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Some simple unsteady unidirectional flows of a binary mixture of incompressible Newtonian fluids
International Journal of Engineering Science 40 (2002) 2023–2040 www.elsevier.com/locate/ijengsci
Some simple unsteady unidirectional flows of a binary mixture of incompressible Newtonian fluids Serdar Barısß
*
Department of Mechanical Engineering, Faculty of Engineering, Istanbul University, Avcılar 34850, Istanbul, Turkey Received 23 January 2002; accepted 12 June 2002
Abstract The problems dealing with some simple unsteady unidirectional flows of a mixture of two incompressible Newtonian fluids are investigated. By using the constitutive equations appeared in the literature for binary mixtures of chemically inert incompressible Newtonian fluids, the equations governing the motion of the binary mixture are reduced to a system of coupled partial differential equations. By means of integral transforms, the exact solutions of these equations are obtained for the following three problems: (i) unsteady Couette flow, (ii) unsteady plane Poiseuille flow, (iii) unsteady axisymmetric Poiseuille flow. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Mixture; Unsteady flow; Fourier sine transform; Hankel transform
1. Introduction The study of a mixture of fluids is always important in view of its applications in various branches of engineering and technology. A familiar example is an emulsion which is the dispersion of one fluid within another fluid. Typical emulsions are oil dispersed within water or water within oil. Another example where the mixture of fluids plays an important role is in multigrade oils. Polymeric type fluids are added to the base oil so as to enhance the lubrication properties of mineral oils [1]. Truesdell [2] was the first to formulate the thermomechanical balance equations for a mixture of general materials. After his pioneering work, a good amount of literature has been generated on the formulation of continuum thermomechanical theories of mixtures. We refer the reader to *
Fax: +90-212-245-0795. E-mail address: [email protected] (S. Barısß).
0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 2 ) 0 0 1 0 9 - X
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S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
the works of Bowen [3], Atkin and Craine [4], Bedford and Drumheller [5], and Rajagopal and Tao [6] regarding the historical development of the theory and detailed analysis of various results on this subject. Some exact solutions for the flow of a binary mixture of incompressible Newtonian fluids were presented by Mills [7], Craine [8], Atkin and Craine [9], Beevers and Craine [10] €ßs [11–16]. Recently, several problems relating to the mechanics of oil and water and G€ o gu emulsions, particularly with regard to applications in lubrication practice, have been considered within the context of the mixture theory by Al-Sharif et al. [17], Chamniprasart et al. [18], and Wang et al. [19]. In the present paper a binary mixture, each constituent of which is a chemically inert incompressible Newtonian fluid, is considered. The balance laws and relevant constitutive equations are presented in Section 2. In the subsequent sections we obtain the exact solutions for some simple unsteady unidirectional flows of the binary mixture under consideration.
2. Basic theory A brief summary of the basic balance laws and the appropriate constitutive theory for a binary mixture of incompressible Newtonian fluids will be given here, for more details the reader should consult Craine [8] and Atkin and Craine [9]. We consider a mixture of two interacting constituents, each of which is regarded as a continuum; we refer to the bth constituent as the continuum sðbÞ . Throughout this paper b takes the values 1 and 2. We assume that each point x within the mixture is occupied simultaneously by one particle from each sðbÞ . If vðbÞ denotes the velocity vector of the bth constituent, the material time derivative DðbÞ =Dt is defined by DðbÞ o ðbÞ o ; ¼ þ vi ot oxi Dt
ð1Þ
where the usual summation convention is applied to repeated Roman subscripts. The mean velocity of the mixture w and the total mass density of the mixture q are given respectively by ð1Þ
ð2Þ
qwk ¼ q1 vk þ q2 vk ;
ð2Þ
q ¼ q1 þ q2 ;
ð3Þ
where qb is the density of sðbÞ at time t, measured per unit volume of mixture. The basic equations for a binary mixture in which the constituents have a common temperature T and do not interact chemically are the following ones: continuity equations: oqb ðbÞ þ ðqb vi Þ;i ¼ 0; ot
ð4Þ
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
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equations of motion: ð1Þ
ð1Þ
ð1Þ
rik;i fk þ q1 Fk
¼ q1
Dð1Þ vk ; Dt
ð2Þ
ð2Þ
ð2Þ
rik;i þ fk þ q2 Fk
¼ q2
Dð2Þ vk ; Dt
ð5Þ
energy equation: qr qi;i q1
Dð1Þ U1 Dð2Þ U2 ð1Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ q2 þ rij vj;i þ rij vj;i þ fj ðvj vj Þ ¼ 0; Dt Dt
ð6Þ
where a comma denotes differentiation with respect to xi . The quantities Ub , rðbÞ and FðbÞ are in turn internal energy per unit mass, partial stress and body force per unit mass of the bth constituent. In addition r the heat supply per unit mass, q the heat flux vector, refer to the mixture as a whole, and f denotes the diffusive force vector. We now turn our attention to the interaction term f. For viscous fluids, in general, this interaction term depends on the relative velocity, the density gradients, the temperature gradients, and possibly other quantities [18]. Such interactions play a very important role in the nature of the solutions (see, for instance [20,21]). In this study we shall assume that the interaction force incorporates only the effect of drag. Consideration of the balance of angular momentum for sðbÞ shows that rðbÞ need not be symmetric although the balance of angular momentum for the mixture results in the symmetry of r, the total stress in the mixture, defined by ð1Þ
ð2Þ
rij ¼ rij þ rij :
ð7Þ
Admissible thermomechanical processes in the mixture must be compatible with an entropy production inequality. If Sb is the partial entropy per unit mass of the bth constituent, then the Clausius–Duhem inequality may be written as follows: Dð1Þ S1 Dð2Þ S2 qr qi q1 þ q2 þ P 0: T Dt Dt T ;i
ð8Þ
In this work we shall concern ourselves with a mixture of two incompressible Newtonian fluids. Let the density of sðbÞ in its reference configuration be qb0 , which in view of the assumed incompressibility is constant. Introducing the quantity /b , is the volume fraction of the bth fluid, and assuming that the mixture does not contain voids, it follows that q1 ¼ /1 q10 ;
q2 ¼ ð1 /1 Þq20
ð9Þ
and hence q1 q þ 2 ¼ 1: q10 q20
ð10Þ
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Using (3) and (10), it can be easily shown that q1 ¼
q10 ðq20 qÞ ; q20 q10
q2 ¼
q20 ðq q10 Þ : q20 q10
ð11Þ
Substituting Eq. (11) into Eq. (4) and eliminating oq=ot between them gives the relation ð1Þ
ð2Þ
ðq20 qÞdii þ ðq q10 Þdii ni ai ¼ 0;
ð12Þ
where ðbÞ
ðbÞ
ðbÞ
2dij ¼ vi;j þ vj;i ;
ð1Þ
ni ¼ q;i ;
ð2Þ
ai ¼ vi vi :
ð13Þ
The derivation of the constitutive equations appropriate to our binary mixture of incompressible Newtonian fluids has been outlined in [9]. If the mixture is considered to be purely mechanical system; that is, thermal effects are ignored, the relevant equations are Ab ¼ Ab ðqÞ;
A ¼ AðqÞ; dA1 p1 ¼ ðq q20 Þ q1 þk ; dq
ð14Þ
dA2 p2 ¼ ðq q10 Þ q2 k ; dq
fk ¼ aak knk ;
ð15Þ ð16Þ
ð1Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð17Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð18Þ
rij ¼ ðp1 þ k1 dkk þ k3 dkk Þdij þ 2l1 dij þ 2l3 dij þ k5 Cij ; rij ¼ ðp2 þ k4 dkk þ k2 dkk Þdij þ 2l4 dij þ 2l2 dij k5 Cij ;
where Ab denotes the partial Helmholtz free energy of the bth constituent, and the Helmholtz free energy of the mixture A (total free energy) is defined by qA ¼ q1 A1 þ q2 A2
ð19Þ
and the coefficients a, k1 ; . . . ; k5 , l1 ; . . . ; l4 are functions of q and satisfy the inequalities a P 0;
k5 P 0;
l1 P 0;
ðl3 þ l4 Þ2 6 4l1 l2 ;
l2 P 0;
k1 þ 23l1 P 0;
k2 þ 23l2 P 0;
2 k3 þ k4 þ 23ðl3 þ l4 Þ 6 4 k1 þ 23l1 k2 þ 23l2 :
ð20Þ
The quantity k is a Lagrange multiplier associated with the constraint (12) and the relative spin C is given by ð1Þ
ð1Þ
ð2Þ
ð2Þ
2Cij ¼ ðvi;j vj;i Þ ðvi;j vj;i Þ:
ð21Þ
Note that as there is only one constraint, that of volume additivity, namely Eq. (10), there can be only one Lagrange multiplier and thus p1 and p2 are not independent. Both of them can be expressed in terms of a single Lagrange multiplier.
