Þ

ð34Þ

X

When b ¼ 0, we go back to the problem of the sphere moving in a viscous fluid with

FHP ! FV ¼

Z

1 2

ðiÞ

A : Aðh Þ

X

The force equation is computed at the Stokes flow field, u ¼ us and therefore the zeroth-order contribution to the force may be given by

F ¼ F0;HP ðus Þ

ð35Þ 0;I

where we have used the result F

F0;HP ðus Þ ¼

1 2

Z

ba

ð1Þ

¼ 0 from our previous calculation. Hence, we may write ð1Þ

e r Aðn1 h Þ : Aðh Þ ¼

X

2 X

n1 w10 e1

ð36Þ

i¼1

where w10 is given by

w10 ¼

1 2

Z

ab

ð1Þ

ð1Þ

e r Aðh Þ : Aðh Þ

ð37Þ

X

We now combine the results of our previous calculations and look at the combined effect of particle heating (and cooling) as well as a power-law nature of the surrounding fluid upon the drag forces that experienced by the sphere. In this case, we write the viscosity as a function of radius, r and also upon the shear-rate, c_ . Hence,

l ¼ lðr; c_ Þ ¼ l1 ðrÞl2 ðc_ Þ The stress tensor corresponding to this problem is:

B.J. Chung, A. Vaidya / International Journal of Engineering Science 48 (2010) 78–100

85

Fig. 2. Contour plot of the viscosity variation due to pressure based on an exponential viscosity–pressure model with a ¼ 1; the values on the contours indicate the magnitude of the viscosity. The reference frame is oriented in the same manner as in Fig. 1.

Fig. 3. Four-, five- and six-sided finite computational volumes constructing a spherical domain.

ab

T ¼ pI þ e r ð1 þ kjA : AjÞm A

ð38Þ

This sort of temperature and shear-rate dependence has been considered before in the several studies before [37,38,52] (also see references cited therein3), however, in the case of external flow problems, one is usually restricted to numerical studies alone in the absence any simplifying assumptions such as the ones we have made in this article. Eq. (38) indicates that there are several components that emerge here, (i) the pure viscous component, (ii) the power-law component, (iii) the contribution

3

In these studies, viscosity has been considered to explicitly depend upon temperature.

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B.J. Chung, A. Vaidya / International Journal of Engineering Science 48 (2010) 78–100

due to the hot (or cold) particle in a viscous fluid and finally, (iv) the combined effect due to the heating (or cooling) and powerlaw nature of the fluid. Adapting the arguments of the previous sections, the net force acting upon the sphere can be given by

Fðus Þ ¼ F0;HPPL

ð39Þ

where

F0;HPPL ðus Þ ¼

1 2

Z

ab

ð1Þ

ð1Þ

ð1Þ

ð1Þ

e r ð1 þ kjAðn1 h Þ : Aðn1 h ÞjÞm Aðn1 h Þ : Aðh Þ

ð40Þ

X

which is much like the force in the pure power-law case but with the extra weight factor.

4. Fluids with pressure-dependent viscosity As far back as 1845 [51] it was shown that the viscosity of several incompressible fluids could have a strong dependence upon pressure. In fact, in recent years, Rajagopal and coworkers (see [4,45–48]) have provided a rigorous methodology whereby constitutive relations for a class of fluids whose viscosity depends on pressure and shear-rate can be obtained. These types of fluids are encountered, for example, in lubrication industry where the fluid is under high pressure [53] and can also be seen to be relevant in geophysical processes where high pressure environments exist. Rajagopal and coworkers, have proposed that

T ¼ pI þ lA

ð41Þ

l ¼ lðp; jAj2 Þ

ð42Þ

where

More specifically, we propose a slight variation of the model that Hron et al. [23] have suggested, namely

T ¼ pI þ lðpÞð1 þ kjAj2 Þm A

ð43Þ

where for m ¼ 0 this model reduces to

T ¼ pI þ lðpÞA

ð44Þ

When m < 0, the fluid is shear-thinning, and when m > 0, it is shear-thickening. Although a variety of models for l ¼ lðpÞ have been proposed [17,23,39], we resort to a specific form of the pressure-dependent viscosity, which was studied by Hron et al. [23], namely:

lðpÞ ¼ eap

ð45Þ

where a is a constant (see Fig. 2). Now repeating the calculations performed above for the force, but now acting on a sphere in a fluid modeled by Eq. (43), we obtain the following non-dimensional equation for force:

a

b

Fig. 4. An axi-symmetric computational spherical fluid domain exterior to a spherical particle representing: (a) entire domain and (b) blow-up of the region near the surface of the sphere.

B.J. Chung, A. Vaidya / International Journal of Engineering Science 48 (2010) 78–100

FPV ¼

1 2

Z

ðiÞ

eap ð1 þ k½A : AÞm A : Aðh Þ

87

ð46Þ

X

where the superscript ‘PV’ is indicative of pressure-dependent viscosity. If a ¼ 0, we retrieve Eq. (27) and if k ¼ 0 we have the ð1Þ force equation corresponding to the model (41). As before, we evaluate the velocity field u at us ¼ n1 h and also the presð1Þ sure field at p ¼ p which is given by Eq. (63). Therefore the Eq. (46) further simplifies to

F0;PV ¼

n1 2

Z X

ð1Þ

ð1Þ

ð1Þ

ð1Þ

ð1Þ

eap ð1 þ kn21 ½Aðh Þ : Aðh ÞÞm Aðh Þ : Aðh Þ:

ð47Þ

5. Computational methods for integration In this section we discuss the theoretical basis for computation of the Eqs. (27), (36), (40) and (47) while varying the parameters k, m and . This section will be devoted specifically to the topics of mesh generation and discretization of the integral equations and to explaining the convergence tests which help choose the appropriate domain size for this unbounded, exterior domain problem. 5.1. Grid generation and discretization of integral equations Integration of Eqs. (27), (36), (40) and (47) over the outer domain of a sphere are performed numerically. Finite volume cells are used to construct a spherical computational domain representing the exterior fluid flow past a sphere. The volumes

Fig. 5. Values of g on numbers of mesh refinement test cases for various domain sizes, Ds ¼ 10; 20; 30; 40; 50; 60 and (a) varying I and set J ¼ 71, K ¼ 41, (b) vary J and fix I ¼ 42 and K ¼ 41 and (c) vary K and let I ¼ 42, J ¼ 71. The symbols 1–6 on the lines in the figures correspond to Ds ¼ 10—60.

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include four-, five- and six-sided cells as shown in Fig. 3. The four-sided cells are used at the center of the sphere while the five-sided cells are formed along the singular line, which occurs at the angle / ¼ 0o and / ¼ po in the spherical coordinate system. To construct a spherical domain having normal vectors which are not aligned with the (x1 ; x2 ; x3 ) coordinates, we employed body–fitted coordinates (f1 ; f2 ; f3 ) so that each cell has the body-fitted coordinates [10,50]. Therefore, the calculations of the gradients are performed through the transformation from the rectangular coordinates to the body-fitted coordinates. The grid points inside the sphere are not considered for the calculation therefore the integration is performed from the surface of the sphere to the outermost boundary of the domain. The axi-symmetric meshed computational domain, representing the fluid surrounding the sphere generated based on the body-fitted coordinates is shown in Fig. 4. A second order approximation on the gradients of vectors based on a finite volume formulation is used to discretize Eq. (27) to evaluate Fi0;PL ðus Þ. The components of gradient of a vector at a grid (l,m,n) is then discretize using the second order central difference method in a form of

@ui @ui @fk uil2;m2;n2  ul1;m1;n1 @fk i ¼  ; @xj @fk @xj 2Dfk @xj

i; j; k ¼ 1; 2; 3

ð48Þ

where ui stands for a component of a vector u, Dð Þ is the grid size for fi and l; m; n are the indices of a nodal point. When k ¼ 1, l2 ¼ l þ 1; l1 ¼ l  1; m2 ¼ m1; n2 ¼ n1 and if k ¼ 2, then m2 ¼ m þ 1; m1 ¼ m  1; l2 ¼ l1; n2 ¼ n1 and so on. The components of the coordinate transformation can be computed using chain rule and Cramer’s rule, for instance,