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Finally, neglecting the body forces, we shall derive the equations governing the flow of a mixture of two incompressible Newtonian fluids. For this purpose, inserting f, rð1Þ , and rð2Þ from Eqs. (16)–(18) into Eq. (5), with the help of Eqs. (13) and (21), one gets the following equations of motion: M1 Dvð1Þ þ M2 Dvð2Þ þ M5 rðr vð1Þ Þ þ M6 rðr vð2Þ Þ þ ðr vð1Þ Þrk1 þ ðr vð2Þ Þrk3 þ ðrvð1Þ ÞT rM1 þ ðrvð2Þ ÞT rM2 þ rvð1Þ rM9 þ rvð2Þ rM10 aðvð1Þ vð2Þ Þ ¼ q1
Dð1Þ vð1Þ þ rp1 krq; Dt
ð22Þ
M3 Dvð1Þ þ M4 Dvð2Þ þ M7 rðr vð1Þ Þ þ M8 rðr vð2Þ Þ þ ðr vð1Þ Þrk4 þ ðr vð2Þ Þrk2 T
T
þ ðrvð1Þ Þ rM3 þ ðrvð2Þ Þ rM4 þ rvð1Þ rM11 þ rvð2Þ rM12 þ aðvð1Þ vð2Þ Þ ¼ q2
Dð2Þ vð2Þ þ rp2 þ krq; Dt
ð23Þ
where M1 ¼ l1 þ
k5 ; 2
M5 ¼ k1 þ l1 M9 ¼ l1
k5 ; 2
M2 ¼ l3 k5 ; 2
k5 ; 2
M3 ¼ l4
M6 ¼ k3 þ l3 þ
M10 ¼ l3 þ
k5 ; 2
k5 ; 2
k5 ; 2
M4 ¼ l2 þ
M7 ¼ k4 þ l4 þ
M11 ¼ l4 þ
k5 ; 2
k5 ; 2
k5 ; 2
M12 ¼ l2
M8 ¼ k2 þ l2
k5 ; 2
ð24Þ
k5 : 2
In the above equations r and D are the gradient and Laplacian operators, respectively. Also, ðbÞ rvðbÞ is the second-order tensor field and its ijth component is taken as vj;i . Note that, under isothermal conditions, the coefficients M1 etc. appearing in Eqs. (22) and (23) depend only on the total density q, and hence spatial coordinates and time. In the subsequent sections, we shall obtain the exact solutions of the above equations for some simple unsteady unidirectional flows of a binary mixture of incompressible Newtonian fluids.
3. Unsteady Couette flow We shall begin with the case of the flow of a binary mixture of incompressible Newtonian fluids between two infinite parallel plates separated by a distance H. The mixture and the two plates are initially at rest. The lower plate is suddenly accelerated from rest and moves in its own plane with a constant velocity U, while the upper plate is held stationary. It is assumed that the flow is entirely driven by the motion of the lower plate, the pressures far upstream and far downstream being kept equal throughout the motion. Thus the pressure gradients in the x-direction are zero. Cartesian coordinates ðx; y; zÞ are used. Let the y-axis be directed normally to the plates, with y ¼ 0 on the lower plate, the x-axis being in the direction of motion of the lower plate.
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We seek solutions in which the velocity vector of the bth fluid and total density are assumed to have the form vðbÞ ¼ fub ðy; tÞ; 0; 0g;
q ¼ qðy; tÞ;
q1 ¼ q1 ðy; tÞ;
q2 ¼ q2 ðy; tÞ:
ð25Þ
Substitution of Eq. (25) into Eq. (4) gives oq1 oq2 ¼ ¼ 0; ot ot
ð26Þ
thus oq=ot ¼ 0 and Eq. (25)2 simplifies to q ¼ qðyÞ. Substituting Eq. (25) into the y-components of the equations of motion (22) and (23), we get k
dq op1 ¼ ; dy oy
k
dq op2 ¼ : dy oy
ð27Þ
With the use of Eqs. (11), (15) and (19), elimination of ok=oy between Eq. (27)1 and (27)2 gives, after some manipulation, ðq q10 Þðq20 qÞ
dq d2 ðqAÞ ¼0 dy dq2
ð28Þ
and since, in general, q 6¼ q10 , q 6¼ q20 and d2 ðqAÞ=dq2 6¼ 0 we deduce that q ¼ q0 is a constant. As a result, the coefficients M1 etc. in Eqs. (22) and (23) are constants. In the light of these arguments, the x-components of the equations of motions (22) and (23) reduce to M1
o2 u1 o2 u2 ou1 þ M aðu1 u2 Þ ¼ q1 ; 2 2 2 oy oy ot
ð29Þ
M3
o2 u1 o2 u2 ou2 þ M þ aðu1 u2 Þ ¼ q2 : 4 2 2 oy oy ot
ð30Þ
It is convenient at this point to introduce dimensionless variables and material constants. If f is used to denote the dimensionless form of a quantity f, it follows that Mi ¼
Mi ; l
qb ¼
qb ; q0
a ¼
aH 2 ; l
t ¼
lt ; q0 H 2
ub ¼
ub ; U
y ¼
y ; H
ð31Þ
where l is a typical viscosity coefficient. The dimensionless governing equations are obtained from Eqs. (29) and (30) by replacing variables and material constants by those given in Eq. (31), so they are not rewritten here. The boundary and initial conditions are ub ð0; tÞ ¼ 1
for t > 0;
ub ðy ; 0Þ ¼ 0 for 0 < y 6 1:
ub ð1; tÞ ¼ 0
for t P 0;
ð32Þ ð33Þ
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
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Throughout this paper, henceforth for convenience, unless stated otherwise, we shall drop the bars that appear over the dimensionless quantities. We first have to transform the problem so that the boundary conditions (32)1 are homogeneous. This can be achieved by decomposing ub ðy; tÞ into the steady plane Couette velocity profile ubS ðyÞ, which are expected to prevail at large times, and the transient component fb ðy; tÞ: ub ðy; tÞ ¼ ubS ðyÞ fb ðy; tÞ:
ð34Þ
The steady-state velocity distribution of the bth fluid is given by (cf. [9]) ubS ðyÞ ¼ 1 y:
ð35Þ
The transient components satisfy the following differential equations M1
o2 f 1 o2 f2 of1 ; þ M aðf1 f2 Þ ¼ q1 2 2 2 oy oy ot
ð36Þ
M3
o2 f 1 o2 f2 of2 ; þ M þ aðf1 f2 Þ ¼ q2 4 2 2 oy oy ot
ð37Þ
that are consistent with the boundary and initial conditions fb ð0; tÞ ¼ fb ð1; tÞ ¼ 0;
ð38Þ
fb ðy; 0Þ ¼ 1 y:
ð39Þ
Finite Fourier sine transform will be used to solve the above two simultaneous partial differential equations with the boundary and initial conditions (38) and (39). The finite Fourier sine transform of a function f ðyÞ defined for 0 < y < a is [22, p. 265] Z a npy ~ f ðyÞ sin dy n ¼ 1; 2; 3; . . . ð40Þ FS ff ðyÞg ¼ f ðnÞ ¼ a 0 with inverse transform FS1 ff~ðnÞg
1 npy 2X ¼ f ðyÞ ¼ f~ðnÞ sin : a n¼1 a
ð41Þ
Application of the Fourier sine transform to Eqs. (36) and (37), taking Eq. (38) into account, gives df~1 ¼ b1 f~1 þ b2 f~2 ; dt
ð42Þ
df~2 ¼ b3 f~1 b4 f~2 ; dt
ð43Þ
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where b1 ¼
a þ M1 n2 p2 ; q1
b2 ¼
a M2 n2 p2 ; q1
b3 ¼
a M3 n2 p2 ; q2
b4 ¼
a þ M4 n2 p2 ; q2
ð44Þ
subject to the transform of Eq. (39), FS ffb ðy; 0Þg ¼ f~b ðn; 0Þ ¼
1 : np
ð45Þ
Solving Eqs. (42) and (43) simultaneously and using the initial conditions (45), we find
pffiffi pffiffi 1 2b2 þ b4 b1 ~ pffiffi expf0:5ðb1 þ b4 Þtg coshð0:5t eÞ þ sinhð0:5t eÞ ; f1 ðn; tÞ ¼ np e
pffiffi pffiffi 1 2b3 þ b1 b4 ~ p ffiffi f2 ðn; tÞ ¼ expf0:5ðb1 þ b4 Þtg coshð0:5t eÞ þ sinhð0:5t eÞ ; np e
ð46Þ ð47Þ
where e ¼ b21 þ 4b2 b3 2b1 b4 þ b24 :
ð48Þ
With the help of Eq. (41), inverting Eqs. (46) and (47) and then substituting the results into Eq. (34), we obtain the following solution for ub ðy; tÞ ub ðy; tÞ ¼ 1 y 2
1 X
f~b ðn; tÞ sinðnpyÞ:
ð49Þ
n¼1
4. Unsteady plane Poiseuille flow In this section we consider the flow of initially motionless a binary mixture of incompressible Newtonian fluids between two infinite parallel plates, separated by a distance H, due to the sudden imposition of constant pressure gradient in the x-direction, namely opb =ox ¼ pX 0 ¼ const. We look for a solution of the form vðbÞ ¼ fvb ðy; tÞ; 0; 0g;
q ¼ qðy; tÞ;
q1 ¼ q1 ðy; tÞ;
q2 ¼ q2 ðy; tÞ:
ð50Þ
As made in the preceding section, it is proved that q is a constant. This is why all the coefficients in Eqs. (22) and (23) are constants. Thus, the dimensionless governing equations are as follows: M1
o2v1 o2v2 ov1 þ M aðv1 v2 Þ ¼ q1 1; 2 2 2 oy oy ot
ð51Þ
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
M3
o2v1 o2v2 ov2 1; þ M þ aðv1 v2 Þ ¼ q2 4 2 2 oy oy ot
2031
ð52Þ
with vb ð0; tÞ ¼ vb ð1; tÞ ¼ 0 vb ðy ; 0Þ ¼ 0
for t P 0;
ð53Þ ð54Þ
for 0 6 y 6 1;
where Mi ¼
Mi ; l
qb ¼
qb ; q0
a ¼
aH 2 ; l
t ¼
lt ; q0 H 2
vb ¼
lvb ; pX 0 H 2
y ¼
y : H
ð55Þ
Note that all of the boundary and initial conditions given in Eqs. (53) and (54) are homogeneous, yet there exist a nontrivial solution, since the partial differential equations (51) and (52) are nonhomogeneous. As in the previous section, we shall attempt to find a solution of the form vb ðy; tÞ ¼ vbS ðyÞ gb ðy; tÞ;
ð56Þ
where vbS ðyÞ is the steady-state velocity distribution of the bth fluid for the plane Poiseuille flow. The procedure for determining vbS ðyÞ is similar to that used in [9], so here we simply state the solutions which are vbS ðyÞ ¼ db
sinhð0:5mf1 ygÞ sinhð0:5myÞ coshð0:5mÞ
þ dyðy 1Þ;
ð57Þ
where 1 d¼ ; g2
2g ðM2 þ M4 Þ d1 ¼ 3 2 ; g2 a
2g ðM1 þ M3 Þ d2 ¼ 3 2 ; g2 a
rffiffiffiffiffiffiffi g2 a ; m¼ g1
ð58Þ
where g1 , g2 and g3 are g1 ¼ M1 M4 M2 M3 ;
g2 ¼ M1 þ M2 þ M3 þ M4 ;
g3 ¼ M1 þ M2 M3 M4 :
ð59Þ
Inserting vb ðy; tÞ from Eq. (56) into Eqs. (51)–(54) yields M1
o2 g1 o2 g2 og1 þ M aðg1 g2 Þ ¼ q1 ; 2 oy 2 oy 2 ot
ð60Þ
M3
o2 g1 o2 g2 og2 þ M þ aðg1 g2 Þ ¼ q2 ; 4 2 2 oy oy ot
ð61Þ
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with the boundary and initial conditions gb ð0; tÞ ¼ gb ð1; tÞ ¼ 0;
ð62Þ
gb ðy; 0Þ ¼ vbS ðyÞ:
ð63Þ
Taking the finite Fourier sine transforms of Eqs. (60) and (61), and employing Eq. (62) result in d~ g1 ¼ b1 g~1 þ b2 g~2 ; dt
ð64Þ
d~ g2 ¼ b3 g~1 b4 g~2 ; dt
ð65Þ
subject to the transform of Eq. (63), 8 m2 db 4d < FS fgb ðy; 0Þg ¼ g~b ðn; 0Þ ¼ npðm2 þ n2 p2 Þ n3 p3 ; : 0;
n ¼ 1; 3; 5; . . .