  @f1 @x2 @x3 @x3 @x2 ¼ J 1  @x1 @f2 @f3 @f2 @f3

ð49Þ

where

J¼

      @x1 @x2 @x3 @x3 @x2 @x2 @x1 @x3 @x3 @x1 @x3 @x1 @x2 @x2 @x1     þ @f1 @f2 @f3 @f2 @f3 @f1 @f2 @f3 @f2 @f3 @f1 @f2 @f3 @f2 @f3

ð50Þ

Again, we use the central difference method to approximate the partial derivatives of xi with respect to fi as

 xil1;m1;n1 @xi xl2;m2;n2  i @fj 2Dfj

ð51Þ

At the outermost boundary of the domain, @ XD and the surface of the sphere, @ XS , the upwind scheme is employed as

 ul;m2;n  ul;m1;n @ui  @f2 i  i ;  @xj f2 Df2 @xj  xl;m2;n  xl;m1;n @xi  i  i  @fj f2 Df2

ð52Þ ð53Þ

7

6 λ=2

Force

5

λ=1.5

4

3

λ=1

2

λ=0.5 λ=0

1

0 -2

-1

0

1

m Fig. 6. Force versus m for varying k for a sphere moving in a power-law fluid.

2

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where m2 ¼ m þ 1 and m1 ¼ m at @ XS and m2 ¼ m and m1 ¼ m  1 at @ XD , respectively. Based on Eqs. (48)–(53), we can write Eq. (27), for example, in the discretized form

F0;PL ðus Þ 

X 1 q 1 þ kjA1;l;m;n : A1;l;m;n j A1;l;m;n : A1;l;m;n DV l;m;n 2 l;m;n

ð54Þ

All other force equations treated in this paper are discretized similarly.

30 ColdParticle

HotParticle

25

20

Force

15

10

5

0

-5 -6

-4

-2

0

β

2

4

6

Fig. 7. Force versus b for a falling sphere in a viscous fluid. The case b < 0 represents a cold particle in fluid and b > 0 corresponds to a hot particle.

3.5

3

β=1

λ=2

Force

2.5

2 λ=1.5

1.5

λ=1 λ=0.5

1

λ=0

0.5

0 -2

-1

0

1

m Fig. 8. Force versus m and k with b ¼ 1:0 for heated sphere in a power-law fluid.

2

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18 16

β=−1

14 λ=2

12

Force

10 8

λ=1.5

6

λ=1

4

λ=0.5

2

λ=0

0 -2 -2

-1

0

1

2

m Fig. 9. Force versus m and k with b ¼ 1:0 for cold sphere in a power-law fluid.

5.2. Mesh refinement and domain size The well known Stokes formula for the value of the drag coefficient for the flow past a sphere is 6p for a ¼ 1 and n1 ¼ 1. To determine a computational domain size and appropriate mesh densities, we performed a mesh refinement study with various domain sizes for the Stokes flow past a sphere, in which the drag coefficient, g ¼ F0;V can be alternatively computed by

g¼

1 2

Z

ð1Þ

ð1Þ

Aðh Þ : Aðh Þ

ð55Þ

X

By comparing the Stokes’ and numerical solution of G, we hope to identify an appropriate domain size and mesh density for ðiÞ our computations. The vector field h in Eq. (55) is the Stokes’ auxiliary field past a sphere, which is provided in Eq. (62). A number of test cases were carried out with I; J; K representing the maximum value of the indices for the directions in the fi coordinates which can be represented by the formula

ðI; J; KÞ ¼ ð12 þ 10n; 41 þ 10n; 11 þ 10nÞ;