ð66Þ
n ¼ 2; 4; 6; . . .
Equations (64) and (65) which satisfy initial conditions (66) are solved by the following analytical expressions g~1 ðn; tÞ ¼ expf0:5ðb1 þ b4 Þtg
pffiffi pffiffi 2b2 g~2 ðn; 0Þ þ ðb4 b1 Þ~ g1 ðn; 0Þ pffiffi
g~1 ðn; 0Þ coshð0:5t eÞ þ sinhð0:5t eÞ ; e g~2 ðn; tÞ ¼ expf0:5ðb1 þ b4 Þtg
pffiffi pffiffi 2b3 g~1 ðn; 0Þ þ ðb1 b4 Þ~ g2 ðn; 0Þ pffiffi
g~2 ðn; 0Þ coshð0:5t eÞ þ sinhð0:5t eÞ : e
ð67Þ
ð68Þ
We now obtain the solution for the velocity of the bth fluid by going back through the various substitutions:
sinhð0:5mf1 ygÞ sinhð0:5myÞ þ dyðy 1Þ vb ðy; tÞ ¼ db coshð0:5mÞ 1 X g~b ð2k 1; tÞ sinfð2k 1Þpyg: 2 ð69Þ k¼1
5. Unsteady axisymmetric Poiseuille flow Finally, we discuss the problem of unsteady flow of a binary mixture of incompressible Newtonian fluids through a circular pipe of radius R. Suppose that the circular pipe is filled with a stationary mixture of two fluids. At the instant t ¼ 0 constant pressure gradient in the z-direction,
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
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namely opb =oz ¼ pZ0 ¼ const:, is imposed and the fluids begin to flow. Cylindrical coordinates ðr; h; zÞ, with the z-axis coinciding with the axis of the circular pipe, is introduced. We shall seek a solution of the form vðbÞ ¼ f0; 0; wb ðr; tÞg;
q ¼ qðr; tÞ;
q1 ¼ q1 ðr; tÞ;
q2 ¼ q2 ðr; tÞ:
ð70Þ
Substitution of Eq. (70) into Eq. (4) gives oqb =ot ¼ 0, thus oq=ot ¼ 0 and Eq. (70)2 simplifies to q ¼ qðrÞ. Substituting Eq. (70) into the r-components of the equations of motion (22) and (23), we get k
dq op1 ¼ ; dr or
k
dq op2 ¼ : dr or
ð71Þ
Elimination of ok=or between Eqs. (71)1 and (71)2 , with the help of Eqs. (11), (15) and (19), leads to ðq q10 Þðq20 qÞ
dq d2 ðqAÞ ¼ 0: dr dq2
ð72Þ
Here, in general, q 6¼ q10 , q 6¼ q20 and d2 ðqAÞ=dq2 6¼ 0, and hence we arrive at the conclusion that the total density q is a constant. Since q has been proved to be a constant, namely q ¼ q0 , the coefficients M1 etc. appearing in equations of motion become constants. Thus, the z-components of the equations of motions (22) and (23) reduce to ! ! 1 1 o 2 1 o o2 w w1 o2 w w2 o w1 2 Þ ¼ q1 1; ð73Þ þ M2 að w1 w þ þ M1 2 2 r or r or or or ot M3
1 1 o o2 w w1 þ 2 r or or
! þ M4
2 1 o o2 w w2 þ 2 r or or
! 2 Þ ¼ q2 þ að w1 w
o w2 1; ot
ð74Þ
with the boundary and initial conditions b ð1; tÞ ¼ 0 w
b ð0; tÞ finite and w
b ðr; 0Þ ¼ 0 w
for 0 6 r 6 1;
for t P 0;
ð75Þ ð76Þ
where Mi ¼
Mi ; l
qb ¼
qb ; q0
a ¼
aR2 ; l
t ¼
lt ; q0 R2
b ¼ w
lwb ; pZ0 R2
r ¼
r : R
ð77Þ
As in the previous section, because the above partial differential equations are nonhomogeneous, we begin by separating off the steady-state solution; that is, we set wb ðr; tÞ ¼ wbS ðrÞ hb ðr; tÞ;
ð78Þ
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S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
where wbS ðrÞ is the solution of M1 M3
d2 w1S 1 dw1S þ r dr dr2 d2 w1S 1 dw1S þ r dr dr2
þ M2
þ M4
d2 w2S 1 dw2S þ r dr dr2 d2 w2S 1 dw2S þ r dr dr2
aðw1S w2S Þ ¼ 1;
ð79Þ
þ aðw1S w2S Þ ¼ 1:
ð80Þ
The solution of the above system of coupled ordinary differential equations, which satisfies the boundary conditions wbS ð1Þ ¼ 0 and remains finite at r ¼ 0, is obtained by employing the procedure used by the authors in Ref. [9]. In order to avoid repetition, the details are omitted and the flow fields are directly given by wbS ðrÞ ¼ cð1 r2 Þ þ cb fI0 ðmrÞ I0 ðmÞg;
ð81Þ
where c¼
1 ; 2g2
c1 ¼
g3 ðM2 þ M4 Þ ; ag22 I0 ðmÞ
c2 ¼
g3 ðM1 þ M3 Þ ; ag22 I0 ðmÞ
ð82Þ
where I0 is the modified Bessel function of the first kind of order zero. With these steady-state solutions, hb ðr; tÞ must satisfy the homogeneous differential equations M1 M3
o2 h1 1 oh1 þ r or or2 o2 h1 1 oh1 þ r or or2
þ M2
þ M4
o2 h2 1 oh2 þ r or or2 o2 h2 1 oh2 þ r or or2
aðh1 h2 Þ ¼ q1
oh1 ; ot
ð83Þ
þ aðh1 h2 Þ ¼ q2
oh2 ; ot
ð84Þ
with hb ð1; tÞ ¼ 0
and hb ð0; tÞ finite;
hb ðr; 0Þ ¼ wbS ðrÞ:
ð85Þ ð86Þ
Finite Hankel transform will be used to solve the above two simultaneous partial differential equations with the boundary and initial conditions (85) and (86). The finite Hankel transform of order n of a function f ðrÞ, defined on 0 < r < a, is [22, p. 318] Z a ^ rf ðrÞJn ðrki Þ dr: ð87Þ Hn ff ðrÞg ¼ f ðki Þ ¼ 0
The associated inverse transform is Hn1 ff^ðki Þg ¼ f ðrÞ ¼
1 2 X Jn ðrki Þ ; f^ðki Þ 2 a2 i¼1 Jnþ1 ðaki Þ
ð88Þ
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
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where Jn is the Bessel function of the first kind of order n, and the summation is taken over all the positive zeros k1 ; k2 ; k3 ; . . . of the Bessel function Jn ðaki Þ. The finite Hankel transforms of order zero of Eqs. (83) and (84), including Eq. (85)1 , respectively, are dh^1 ¼ v1 h^1 þ v2 h^2 ; dt
ð89Þ
dh^2 ¼ v3 h^1 v4 h^2 ; dt
ð90Þ
where v1 ¼
a þ M1 ki2 ; q1
v2 ¼
a M2 ki2 ; q1
v3 ¼
a M3 ki2 ; q2
v4 ¼
a þ M4 ki2 ; q2
ð91Þ
subject to the transform of Eq. (86), H0 fhb ðr; 0Þg ¼ h^b ðki ; 0Þ ¼
mcb mI0 ðmÞJ1 ðki Þ 2cJ2 ðki Þ þ I1 ðmÞJ0 ðki Þ : 2 2 ki ki2 ki þ m
ð92Þ
Since ki are the roots of J0 ðki Þ, we find 2cJ2 ðki Þ m2 cb I0 ðmÞJ1 ðki Þ h^b ðki ; 0Þ ¼ : ki ðki2 þ m2 Þ ki2
ð93Þ
The solution of the simultaneous equations (89) and (90) satisfying the initial conditions (93) can be written as h^1 ðki ; tÞ ¼ expf0:5ðv1 þ v4 Þtg ( ) ^ ^ pffiffiffiffi p ffiffiffi ffi 2v ðk ; 0Þ þ ðv v Þ h ðk ; 0Þ h 2 i 1 i 1 pffiffiffi4ffi
h^1 ðki ; 0Þ coshð0:5t /Þ þ 2 sinhð0:5t /Þ ; ð94Þ / h^2 ðki ; tÞ ¼ expf0:5ðv1 þ v4 Þtg ( ) ^ ^ pffiffiffiffi p ffiffiffi ffi 2v ðk ; 0Þ þ ðv v Þ h ðk ; 0Þ h 1 i 2 i 4 pffiffiffi1ffi
h^2 ðki ; 0Þ coshð0:5t /Þ þ 3 sinhð0:5t /Þ ; ð95Þ / where / ¼ v21 þ 4v2 v3 2v1 v4 þ v24 :
ð96Þ
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S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
By means of Eq. (88), taking the inversions of Eqs. (94) and (95) and then inserting the results into Eq. (78), we get wb ðr; tÞ ¼ cð1 r2 Þ þ cb fI0 ðmrÞ I0 ðmÞg 2
1 X i¼1
J0 ðrki Þ h^b ðki ; tÞ 2 ; J1 ðki Þ
ð97Þ
where the summation is taken over all the positive zeros k1 ; k2 ; k3 ; . . . of the Bessel function J0 ðki Þ.