n ¼ 0; 1; 2; 3; 4

ð56Þ

All possible combinations, 15 in total, of these values were considered for the mesh density tests. The values, g obtained for the different cases are also evaluated for varying domain sizes, Ds ¼ 10; 20; 30; 40; 50; 60. Note that Ds refers to the ratio of the actual radius of the domain to the radius of the sphere. Fig. 5 illustrates that the value g converges well as I; J; K increase and as Ds increases. The error between the Stokes and numerical g was within 2% allowing us to choose I ¼ 42; J ¼ 71; K ¼ 41 and Ds ¼ 60 for the present study. Note that the geometric ratio (Gr ¼ 1:1) in J-direction was employed to construct a much finer mesh in the region near the surface of sphere where rapid changing velocity gradients are present as shown in Fig. 4. 6. Results and discussion The results of our computations are displayed in Fig. 6. In the following graphs, we plot the relative force, i.e. F 1 ¼ Fðus Þ=6p4 with n1 ¼ 1 (without loss of generality) versus the parameters m; ; k and a. The first of these plots, Fig. 6 shows the variation of F 1 versus m for the cases of shear-thickening and shear-thinning fluids while also varying k. For any fixed k, the coefficient increases monotonically and rapidly with increasing m. As is to be expected, when m < 0, F 1 < 1 and for m > 0, force increases beyond 1 much more rapidly than in the case of a shear thinning fluid. The variation of k has the result of amplifying the effects of shear-thickening or thinning in a linear manner. It is therefore best to interpret k as a concentration parameter, which accounts for the concentration of the actual polymeric molecules in the background solute. When k ¼ 0 we must therefore revert to the case of a constant viscosity solution, as is observed in our calculations.

4

In Figs. 6–8, 910, 11, 16 and 17 we refer to F 1 simply as ‘Force’.

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In Fig. 7, we plot the variation of F 1 with b for the cases of hot and cold particles corresponding to b > 0 and b < 0, respectively. When b ¼ 0 the particle is in thermal equilibrium with the surroundings. The effect of increasing b results in monotonically decreasing F 1 . We observe that limb!þ1 F 1 ¼ 0 and limb!0 F 1 ¼ 1. In the case when the particle is colder than the surroundings, it experiences a very large drag as seen by the very steep gradient of F 1 with respect to b which reduces considerably when the particle becomes hotter than the surrounding fluid. Figs. 8 and 9 show the results of the combined case of particle heating and power law nature of the fluid. Figs. 8 and 9 are much like Fig. 6 and show the variation of F 1 with m for the cases of hot and cold particles, b ¼ 1:0 and b ¼ 1:0, respectively, but for varying k. In comparison with Fig. 6, we observe essentially the same profiles but with the values of F 1 slightly diminished in the case of the hot particle and much higher for the cold particle. In the special case, k ¼ 2 and m ¼ 1, for instance in Fig. 8, F 1 ¼ 1:25 whereas, when b ¼ 0, F 1 ¼ 2:25 (approximately) and for b ¼ 1:0, F 1 ¼ 4 (approximately) for the same values of k and m. The values of the force can therefore increase over 300% with reduction in temperature of the particle compared to the surroundings. This seems to indicate that particle heating or cooling can be a significant source of drag reduction and hence a useful control parameter in sedimentation problems. It must be noted that our analysis is based after all on certain important assumptions (see Section 3) and hence variations of b far from zero needs to be considered with care. However, we argue that the fact that we have managed to provide a simple and coherent analysis of drag in rather complex nonlinear situations merits attention, at least qualitatively. Figs. 10 and 11 denote the F 1 versus parameter m for fluids where viscosity varies with pressure. Fig. 10 shows the variation of F 1 with m for changing values of 1 6 a 6 5 and with fixed k ¼ 0:1. As a increases, the F 1 also increases rapidly since increasing a has essentially the effect of increasing viscosity. Fig. 11 is drawn for fixed a ¼ 1:0 and shows a similar variation with m as in Figs. 6, 8 and 9. The values are now magnified a little more than in the earlier cases. In Figs. 12–17, we discuss the impact of the moving sphere on the fluid. Figs. 12–15, in particular show the local variation of stress density in various cases using a contour plot (2:75 < x < 2:75, 2:125 < z < 2:125). In these plots, the reference frame is oriented in the same manner as shown in Fig. 1. By stress density (denoted r henceforth), we refer to the integrand in the Eqs. (27), (36), (40), (47), corresponding to Figs. 12–15, respectively. Note that we have chosen specific values of parameters in these plots (m ¼ 0:5; a ¼ 1:0; k ¼ 0:5 and b ¼ 1:0) for different situations. The results are, however, qualitatively the same for other values of these parameters as well. The value of r is seen to be a maximum along the top and bottom portions of the sphere for Figs. 12–14. Interestingly, in Fig. 15 the maximum value of r appears shifted to a different point on the surface of the sphere, along the x direction. The minimum of r occurs on the stagnation points of the sphere in all of the cases. The variation of r in the fluid domain is different for the different cases, however, it is dramatically different for the pressure-dependent viscosity case where r is nearly zero everywhere to the left of the sphere. In Figs. 16 and 17, we look at the decay rate of r along the maximum direction (without loss of generality) for the cases of shear thinning (m ¼ 0:5) and shear-thickening(m ¼ 0:5) liquids, respectively, for a ¼ 1:0; k ¼ 0:5 and b ¼ 1:0. In this case we plot the hot particle case alone. As also seen in the contour plots, the pressure-dependent viscosity model has the maximum impact (in terms of force and r) upon the sphere and also on the fluid; its value on the sphere surface is a maximum,