6. Numerical results and discussion In this paper some simple unsteady unidirectional flows relevant to a binary mixture of chemically inert incompressible Newtonian fluids are studied theoretically. Exact solutions for the system of coupled partial differential equations governing the velocity fields are obtained by using the finite Fourier sine and finite Hankel transforms. These solutions would be useful in checking the validity of complicated flows of such mixtures. In order to make predictions based on foregoing analysis, it is necessary to know all of the material functions in the constitutive equations. Determination of these functions for a mixture is much more difficult than that for a single continuum. Since the flow of mixtures is of great technical importance, a good amount of literature has grown up around the problem of determining these functions. For example, employing results obtained from the kinetic theory of fluids, Sampaio and Williams [23] were able to derive formulae for l1 , l2 , l3 and l4 in terms of the viscosities of the unmixed fluids and the volume fractions in the case of k5 ¼ 0. In this work we benefit from the formulae suggested in [23] with a view to assigning the reasonable values to M 1 , M 2 , M 3 and M 4 . To achieve this, at the outset we assume that the densities of unmixed fluids and the volume fractions are known. With the aid of Eqs. (3) and (9), knowledge of these quantities enables q1 , q2 and q0 to be calculated [10]. Later, the material coefficients l1 , l2 , l3 and l4 can be determined by using the formulae proposed in [23]. For the purpose of simulations, the following values are given to the dimensionless parameters: M 1 ¼ 0:32;
M 2 ¼ M 3 ¼ 0:22;
M 4 ¼ 0:68;
q1 ¼ 0:65;
q2 ¼ 0:35:
ð98Þ
In Figs. 1–8, for comparison, the velocity distributions for pure Newtonian fluid and constituents of binary mixture are plotted as a function of position for various values of a and t, keeping the remaining parameters fixed at the values given in Eq. (98). For the solutions corresponding to pure Newtonian fluid we refer the reader to Ref. [24, p. 239–250]. From these figures we observe how the velocity profiles grow with increasing time and approach asymptotically the steady-state velocity profiles. It is important to bear in mind that the assumptions of Eqs. (34), (56) and (78) are convenient, since the unsteady problems investigated here approach the steady solutions as t ! 1. On comparing Fig. 1 with Fig. 3 or Fig. 6 with Fig. 8, we arrive at the conclusion that with an increase in the coefficient of interaction a, characterized by the drag force between the two constituents, the mixture tends to behave as a single continuum. As a result, it is not difficult to predict that the fluid particles belonging to both constituents have the same velocity at a given point in the mixture as a ! 1.
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
2037
Fig. 1. Velocity profiles of Couette flow for a ¼ 10, t ¼ 0:01 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 2. Velocity profiles of Couette flow for a ¼ 10, t ¼ 0:2 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 3. Velocity profiles of Couette flow for a ¼ 150, t ¼ 0:01 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
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Fig. 4. Velocity profiles of plane Poiseuille flow for a ¼ 10, t ¼ 0:01 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 5. Velocity profiles of plane Poiseuille flow for a ¼ 10, t ¼ 0:25 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 6. Velocity profiles of axisymmetric Poiseuille flow for a ¼ 10, t ¼ 0:02 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
2039
Fig. 7. Velocity profiles of axisymmetric Poiseuille flow for a ¼ 10, t ¼ 0:45 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 8. Velocity profiles of axisymmetric Poiseuille flow for a ¼ 30, t ¼ 0:02 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Finally we shall discuss reliability of the solutions given in Eqs. (49), (69) and (97). The velocity fields corresponding to the flows under consideration are in the form of series. As expected, these series are rapidly convergent for large values of time but slowly convergent for small values of time. It is important to note that series solutions (49), (69) and (97) can also be used for small values of time provided number of terms in the series expansions is enough to yield satisfactory accuracy. For example, for unsteady Couette flow, in the case of t ¼ 0:2, the fourth term is the first term in the series (49), absolute value of which is less than 108 . Therefore the sum of the first four term will give the velocity values of the bth fluid with an error of less than 108 . On the other hand, it is necessary to take account of the first 21 term for the same order of accuracy in the case of t ¼ 0:01. References [1] F. Dai, M.M. Khonsari, A theory of hydrodynamic lubrication involving the mixture of two fluids, J. Appl. Mech. 61 (1994) 634–641. [2] C. Truesdell, Sulle basi della thermomeccanica, Rend. Lincei 22 (8) (1957) 33–38, 158–166.
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[3] R.M. Bowen, Theory of mixtures, in: A.C. Eringen (Ed.), Continuum Physics, vol. III, Academic Press, New York, 1976. [4] R.J. Atkin, R.E. Craine, Continuum theories of mixtures. Basic theory and historical development, Quart. J. Mech. Appl. Math. 29 (1976) 209–244. [5] A. Bedford, D.S. Drumheller, Theory of immiscible and structured mixtures, Int. J. Engng. Sci. 21 (1983) 863–960. [6] K.R. Rajagopal, L. Tao, Mechanics of Mixtures, World Scientific Publishing, 1995. [7] N. Mills, Incompressible mixtures of Newtonian fluids, Int. J. Engng. Sci. 4 (1966) 97–112. [8] R.E. Craine, Oscillations of a plate in a binary mixture of incompressible Newtonian fluids, Int. J. Engng. Sci. 9 (1971) 1177–1192. [9] R.J. Atkin, R.E. Craine, Continuum theories of mixtures: Applications, J. Inst. Maths. Appl. 17 (1976) 153–207. [10] C.E. Beevers, R.E. Craine, On the determination of response functions for a binary mixture of incompressible Newtonian fluids, Int. J. Engng. Sci. 20 (1982) 737–745. €ßs , The influence of vibrating plates on Poiseuille flow of a binary mixture, Int. J. Engng. Sci. 26 (1988) [11] M.S ogu ß . G€ 313–323. €ßs , The influence of a vibrating plate on the flow between parallel plates of a binary mixture, Int. J. [12] M.S ogu ß . G€ Engng. Sci. 29 (1991) 1651–1659. €ßs , The influence of a longitudinal vibrating pipe on Poiseuille flow of a binary mixture, Int. J. Engng. [13] M.S ogu ß . G€ Sci. 30 (1992) 141–151. €ßs , The steady flow of a binary mixture between two rotating parallel non-coaxial disks, Int. J. Engng. [14] M.S ogu ß . G€ Sci. 30 (1992) 665–677. € ßs , The velocity profile in pulsatile flow of a binary mixture, Int. J. Engng. Sci. 32 (1994) 705–714. [15] M.S ogu ß . G€ €ßs , The steady flow of a binary mixture between two rotating parallel coaxial disks, Int. J. Engng. Sci. 33 [16] M.S ogu ß . G€ (1995) 611–624. [17] A. Al-Sharif, K. Chamniprasart, K.R. Rajagopal, A.Z. Szeri, Lubrication with binary mixtures: Liquid–liquid emulsion, J. Tribol. 115 (1993) 46–55. [18] K. Chamniprasart, A. Al-Sharif, K.R. Rajagopal, A.Z. Szeri, Lubrication with binary mixtures: Bubbly oil, J. Tribol. 115 (1993) 253–260. [19] S.H. Wang, A. Al-Sharif, K.R. Rajagopal, A.Z. Szeri, Lubrication with binary mixtures: Liquid–liquid emulsion in an EHL conjunction, J. Tribol. 115 (1993) 515–522. [20] G. Johnson, M. Massoudi, K.R. Rajagopal, Flow of a fluid–solid mixture between flat plates, Chem. Engng. Sci. 46 (1991) 1713–1723. [21] G. Johnson, M. Massoudi, K.R. Rajagopal, Flow of a fluid infused with solid particles through a pipe, Int. J. Engng. Sci. 29 (1991) 649–661. [22] L. Debnath, Integral Transforms and their Applications, CRC Press, 1995. [23] R. Sampaio, W.O. Williams, On the viscosities of liquid mixtures, Z. Angew. Math. Phys. (ZAMP) 28 (1977) 607– 613. [24] T.C. Papanastasiou, G.C. Georgiou, A.N. Alexandrou, Viscous Fluid Flow, CRC Press, 2000.