8 7

α=5

6

Force

5 4 3 α=3

2

α=1

1 0 -3

-2

-1

0

1

2

3

m Fig. 10. Force versus m for different a and k ¼ 0:1 for a moving sphere in a power-law fluid with pressure-dependent viscosity.

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8

α=1 6

Force

λ=2

4

λ=1.5 λ=1 λ=0.5

2

λ=0

0 -2

-1

0

1

2

m Fig. 11. Force versus m and k for a ¼ 0:1 for a moving sphere in a fluid with pressure-dependent viscosity.

Fig. 12. Variation of stresses due to a moving sphere in a shear-thinning power-law fluid with m ¼ 0:5 and k ¼ 0:5. The values of the contour lines indicate the magnitude of the stress density, r.

when compared to the other cases and its decay rate is the slowest implying that its domain of influence in the fluid is the largest. We can also see from Fig. 16 that r, for pure hot particle is larger than when it is immersed in a shear-thinning fluid and its domain of influence in the fluid is also larger. When the fluid is shear-thickening, Fig. 17 indicates that r for the hot

B.J. Chung, A. Vaidya / International Journal of Engineering Science 48 (2010) 78–100

93

Fig. 13. Variation of stresses due to a moving, heated sphere in a viscous fluid with b ¼ 1:0. The values of the contour lines indicate the magnitude of the stress density, r.

Fig. 14. Variation of stresses due to a heated sphere in a power-law fluid with m ¼ 0:5, k ¼ 0:5 and b ¼ 1:0. The values of the contour lines indicate the magnitude of the stress density, r.

particle with the power-law fluid dominates over the hot particle in a viscous fluid. Another relevant information that we note in Figs. 16 and 17 is that in each of the cases we have studied r becomes vanishingly small at about 3:5a, where a

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Fig. 15. Variation of stresses for a moving sphere in a power-law fluid with pressure-dependent viscosity with m ¼ 0:5, k ¼ 0:5 and a ¼ 1:0. The values of the contour lines indicate the magnitude of the stress density, r.

3

PL HP P+PL HP+PL

2.5

Stress Density

2

β=1.0 λ=0.5 m=-0.5

1.5

1

0.5

0 1

1.5

2

2.5

3

3.5

r Fig. 16. Comparative radial decay of stresses for various models with m ¼ 0:5, a ¼ 1:0 and k ¼ 0:5. The direction of r is chosen to coincide with the direction of the maximum stresses induced in the fluid.

is the radius of the sphere. In the case of the pressure-dependent viscosity model, we also observe that the position of maximum stress density strongly depends upon a. In Fig. 18, we plot the angle, h where the maximum r occurs versus a. The angle h is defined in the clockwise direction from the negative-x to the z axis. It is seen that h decreases exponentially with increasing a.

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6 5.5 PL HP P+PL HP+PL

5 4.5

Stress Density

4

β = 1.0 λ = 0.5 m = 0.5

3.5 3 2.5 2 1.5 1 0.5 0 1

1.5

2

r

2.5

3

3.5

Fig. 17. Comparative radial decay of stresses for various models with m ¼ 0:5, a ¼ 1:0 and k ¼ 0:5. The direction of r is chosen to coincide with the direction of the maximum stresses induced in the fluid.

λ=0.5 m=-0.5

θo

80

60

40

20 0

2

4

α

6

8

10

Fig. 18. Angle where stress density is a local maximum versus the parameter a for a shear-thining fluid with pressure-dependent viscosity with m ¼ 0:5 and k ¼ 0:5.