Some simple unsteady unidirectional flows of a binary mixture of incompressible Newtonian fluids Serdar Barısß
*
Department of Mechanical Engineering, Faculty of Engineering, Istanbul University, Avcılar 34850, Istanbul, Turkey Received 23 January 2002; accepted 12 June 2002
Abstract The problems dealing with some simple unsteady unidirectional flows of a mixture of two incompressible Newtonian fluids are investigated. By using the constitutive equations appeared in the literature for binary mixtures of chemically inert incompressible Newtonian fluids, the equations governing the motion of the binary mixture are reduced to a system of coupled partial differential equations. By means of integral transforms, the exact solutions of these equations are obtained for the following three problems: (i) unsteady Couette flow, (ii) unsteady plane Poiseuille flow, (iii) unsteady axisymmetric Poiseuille flow. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Mixture; Unsteady flow; Fourier sine transform; Hankel transform
1. Introduction The study of a mixture of fluids is always important in view of its applications in various branches of engineering and technology. A familiar example is an emulsion which is the dispersion of one fluid within another fluid. Typical emulsions are oil dispersed within water or water within oil. Another example where the mixture of fluids plays an important role is in multigrade oils. Polymeric type fluids are added to the base oil so as to enhance the lubrication properties of mineral oils [1]. Truesdell [2] was the first to formulate the thermomechanical balance equations for a mixture of general materials. After his pioneering work, a good amount of literature has been generated on the formulation of continuum thermomechanical theories of mixtures. We refer the reader to *
Fax: +90-212-245-0795. E-mail address: [email protected] (S. Barısß).
0020-7225/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 0 2 ) 0 0 1 0 9 - X
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the works of Bowen [3], Atkin and Craine [4], Bedford and Drumheller [5], and Rajagopal and Tao [6] regarding the historical development of the theory and detailed analysis of various results on this subject. Some exact solutions for the flow of a binary mixture of incompressible Newtonian fluids were presented by Mills [7], Craine [8], Atkin and Craine [9], Beevers and Craine [10] €ßs [11–16]. Recently, several problems relating to the mechanics of oil and water and G€ o gu emulsions, particularly with regard to applications in lubrication practice, have been considered within the context of the mixture theory by Al-Sharif et al. [17], Chamniprasart et al. [18], and Wang et al. [19]. In the present paper a binary mixture, each constituent of which is a chemically inert incompressible Newtonian fluid, is considered. The balance laws and relevant constitutive equations are presented in Section 2. In the subsequent sections we obtain the exact solutions for some simple unsteady unidirectional flows of the binary mixture under consideration.
2. Basic theory A brief summary of the basic balance laws and the appropriate constitutive theory for a binary mixture of incompressible Newtonian fluids will be given here, for more details the reader should consult Craine [8] and Atkin and Craine [9]. We consider a mixture of two interacting constituents, each of which is regarded as a continuum; we refer to the bth constituent as the continuum sðbÞ . Throughout this paper b takes the values 1 and 2. We assume that each point x within the mixture is occupied simultaneously by one particle from each sðbÞ . If vðbÞ denotes the velocity vector of the bth constituent, the material time derivative DðbÞ =Dt is defined by DðbÞ o ðbÞ o ; ¼ þ vi ot oxi Dt
ð1Þ
where the usual summation convention is applied to repeated Roman subscripts. The mean velocity of the mixture w and the total mass density of the mixture q are given respectively by ð1Þ
ð2Þ
qwk ¼ q1 vk þ q2 vk ;
ð2Þ
q ¼ q1 þ q2 ;
ð3Þ
where qb is the density of sðbÞ at time t, measured per unit volume of mixture. The basic equations for a binary mixture in which the constituents have a common temperature T and do not interact chemically are the following ones: continuity equations: oqb ðbÞ þ ðqb vi Þ;i ¼ 0; ot
ð4Þ
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
2025
equations of motion: ð1Þ
ð1Þ
ð1Þ
rik;i fk þ q1 Fk
¼ q1
Dð1Þ vk ; Dt
ð2Þ
ð2Þ
ð2Þ
rik;i þ fk þ q2 Fk
¼ q2
Dð2Þ vk ; Dt
ð5Þ
energy equation: qr qi;i q1
Dð1Þ U1 Dð2Þ U2 ð1Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ q2 þ rij vj;i þ rij vj;i þ fj ðvj vj Þ ¼ 0; Dt Dt
ð6Þ
where a comma denotes differentiation with respect to xi . The quantities Ub , rðbÞ and FðbÞ are in turn internal energy per unit mass, partial stress and body force per unit mass of the bth constituent. In addition r the heat supply per unit mass, q the heat flux vector, refer to the mixture as a whole, and f denotes the diffusive force vector. We now turn our attention to the interaction term f. For viscous fluids, in general, this interaction term depends on the relative velocity, the density gradients, the temperature gradients, and possibly other quantities [18]. Such interactions play a very important role in the nature of the solutions (see, for instance [20,21]). In this study we shall assume that the interaction force incorporates only the effect of drag. Consideration of the balance of angular momentum for sðbÞ shows that rðbÞ need not be symmetric although the balance of angular momentum for the mixture results in the symmetry of r, the total stress in the mixture, defined by ð1Þ
ð2Þ
rij ¼ rij þ rij :
ð7Þ
Admissible thermomechanical processes in the mixture must be compatible with an entropy production inequality. If Sb is the partial entropy per unit mass of the bth constituent, then the Clausius–Duhem inequality may be written as follows: Dð1Þ S1 Dð2Þ S2 qr qi q1 þ q2 þ P 0: T Dt Dt T ;i
ð8Þ
In this work we shall concern ourselves with a mixture of two incompressible Newtonian fluids. Let the density of sðbÞ in its reference configuration be qb0 , which in view of the assumed incompressibility is constant. Introducing the quantity /b , is the volume fraction of the bth fluid, and assuming that the mixture does not contain voids, it follows that q1 ¼ /1 q10 ;
q2 ¼ ð1 /1 Þq20
ð9Þ
and hence q1 q þ 2 ¼ 1: q10 q20
ð10Þ
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Using (3) and (10), it can be easily shown that q1 ¼
q10 ðq20 qÞ ; q20 q10
q2 ¼
q20 ðq q10 Þ : q20 q10
ð11Þ
Substituting Eq. (11) into Eq. (4) and eliminating oq=ot between them gives the relation ð1Þ
ð2Þ
ðq20 qÞdii þ ðq q10 Þdii ni ai ¼ 0;
ð12Þ
where ðbÞ
ðbÞ
ðbÞ
2dij ¼ vi;j þ vj;i ;
ð1Þ
ni ¼ q;i ;
ð2Þ
ai ¼ vi vi :
ð13Þ
The derivation of the constitutive equations appropriate to our binary mixture of incompressible Newtonian fluids has been outlined in [9]. If the mixture is considered to be purely mechanical system; that is, thermal effects are ignored, the relevant equations are Ab ¼ Ab ðqÞ;
A ¼ AðqÞ; dA1 p1 ¼ ðq q20 Þ q1 þk ; dq
ð14Þ
dA2 p2 ¼ ðq q10 Þ q2 k ; dq
fk ¼ aak knk ;
ð15Þ ð16Þ
ð1Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð17Þ
ð2Þ
ð1Þ
ð2Þ
ð1Þ
ð2Þ
ð18Þ
rij ¼ ðp1 þ k1 dkk þ k3 dkk Þdij þ 2l1 dij þ 2l3 dij þ k5 Cij ; rij ¼ ðp2 þ k4 dkk þ k2 dkk Þdij þ 2l4 dij þ 2l2 dij k5 Cij ;
where Ab denotes the partial Helmholtz free energy of the bth constituent, and the Helmholtz free energy of the mixture A (total free energy) is defined by qA ¼ q1 A1 þ q2 A2
ð19Þ
and the coefficients a, k1 ; . . . ; k5 , l1 ; . . . ; l4 are functions of q and satisfy the inequalities a P 0;
k5 P 0;
l1 P 0;
ðl3 þ l4 Þ2 6 4l1 l2 ;
l2 P 0;
k1 þ 23l1 P 0;
k2 þ 23l2 P 0;
2 k3 þ k4 þ 23ðl3 þ l4 Þ 6 4 k1 þ 23l1 k2 þ 23l2 :
ð20Þ
The quantity k is a Lagrange multiplier associated with the constraint (12) and the relative spin C is given by ð1Þ
ð1Þ
ð2Þ
ð2Þ
2Cij ¼ ðvi;j vj;i Þ ðvi;j vj;i Þ:
ð21Þ
Note that as there is only one constraint, that of volume additivity, namely Eq. (10), there can be only one Lagrange multiplier and thus p1 and p2 are not independent. Both of them can be expressed in terms of a single Lagrange multiplier.