7. Conclusions In summary, this paper discusses the effect of drag on a spherical particle due to combined effects of power-law viscosity, surface heating (or cooling) and also pressure-dependent viscosity. The force is written down as contributions of the viscous and inertial terms for bodies of arbitrary shape and is then evaluated for slow flows by taking the body to be a sphere and the fluid velocity to be the creeping flow velocity. This approach permits us to tackle highly nonlinear problems which are not easily achieved by standard perturbation techniques and compared to fully numerical approaches, the above technique is computationally much less expensive. Our paper also advances beyond looking at simple power-law fluids and also dis-

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cusses the complex effects of particle heating and pressure-dependent viscosity effects whose study has recently been revived. In fact, a novelty of this paper is the combined effect of these factors. The expression for force derived in this paper allows us to simultaneously look at the qualitative and quantitative effect of the stresses generated locally in the fluid which, as far as we know, has not been studied in the previously literature. Our Figs. 16 and 17 summarize this part of our work by showing the decay of the stress density radially outwards from the sphere for the various cases, for both shear-thinning and shear-thickening fluids. Among the interesting observations that we make is the effect of particle heating or cooling. We notice that controlling the surface temperature of the body can effect in reducing or increasing the effective drag experienced by the particle. Though this problem is formulated for an unbounded domain, our results can be applied to estimate distances at which wall effects become prominent. Since the stresses generated in the fluid locally seem to almost vanish at about four times the radius of the sphere (see Figs. 16 and 17), the effect of the motion of a sphere on a possible wall placed at this distance will be negligible. So quantitatively our calculations indicate a critical distance beyond which wall effects can be ignored. Ambari et al. [2] have experimentally studied the effect of walls on a translating sphere in a viscous fluid at Reynolds numbers of order 103 . They observe that wall effects become small when the distance between the sphere surface and the wall becomes more than the radius of the sphere. Their experiment shows that in the vicinity of the wall, the sphere begins to rotate as well and so their estimation of force is higher than that obtained by O’Neill’s [43,44] exact solution in absence of rotation for a viscous fluid. Our data indicates the critical distance of vanishing wall effect even in the case of power-law fluids is quite close to that of O’Neill. The primary limitation of this study is the potential difficulty of finding an analytical flow field around the body and hence the assumption of the Stokes flow for each case which results in the vanishing of the inertial and viscoelastic contributions of the force. Some obvious extensions of our work, that we are currently looking into, include bodies of different shapes such as oblate and prolate spheroids. The effect of rotation on the drag in these complex fluids is also an interesting question. We also intend to conduct experiments on bodies with heated surfaces to verify our theory. The difficulty in doing rigorous analysis of these highly nonlinear problems points us instead to possible numerical resolution in cases of non-zero Re. Acknowledgement The author, A.V., wishes to thank Dr. Rama Govindarajan, JNCASR, Bangalore for her invitation and hospitality during the summer of 2006 where this project was started and for her very valuable comments. The authors would like to thank Prof. Rajagopal and the referee for their helpful comments to improve the paper.

Appendix A. Expression for force Consider the non-dimensional version of the linear momentum equation in indicial form is given by:

Re ui @ i uj ¼ @ i T ij

ð57Þ

where T ij is the stress tensor. If we now multiply this equation with the auxiliary field main, we have

Re

Z X

ðkÞ

ui @ i uj hj ¼ ðkÞ

where we define ej

Z X

ðkÞ

@ i T ij hj ¼

Z

ðkÞ

@X

ni T ij hj 

Z X

X

ðkÞ

T ij @ i hj ¼

¼ dkj ek , for j ¼ 1; 2; 3 and Dij ðuÞ ¼

cous and pressure components. Using the fact

Re

Z

ðkÞ

ui @ i uj hj ¼

Z

ni T ij ej  2

@X

Z X

ðkÞ @ j hj

@ i uj þ@ j ui . 2

Z @X

ðkÞ

ni T ij ej 

Z X

ðkÞ hj

and integrate over the fluid do-

ðkÞ

ðpdij þ 2lðxm ÞDij ðuÞÞ@ i hj

ð58Þ

Also, note that we have expanded the stress tensor into its vis-

¼ 0, the Eq. (58) can be written as

lðxm ÞDij ðuÞ@ i hðkÞ j :

ð59Þ

In Eq. (59), we note that the first term on the right hand side is nothing but the force imposed by the fluid upon the body. The second term on the right hand side can be further simplified by invoking the property of symmetric tensors whereby

Re

Z X

u  ru  h

ðkÞ

¼ F

Z

lðxm ÞDðuÞ : DðhðkÞ Þ:

ð60Þ

X

Appendix B. Symmetry of creeping flow The central argument in this paper lies in evaluating the net forces imposed upon a sedimenting body by the surrounding fluid. Since the fluid models under consideration in this paper contain several nonlinear terms, it is best to first simplify the

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expression for torque before trying to manipulate it. For this purpose, it is perhaps best to make use of the symmetries of the sedimenting body, B and those of the Stokes flow around the body. This motivates the definitions below. Definition 1 (Rotational symmetry). We say that a body B has rotational symmetry about an axis, say x1 , if and only if:

ðx1 ; x2 ; x3 Þ 2 R ) ðx1 ; x2 ; x3 Þ; ðx1 ; x2 ; x3 Þ 2 R Definition 2 (Symmetry operators). We define certain new symmetry classes next. We define the operators Pi , i ¼ 1; 2; 3 such that:

P1 f ðx1 ; x2 ; x3 Þ :¼ f ðx1 ; x2 ; x3 Þ P2 f ðx1 ; x2 ; x3 Þ :¼ f ðx1 ; x2 ; x3 Þ P3 f ðx1 ; x2 ; x3 Þ :¼ f ðx1 ; x2 ; x3 Þ P4 f ðx1 ; x2 ; x3 Þ :¼ f ðx1 ; x2 ; x3 Þ Definition 3 (Symmetry class for scalar functions). Suppose / ¼ /ðx1 ; x2 ; x3 Þ is a scalar field. Then, we define the following symmetry classes:

Cs1 :¼ f/ : P4 / ¼ /; P3 / ¼ /g Cs2 :¼ f/ : P4 / ¼ /; P3 / ¼ /g Cs3 :¼ f/ : P4 / ¼ /; P3 / ¼ /g Cs4 :¼ f/ : P1 / ¼ /; P2 / ¼ /; P3 / ¼ /g Cs5 :¼ f/ : P1 / ¼ /; P2 / ¼ /; P3 / ¼ /g Cs6 :¼ f/ : P1 / ¼ /; P2 / ¼ /; P3 / ¼ /g  ¼ fCs : k ¼ 4; 5; 6g. We define the set C k Remark 1. Note that in particular, if we consider two scalar functions, /1 and /2 , such that /1 2 Cs1 and /2 2 Csm , where m ¼ 2; 3, then, it follows that:

Z

/1 /2 ¼ 0

X

 then where X is the fluid domain exterior to the symmetric body. Furthermore, if /3 2 C,

Z

/3 ¼ 0

X

Definition 4 (Symmetry class for vector fields). Suppose w ¼ ðw1 ; w2 ; w3 Þ is a vector field, then we define the following classes:

Cv1 :¼ fw : w1 ¼ P1 w1 ¼ P2 w1 ¼ P3 w1 ; w2 ¼ P1 w2 ¼ P2 w2 ¼ P3 w2 ; w3 ¼ P1 w3 ¼ P2 w3 ¼ P3 w3 g Cv2 :¼ fw : w1 ¼ P1 w1 ¼ P2 w1 ¼ P3 w1 ; w2 ¼ P1 w2 ¼ P2 w2 ¼ P3 w2 ; w3 ¼ P1 w3 ¼ P2 w3 ¼ P3 w3 g

ðiÞ

We introduce a set of translational auxiliary fields (h ; pðiÞ Þ in order to decouple the motion of the fluid from that of the body [19]. These fields represent the translational Stokes velocity fields of the flow along the ei direction and satisfy the following equations: ðiÞ

Dh ¼ rpðiÞ ðiÞ

div h

¼0

ðiÞ

lim h ðxÞ ¼ 0

jxj!1 ðiÞ

h ðyÞ ¼ ei ;

y2R

ð61Þ

for i ¼ 1; 2; 3 and can be determined simply from the geometric properties of B. Remark 2. In fact, the auxiliary fields are known for certain simple geometries such as spheres, prolate and oblate spheroids [11]. The auxiliary field for a sphere of radius ‘a’, along the e1 axis is given by (see Fig. 1),