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Finally, neglecting the body forces, we shall derive the equations governing the flow of a mixture of two incompressible Newtonian fluids. For this purpose, inserting f, rð1Þ , and rð2Þ from Eqs. (16)–(18) into Eq. (5), with the help of Eqs. (13) and (21), one gets the following equations of motion: M1 Dvð1Þ þ M2 Dvð2Þ þ M5 rðr vð1Þ Þ þ M6 rðr vð2Þ Þ þ ðr vð1Þ Þrk1 þ ðr vð2Þ Þrk3 þ ðrvð1Þ ÞT rM1 þ ðrvð2Þ ÞT rM2 þ rvð1Þ rM9 þ rvð2Þ rM10 aðvð1Þ vð2Þ Þ ¼ q1
Dð1Þ vð1Þ þ rp1 krq; Dt
ð22Þ
M3 Dvð1Þ þ M4 Dvð2Þ þ M7 rðr vð1Þ Þ þ M8 rðr vð2Þ Þ þ ðr vð1Þ Þrk4 þ ðr vð2Þ Þrk2 T
T
þ ðrvð1Þ Þ rM3 þ ðrvð2Þ Þ rM4 þ rvð1Þ rM11 þ rvð2Þ rM12 þ aðvð1Þ vð2Þ Þ ¼ q2
Dð2Þ vð2Þ þ rp2 þ krq; Dt
ð23Þ
where M1 ¼ l1 þ
k5 ; 2
M5 ¼ k1 þ l1 M9 ¼ l1
k5 ; 2
M2 ¼ l3 k5 ; 2
k5 ; 2
M3 ¼ l4
M6 ¼ k3 þ l3 þ
M10 ¼ l3 þ
k5 ; 2
k5 ; 2
k5 ; 2
M4 ¼ l2 þ
M7 ¼ k4 þ l4 þ
M11 ¼ l4 þ
k5 ; 2
k5 ; 2
k5 ; 2
M12 ¼ l2
M8 ¼ k2 þ l2
k5 ; 2
ð24Þ
k5 : 2
In the above equations r and D are the gradient and Laplacian operators, respectively. Also, ðbÞ rvðbÞ is the second-order tensor field and its ijth component is taken as vj;i . Note that, under isothermal conditions, the coefficients M1 etc. appearing in Eqs. (22) and (23) depend only on the total density q, and hence spatial coordinates and time. In the subsequent sections, we shall obtain the exact solutions of the above equations for some simple unsteady unidirectional flows of a binary mixture of incompressible Newtonian fluids.
3. Unsteady Couette flow We shall begin with the case of the flow of a binary mixture of incompressible Newtonian fluids between two infinite parallel plates separated by a distance H. The mixture and the two plates are initially at rest. The lower plate is suddenly accelerated from rest and moves in its own plane with a constant velocity U, while the upper plate is held stationary. It is assumed that the flow is entirely driven by the motion of the lower plate, the pressures far upstream and far downstream being kept equal throughout the motion. Thus the pressure gradients in the x-direction are zero. Cartesian coordinates ðx; y; zÞ are used. Let the y-axis be directed normally to the plates, with y ¼ 0 on the lower plate, the x-axis being in the direction of motion of the lower plate.
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We seek solutions in which the velocity vector of the bth fluid and total density are assumed to have the form vðbÞ ¼ fub ðy; tÞ; 0; 0g;
q ¼ qðy; tÞ;
q1 ¼ q1 ðy; tÞ;
q2 ¼ q2 ðy; tÞ:
ð25Þ
Substitution of Eq. (25) into Eq. (4) gives oq1 oq2 ¼ ¼ 0; ot ot
ð26Þ
thus oq=ot ¼ 0 and Eq. (25)2 simplifies to q ¼ qðyÞ. Substituting Eq. (25) into the y-components of the equations of motion (22) and (23), we get k
dq op1 ¼ ; dy oy
k
dq op2 ¼ : dy oy
ð27Þ
With the use of Eqs. (11), (15) and (19), elimination of ok=oy between Eq. (27)1 and (27)2 gives, after some manipulation, ðq q10 Þðq20 qÞ
dq d2 ðqAÞ ¼0 dy dq2
ð28Þ
and since, in general, q 6¼ q10 , q 6¼ q20 and d2 ðqAÞ=dq2 6¼ 0 we deduce that q ¼ q0 is a constant. As a result, the coefficients M1 etc. in Eqs. (22) and (23) are constants. In the light of these arguments, the x-components of the equations of motions (22) and (23) reduce to M1
o2 u1 o2 u2 ou1 þ M aðu1 u2 Þ ¼ q1 ; 2 2 2 oy oy ot
ð29Þ
M3
o2 u1 o2 u2 ou2 þ M þ aðu1 u2 Þ ¼ q2 : 4 2 2 oy oy ot
ð30Þ
It is convenient at this point to introduce dimensionless variables and material constants. If f is used to denote the dimensionless form of a quantity f, it follows that Mi ¼
Mi ; l
qb ¼
qb ; q0
a ¼
aH 2 ; l
t ¼
lt ; q0 H 2
ub ¼
ub ; U
y ¼
y ; H
ð31Þ
where l is a typical viscosity coefficient. The dimensionless governing equations are obtained from Eqs. (29) and (30) by replacing variables and material constants by those given in Eq. (31), so they are not rewritten here. The boundary and initial conditions are ub ð0; tÞ ¼ 1
for t > 0;
ub ðy ; 0Þ ¼ 0 for 0 < y 6 1:
ub ð1; tÞ ¼ 0
for t P 0;
ð32Þ ð33Þ
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
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Throughout this paper, henceforth for convenience, unless stated otherwise, we shall drop the bars that appear over the dimensionless quantities. We first have to transform the problem so that the boundary conditions (32)1 are homogeneous. This can be achieved by decomposing ub ðy; tÞ into the steady plane Couette velocity profile ubS ðyÞ, which are expected to prevail at large times, and the transient component fb ðy; tÞ: ub ðy; tÞ ¼ ubS ðyÞ fb ðy; tÞ:
ð34Þ
The steady-state velocity distribution of the bth fluid is given by (cf. [9]) ubS ðyÞ ¼ 1 y:
ð35Þ
The transient components satisfy the following differential equations M1
o2 f 1 o2 f2 of1 ; þ M aðf1 f2 Þ ¼ q1 2 2 2 oy oy ot
ð36Þ
M3
o2 f 1 o2 f2 of2 ; þ M þ aðf1 f2 Þ ¼ q2 4 2 2 oy oy ot
ð37Þ
that are consistent with the boundary and initial conditions fb ð0; tÞ ¼ fb ð1; tÞ ¼ 0;
ð38Þ
fb ðy; 0Þ ¼ 1 y:
ð39Þ
Finite Fourier sine transform will be used to solve the above two simultaneous partial differential equations with the boundary and initial conditions (38) and (39). The finite Fourier sine transform of a function f ðyÞ defined for 0 < y < a is [22, p. 265] Z a npy ~ f ðyÞ sin dy n ¼ 1; 2; 3; . . . ð40Þ FS ff ðyÞg ¼ f ðnÞ ¼ a 0 with inverse transform FS1 ff~ðnÞg
1 npy 2X ¼ f ðyÞ ¼ f~ðnÞ sin : a n¼1 a
ð41Þ
Application of the Fourier sine transform to Eqs. (36) and (37), taking Eq. (38) into account, gives df~1 ¼ b1 f~1 þ b2 f~2 ; dt
ð42Þ
df~2 ¼ b3 f~1 b4 f~2 ; dt
ð43Þ
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S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
where b1 ¼
a þ M1 n2 p2 ; q1
b2 ¼
a M2 n2 p2 ; q1
b3 ¼
a M3 n2 p2 ; q2
b4 ¼
a þ M4 n2 p2 ; q2
ð44Þ
subject to the transform of Eq. (39), FS ffb ðy; 0Þg ¼ f~b ðn; 0Þ ¼
1 : np
ð45Þ
Solving Eqs. (42) and (43) simultaneously and using the initial conditions (45), we find
pffiffi pffiffi 1 2b2 þ b4 b1 ~ pffiffi expf0:5ðb1 þ b4 Þtg coshð0:5t eÞ þ sinhð0:5t eÞ ; f1 ðn; tÞ ¼ np e
pffiffi pffiffi 1 2b3 þ b1 b4 ~ p ffiffi f2 ðn; tÞ ¼ expf0:5ðb1 þ b4 Þtg coshð0:5t eÞ þ sinhð0:5t eÞ ; np e
ð46Þ ð47Þ
where e ¼ b21 þ 4b2 b3 2b1 b4 þ b24 :
ð48Þ
With the help of Eq. (41), inverting Eqs. (46) and (47) and then substituting the results into Eq. (34), we obtain the following solution for ub ðy; tÞ ub ðy; tÞ ¼ 1 y 2
1 X
f~b ðn; tÞ sinðnpyÞ:
ð49Þ
n¼1
4. Unsteady plane Poiseuille flow In this section we consider the flow of initially motionless a binary mixture of incompressible Newtonian fluids between two infinite parallel plates, separated by a distance H, due to the sudden imposition of constant pressure gradient in the x-direction, namely opb =ox ¼ pX 0 ¼ const. We look for a solution of the form vðbÞ ¼ fvb ðy; tÞ; 0; 0g;
q ¼ qðy; tÞ;
q1 ¼ q1 ðy; tÞ;
q2 ¼ q2 ðy; tÞ:
ð50Þ
As made in the preceding section, it is proved that q is a constant. This is why all the coefficients in Eqs. (22) and (23) are constants. Thus, the dimensionless governing equations are as follows: M1
o2v1 o2v2 ov1 þ M aðv1 v2 Þ ¼ q1 1; 2 2 2 oy oy ot
ð51Þ
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
M3
o2v1 o2v2 ov2 1; þ M þ aðv1 v2 Þ ¼ q2 4 2 2 oy oy ot
2031
ð52Þ
with vb ð0; tÞ ¼ vb ð1; tÞ ¼ 0 vb ðy ; 0Þ ¼ 0
for t P 0;
ð53Þ ð54Þ
for 0 6 y 6 1;
where Mi ¼
Mi ; l
qb ¼
qb ; q0
a ¼
aH 2 ; l
t ¼
lt ; q0 H 2
vb ¼
lvb ; pX 0 H 2
y ¼
y : H
ð55Þ
Note that all of the boundary and initial conditions given in Eqs. (53) and (54) are homogeneous, yet there exist a nontrivial solution, since the partial differential equations (51) and (52) are nonhomogeneous. As in the previous section, we shall attempt to find a solution of the form vb ðy; tÞ ¼ vbS ðyÞ gb ðy; tÞ;
ð56Þ
where vbS ðyÞ is the steady-state velocity distribution of the bth fluid for the plane Poiseuille flow. The procedure for determining vbS ðyÞ is similar to that used in [9], so here we simply state the solutions which are vbS ðyÞ ¼ db
sinhð0:5mf1 ygÞ sinhð0:5myÞ coshð0:5mÞ
þ dyðy 1Þ;
ð57Þ
where 1 d¼ ; g2
2g ðM2 þ M4 Þ d1 ¼ 3 2 ; g2 a
2g ðM1 þ M3 Þ d2 ¼ 3 2 ; g2 a
rffiffiffiffiffiffiffi g2 a ; m¼ g1
ð58Þ
where g1 , g2 and g3 are g1 ¼ M1 M4 M2 M3 ;
g2 ¼ M1 þ M2 þ M3 þ M4 ;
g3 ¼ M1 þ M2 M3 M4 :
ð59Þ
Inserting vb ðy; tÞ from Eq. (56) into Eqs. (51)–(54) yields M1
o2 g1 o2 g2 og1 þ M aðg1 g2 Þ ¼ q1 ; 2 oy 2 oy 2 ot
ð60Þ
M3
o2 g1 o2 g2 og2 þ M þ aðg1 g2 Þ ¼ q2 ; 4 2 2 oy oy ot
ð61Þ
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with the boundary and initial conditions gb ð0; tÞ ¼ gb ð1; tÞ ¼ 0;
ð62Þ
gb ðy; 0Þ ¼ vbS ðyÞ:
ð63Þ
Taking the finite Fourier sine transforms of Eqs. (60) and (61), and employing Eq. (62) result in d~ g1 ¼ b1 g~1 þ b2 g~2 ; dt
ð64Þ
d~ g2 ¼ b3 g~1 b4 g~2 ; dt
ð65Þ
subject to the transform of Eq. (63), 8 m2 db 4d < FS fgb ðy; 0Þg ¼ g~b ðn; 0Þ ¼ npðm2 þ n2 p2 Þ n3 p3 ; : 0;
n ¼ 1; 3; 5; . . .
ð66Þ
n ¼ 2; 4; 6; . . .
Equations (64) and (65) which satisfy initial conditions (66) are solved by the following analytical expressions g~1 ðn; tÞ ¼ expf0:5ðb1 þ b4 Þtg
pffiffi pffiffi 2b2 g~2 ðn; 0Þ þ ðb4 b1 Þ~ g1 ðn; 0Þ pffiffi
g~1 ðn; 0Þ coshð0:5t eÞ þ sinhð0:5t eÞ ; e g~2 ðn; tÞ ¼ expf0:5ðb1 þ b4 Þtg
pffiffi pffiffi 2b3 g~1 ðn; 0Þ þ ðb1 b4 Þ~ g2 ðn; 0Þ pffiffi
g~2 ðn; 0Þ coshð0:5t eÞ þ sinhð0:5t eÞ : e
ð67Þ
ð68Þ
We now obtain the solution for the velocity of the bth fluid by going back through the various substitutions:
sinhð0:5mf1 ygÞ sinhð0:5myÞ þ dyðy 1Þ vb ðy; tÞ ¼ db coshð0:5mÞ 1 X g~b ð2k 1; tÞ sinfð2k 1Þpyg: 2 ð69Þ k¼1
5. Unsteady axisymmetric Poiseuille flow Finally, we discuss the problem of unsteady flow of a binary mixture of incompressible Newtonian fluids through a circular pipe of radius R. Suppose that the circular pipe is filled with a stationary mixture of two fluids. At the instant t ¼ 0 constant pressure gradient in the z-direction,
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
2033
namely opb =oz ¼ pZ0 ¼ const:, is imposed and the fluids begin to flow. Cylindrical coordinates ðr; h; zÞ, with the z-axis coinciding with the axis of the circular pipe, is introduced. We shall seek a solution of the form vðbÞ ¼ f0; 0; wb ðr; tÞg;
q ¼ qðr; tÞ;
q1 ¼ q1 ðr; tÞ;
q2 ¼ q2 ðr; tÞ:
ð70Þ
Substitution of Eq. (70) into Eq. (4) gives oqb =ot ¼ 0, thus oq=ot ¼ 0 and Eq. (70)2 simplifies to q ¼ qðrÞ. Substituting Eq. (70) into the r-components of the equations of motion (22) and (23), we get k
dq op1 ¼ ; dr or
k
dq op2 ¼ : dr or
ð71Þ
Elimination of ok=or between Eqs. (71)1 and (71)2 , with the help of Eqs. (11), (15) and (19), leads to ðq q10 Þðq20 qÞ
dq d2 ðqAÞ ¼ 0: dr dq2
ð72Þ
Here, in general, q 6¼ q10 , q 6¼ q20 and d2 ðqAÞ=dq2 6¼ 0, and hence we arrive at the conclusion that the total density q is a constant. Since q has been proved to be a constant, namely q ¼ q0 , the coefficients M1 etc. appearing in equations of motion become constants. Thus, the z-components of the equations of motions (22) and (23) reduce to ! ! 1 1 o 2 1 o o2 w w1 o2 w w2 o w1 2 Þ ¼ q1 1; ð73Þ þ M2 að w1 w þ þ M1 2 2 r or r or or or ot M3
1 1 o o2 w w1 þ 2 r or or
! þ M4
2 1 o o2 w w2 þ 2 r or or
! 2 Þ ¼ q2 þ að w1 w
o w2 1; ot
ð74Þ
with the boundary and initial conditions b ð1; tÞ ¼ 0 w
b ð0; tÞ finite and w
b ðr; 0Þ ¼ 0 w
for 0 6 r 6 1;
for t P 0;
ð75Þ ð76Þ
where Mi ¼
Mi ; l
qb ¼
qb ; q0
a ¼
aR2 ; l
t ¼
lt ; q0 R2
b ¼ w
lwb ; pZ0 R2
r ¼
r : R
ð77Þ
As in the previous section, because the above partial differential equations are nonhomogeneous, we begin by separating off the steady-state solution; that is, we set wb ðr; tÞ ¼ wbS ðrÞ hb ðr; tÞ;
ð78Þ
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where wbS ðrÞ is the solution of M1 M3
d2 w1S 1 dw1S þ r dr dr2 d2 w1S 1 dw1S þ r dr dr2
þ M2
þ M4
d2 w2S 1 dw2S þ r dr dr2 d2 w2S 1 dw2S þ r dr dr2
aðw1S w2S Þ ¼ 1;
ð79Þ
þ aðw1S w2S Þ ¼ 1:
ð80Þ
The solution of the above system of coupled ordinary differential equations, which satisfies the boundary conditions wbS ð1Þ ¼ 0 and remains finite at r ¼ 0, is obtained by employing the procedure used by the authors in Ref. [9]. In order to avoid repetition, the details are omitted and the flow fields are directly given by wbS ðrÞ ¼ cð1 r2 Þ þ cb fI0 ðmrÞ I0 ðmÞg;
ð81Þ
where c¼
1 ; 2g2
c1 ¼
g3 ðM2 þ M4 Þ ; ag22 I0 ðmÞ
c2 ¼
g3 ðM1 þ M3 Þ ; ag22 I0 ðmÞ
ð82Þ
where I0 is the modified Bessel function of the first kind of order zero. With these steady-state solutions, hb ðr; tÞ must satisfy the homogeneous differential equations M1 M3
o2 h1 1 oh1 þ r or or2 o2 h1 1 oh1 þ r or or2
þ M2
þ M4
o2 h2 1 oh2 þ r or or2 o2 h2 1 oh2 þ r or or2
aðh1 h2 Þ ¼ q1
oh1 ; ot
ð83Þ
þ aðh1 h2 Þ ¼ q2
oh2 ; ot
ð84Þ
with hb ð1; tÞ ¼ 0
and hb ð0; tÞ finite;
hb ðr; 0Þ ¼ wbS ðrÞ:
ð85Þ ð86Þ
Finite Hankel transform will be used to solve the above two simultaneous partial differential equations with the boundary and initial conditions (85) and (86). The finite Hankel transform of order n of a function f ðrÞ, defined on 0 < r < a, is [22, p. 318] Z a ^ rf ðrÞJn ðrki Þ dr: ð87Þ Hn ff ðrÞg ¼ f ðki Þ ¼ 0
The associated inverse transform is Hn1 ff^ðki Þg ¼ f ðrÞ ¼
1 2 X Jn ðrki Þ ; f^ðki Þ 2 a2 i¼1 Jnþ1 ðaki Þ
ð88Þ
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
2035
where Jn is the Bessel function of the first kind of order n, and the summation is taken over all the positive zeros k1 ; k2 ; k3 ; . . . of the Bessel function Jn ðaki Þ. The finite Hankel transforms of order zero of Eqs. (83) and (84), including Eq. (85)1 , respectively, are dh^1 ¼ v1 h^1 þ v2 h^2 ; dt
ð89Þ
dh^2 ¼ v3 h^1 v4 h^2 ; dt
ð90Þ
where v1 ¼
a þ M1 ki2 ; q1
v2 ¼
a M2 ki2 ; q1
v3 ¼
a M3 ki2 ; q2
v4 ¼
a þ M4 ki2 ; q2
ð91Þ
subject to the transform of Eq. (86), H0 fhb ðr; 0Þg ¼ h^b ðki ; 0Þ ¼
mcb mI0 ðmÞJ1 ðki Þ 2cJ2 ðki Þ þ I1 ðmÞJ0 ðki Þ : 2 2 ki ki2 ki þ m
ð92Þ
Since ki are the roots of J0 ðki Þ, we find 2cJ2 ðki Þ m2 cb I0 ðmÞJ1 ðki Þ h^b ðki ; 0Þ ¼ : ki ðki2 þ m2 Þ ki2
ð93Þ
The solution of the simultaneous equations (89) and (90) satisfying the initial conditions (93) can be written as h^1 ðki ; tÞ ¼ expf0:5ðv1 þ v4 Þtg ( ) ^ ^ pffiffiffiffi p ffiffiffi ffi 2v ðk ; 0Þ þ ðv v Þ h ðk ; 0Þ h 2 i 1 i 1 pffiffiffi4ffi
h^1 ðki ; 0Þ coshð0:5t /Þ þ 2 sinhð0:5t /Þ ; ð94Þ / h^2 ðki ; tÞ ¼ expf0:5ðv1 þ v4 Þtg ( ) ^ ^ pffiffiffiffi p ffiffiffi ffi 2v ðk ; 0Þ þ ðv v Þ h ðk ; 0Þ h 1 i 2 i 4 pffiffiffi1ffi
h^2 ðki ; 0Þ coshð0:5t /Þ þ 3 sinhð0:5t /Þ ; ð95Þ / where / ¼ v21 þ 4v2 v3 2v1 v4 þ v24 :
ð96Þ
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By means of Eq. (88), taking the inversions of Eqs. (94) and (95) and then inserting the results into Eq. (78), we get wb ðr; tÞ ¼ cð1 r2 Þ þ cb fI0 ðmrÞ I0 ðmÞg 2
1 X i¼1
J0 ðrki Þ h^b ðki ; tÞ 2 ; J1 ðki Þ
ð97Þ
where the summation is taken over all the positive zeros k1 ; k2 ; k3 ; . . . of the Bessel function J0 ðki Þ.