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h

ð1Þ

pð1Þ

   a a2 3ax a2 ¼ 1 3þ 2 e1 þ 2 ð 2  1Þer 4R R 4R R 3ax ¼ 3 2R

ð62Þ ð63Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 where R ¼ x2 þ y2 þ z2 and er ¼ ðyey þ zez Þ=ðy2 þ z2 Þ2 . ðiÞ

Remark 3. Having defined the auxiliary fields, we now state some essential properties of these fields. The field h some interesting symmetry properties, namely

h

ð1Þ

2 Cv1 ;

h

ð2Þ

2 Cv2

possess

ð64Þ

A rigorous proof of this result can be found in [19,18]. Remark 4. Using the symmetries of the Stokes flow field, introduced in Section 2, we may analyze Eq. (20) in order to simplify it. We first evaluate the inertial component of the force. At u ¼ us , the complete form of this force is as follows:

F0;I i ¼

Z

 ð1Þ ð1Þ ðiÞ n1 h Þ  gradðn1 h h

ð65Þ

X

where i ¼ 1; 2; 3. If we refer to the integrand in the Eq. (65) as ðf I Þi , then it is easily verified upon using the Definition 3 that

 ðf I Þi 2 C and therefore using the Remark 1, we have that Fi0;I ¼ 0 for i ¼ 1; 2; 3. Let us now consider the case of a Second order viscoelastic fluid whose constitutive equation is given by

T ¼ pI þ lA1 þ a1 A2 þ a2 A21

ð66Þ

where p is the indeterminate part of the stress tensor due to the constraint of incompressibility, l is the coefficient of viscosity and a1 and a2 are material moduli which are usually referred to as the normal stress coefficients. The kinematical tensors A1 and A2 are defined through

A1 ¼ L þ LT

ð67Þ

d A2 ¼ A1 þ A1 L þ LT A1 dt L ¼ grad u

ð68Þ ð69Þ

The contribution to force due to the normal stress part of the stress is given in non-dimensional form by

FV ¼

1 2

Z

S2 : A1

ð70Þ

X

where S2 ¼ A2 þ A21 and

 ¼ aa . 2 1

Remark 5. If we evaluate the viscoelastic contribution to the force at u ¼ us , we have

F0;V ¼

1 2

Z X

ðA2 ðus Þ þ A21 ðus ÞÞ : A1 ðus Þ

ð71Þ

A simple calculation, similar to that in Remark 4, shows that the integrand in Eq. (71), denoted fiV , has the following symmetry:

fiV 2 Cs2 and it therefore follows from Remark 4 that F0;V ¼ 0. Therefore, the second order fluid model has no viscoelastic contribution to the net force on a sedimenting sphere at the creeping flow approximation, based on our calculations. An interesting observation emerges if we extend our calculation to the case of a third order viscoelastic fluid. The constitutive equation for a fluid of third order is given by [55]

T ¼ pI þ lA1 þ S2 þ S3

ð72Þ

where S2 is given above and

S3 ¼ b1 A3 þ b2 ðA2 A1 þ A1 A2 Þ þ b3 ðtrA2 ÞA1

ð73Þ

and A3 is given by

A3 ¼

d A2 þ A2 L þ LT A2 : dt

ð74Þ

B.J. Chung, A. Vaidya / International Journal of Engineering Science 48 (2010) 78–100

99

Fosdick and Rajagopal [16] showed that for an incompressible thermodynamically compatible fluid of grade three, this model reduces to:

T ¼ pI þ ½l þ b trA21 A1 þ a1 A2 þ a2 A21

ð75Þ

where

l>0 a1 P 0 pffiffiffiffiffiffiffiffiffiffiffiffi ja1 þ a2 j 6 24lb bP0 it can be seen this can also be considered as a generalization of the standard second grade fluid model with an effective viscosity, leff , given by

leff ¼ l þ b trA21

ð76Þ

We make two immediate inferences now: (a) the viscoelastic part of this model based on Remark 5, has no contribution to the force on a sphere within our approximation. However, the viscous part of this model is certainly non-zero. (b) The effective viscosity, leff is seen to be a specific case (set m ¼ 1 and b ¼ j in Eq. (10)) of the general viscosity function considered in this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

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