6. Numerical results and discussion In this paper some simple unsteady unidirectional flows relevant to a binary mixture of chemically inert incompressible Newtonian fluids are studied theoretically. Exact solutions for the system of coupled partial differential equations governing the velocity fields are obtained by using the finite Fourier sine and finite Hankel transforms. These solutions would be useful in checking the validity of complicated flows of such mixtures. In order to make predictions based on foregoing analysis, it is necessary to know all of the material functions in the constitutive equations. Determination of these functions for a mixture is much more difficult than that for a single continuum. Since the flow of mixtures is of great technical importance, a good amount of literature has grown up around the problem of determining these functions. For example, employing results obtained from the kinetic theory of fluids, Sampaio and Williams [23] were able to derive formulae for l1 , l2 , l3 and l4 in terms of the viscosities of the unmixed fluids and the volume fractions in the case of k5 ¼ 0. In this work we benefit from the formulae suggested in [23] with a view to assigning the reasonable values to M 1 , M 2 , M 3 and M 4 . To achieve this, at the outset we assume that the densities of unmixed fluids and the volume fractions are known. With the aid of Eqs. (3) and (9), knowledge of these quantities enables q1 , q2 and q0 to be calculated [10]. Later, the material coefficients l1 , l2 , l3 and l4 can be determined by using the formulae proposed in [23]. For the purpose of simulations, the following values are given to the dimensionless parameters: M 1 ¼ 0:32;
M 2 ¼ M 3 ¼ 0:22;
M 4 ¼ 0:68;
q1 ¼ 0:65;
q2 ¼ 0:35:
ð98Þ
In Figs. 1–8, for comparison, the velocity distributions for pure Newtonian fluid and constituents of binary mixture are plotted as a function of position for various values of a and t, keeping the remaining parameters fixed at the values given in Eq. (98). For the solutions corresponding to pure Newtonian fluid we refer the reader to Ref. [24, p. 239–250]. From these figures we observe how the velocity profiles grow with increasing time and approach asymptotically the steady-state velocity profiles. It is important to bear in mind that the assumptions of Eqs. (34), (56) and (78) are convenient, since the unsteady problems investigated here approach the steady solutions as t ! 1. On comparing Fig. 1 with Fig. 3 or Fig. 6 with Fig. 8, we arrive at the conclusion that with an increase in the coefficient of interaction a, characterized by the drag force between the two constituents, the mixture tends to behave as a single continuum. As a result, it is not difficult to predict that the fluid particles belonging to both constituents have the same velocity at a given point in the mixture as a ! 1.
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
2037
Fig. 1. Velocity profiles of Couette flow for a ¼ 10, t ¼ 0:01 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 2. Velocity profiles of Couette flow for a ¼ 10, t ¼ 0:2 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 3. Velocity profiles of Couette flow for a ¼ 150, t ¼ 0:01 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
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Fig. 4. Velocity profiles of plane Poiseuille flow for a ¼ 10, t ¼ 0:01 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 5. Velocity profiles of plane Poiseuille flow for a ¼ 10, t ¼ 0:25 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 6. Velocity profiles of axisymmetric Poiseuille flow for a ¼ 10, t ¼ 0:02 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
S. Barısß / International Journal of Engineering Science 40 (2002) 2023–2040
2039
Fig. 7. Velocity profiles of axisymmetric Poiseuille flow for a ¼ 10, t ¼ 0:45 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Fig. 8. Velocity profiles of axisymmetric Poiseuille flow for a ¼ 30, t ¼ 0:02 (––: pure Newtonian fluid, - - -: fluid 1, : fluid 2).
Finally we shall discuss reliability of the solutions given in Eqs. (49), (69) and (97). The velocity fields corresponding to the flows under consideration are in the form of series. As expected, these series are rapidly convergent for large values of time but slowly convergent for small values of time. It is important to note that series solutions (49), (69) and (97) can also be used for small values of time provided number of terms in the series expansions is enough to yield satisfactory accuracy. For example, for unsteady Couette flow, in the case of t ¼ 0:2, the fourth term is the first term in the series (49), absolute value of which is less than 108 . Therefore the sum of the first four term will give the velocity values of the bth fluid with an error of less than 108 . On the other hand, it is necessary to take account of the first 21 term for the same order of accuracy in the case of t ¼ 0:01. References [1] F. Dai, M.M. Khonsari, A theory of hydrodynamic lubrication involving the mixture of two fluids, J. Appl. Mech. 61 (1994) 634–641. [2] C. Truesdell, Sulle basi della thermomeccanica, Rend. Lincei 22 (8) (1957) 33–38, 158–166.
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[3] R.M. Bowen, Theory of mixtures, in: A.C. Eringen (Ed.), Continuum Physics, vol. III, Academic Press, New York, 1976. [4] R.J. Atkin, R.E. Craine, Continuum theories of mixtures. Basic theory and historical development, Quart. J. Mech. Appl. Math. 29 (1976) 209–244. [5] A. Bedford, D.S. Drumheller, Theory of immiscible and structured mixtures, Int. J. Engng. Sci. 21 (1983) 863–960. [6] K.R. Rajagopal, L. Tao, Mechanics of Mixtures, World Scientific Publishing, 1995. [7] N. Mills, Incompressible mixtures of Newtonian fluids, Int. J. Engng. Sci. 4 (1966) 97–112. [8] R.E. Craine, Oscillations of a plate in a binary mixture of incompressible Newtonian fluids, Int. J. Engng. Sci. 9 (1971) 1177–1192. [9] R.J. Atkin, R.E. Craine, Continuum theories of mixtures: Applications, J. Inst. Maths. Appl. 17 (1976) 153–207. [10] C.E. Beevers, R.E. Craine, On the determination of response functions for a binary mixture of incompressible Newtonian fluids, Int. J. Engng. Sci. 20 (1982) 737–745. €ßs , The influence of vibrating plates on Poiseuille flow of a binary mixture, Int. J. Engng. Sci. 26 (1988) [11] M.S ogu ß . G€ 313–323. €ßs , The influence of a vibrating plate on the flow between parallel plates of a binary mixture, Int. J. [12] M.S ogu ß . G€ Engng. Sci. 29 (1991) 1651–1659. €ßs , The influence of a longitudinal vibrating pipe on Poiseuille flow of a binary mixture, Int. J. Engng. [13] M.S ogu ß . G€ Sci. 30 (1992) 141–151. €ßs , The steady flow of a binary mixture between two rotating parallel non-coaxial disks, Int. J. Engng. [14] M.S ogu ß . G€ Sci. 30 (1992) 665–677. € ßs , The velocity profile in pulsatile flow of a binary mixture, Int. J. Engng. Sci. 32 (1994) 705–714. [15] M.S ogu ß . G€ €ßs , The steady flow of a binary mixture between two rotating parallel coaxial disks, Int. J. Engng. Sci. 33 [16] M.S ogu ß . G€ (1995) 611–624. [17] A. Al-Sharif, K. Chamniprasart, K.R. Rajagopal, A.Z. Szeri, Lubrication with binary mixtures: Liquid–liquid emulsion, J. Tribol. 115 (1993) 46–55. [18] K. Chamniprasart, A. Al-Sharif, K.R. Rajagopal, A.Z. Szeri, Lubrication with binary mixtures: Bubbly oil, J. Tribol. 115 (1993) 253–260. [19] S.H. Wang, A. Al-Sharif, K.R. Rajagopal, A.Z. Szeri, Lubrication with binary mixtures: Liquid–liquid emulsion in an EHL conjunction, J. Tribol. 115 (1993) 515–522. [20] G. Johnson, M. Massoudi, K.R. Rajagopal, Flow of a fluid–solid mixture between flat plates, Chem. Engng. Sci. 46 (1991) 1713–1723. [21] G. Johnson, M. Massoudi, K.R. Rajagopal, Flow of a fluid infused with solid particles through a pipe, Int. J. Engng. Sci. 29 (1991) 649–661. [22] L. Debnath, Integral Transforms and their Applications, CRC Press, 1995. [23] R. Sampaio, W.O. Williams, On the viscosities of liquid mixtures, Z. Angew. Math. Phys. (ZAMP) 28 (1977) 607– 613. [24] T.C. Papanastasiou, G.C. Georgiou, A.N. Alexandrou, Viscous Fluid Flow, CRC Press, 2000.