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The velocity profile in pulsatile flow of a binary mixture
Int. 1. EngngSci. Vol. 32. No. 4, pp. 705-714, 1994 Copyright@) 1994 Elscvicr ScienceLtd Printed in Great Britain. All rightsreserved 0020-7225/W $6.00 + 0.00
Pergaaon
LETTERS IN APPLIED
AND ENGINEERING
THE VELOCITY PROFILE IN PULSATILE BINARY MIXTURE
SCIENCES
FLOW OF A
M. SINASI GbG;ijS Division of Mechanics, Faculty of Mechanical Engineering, Istanbul Technical University, Giimiigsuyu, Istanbul, Turkey Absiract-A mixture of two incompressible Newtonian fluids is considered, and the flow induced by the simple harmonic motion of pressure-gradient on poiseuille flow in a uniform circular pipe is then investigated. Explicit results are obtained in the limiting cases in which a suitable non-dimensionalized frequency is very small or every large. The flow is not steady, and thus, it is to be anticipated that the velocity profile will not be of the same parabolic form that is found in steady laminar flow.
1.
INTRODUCTION
Recently there have been two alternative approaches to the theory of non-linear diffusion and flow of mixture of continuous media. In the first theory, given by Truesdell and Toupin [l], equations of mass, momentum, moment of momentum and energy balance are basic postulates for each ingredient in the mixture. The second theory, given by Green and Naghdi [2], uses only an equation of energy balance and an entropy production inequality as basic postulates. Coincidentally, a paper by Eringen and Ingram has been published which presents a unified approach for the derivation of basic conservation equations for a chemically reacting continuum [3]. Several other papers also allow for the interchange of mass between the constituents of a mixture. In all cases there are similarities with the formulation suggested in Atkin and Craine [4] but differences occur, usually in the thermodynamic arguments adopted. There is now considerable agreement amongst the various authors on the basic balance equations for mixtures with either single or multiple temperatures, and a number of problems have been solved. These solutions arise from applications to a wide range of mixtures and, in all the cases so far considered, the non-linear theories can be readily simplified to the appropriate classicat ones. Lastly we mention the time-dependent problems which have been investigated by Craine [S, 61 and Gogus (7-111. In the work described here a binary mixture of incompressible Newtonian fluids is considered. In Sections 2 and 3 we use the governing field equations of the continuum theory of mixtures and notations given by Craine [5,6] and Mills [12]. The appropriate constitutive theory developed by the authors mentioned above is given in Section 3 [13,14]. in Sections 4, 5 and 6 of the present paper, the flow induced by the simple harmonic motion of pressure-gradient on poiseuille flow in a uniform circular pipe is investigated and an exact solution is obtained, but explicit results are obtainable only in the low- and high-frequency limits. Throughout in this work, it will be assumed that body forces are absent and that the motion of the mixed fluid takes place under isothermal conditions.
2.
BASIC
THEORY
The governing equations are summarized in this section, for more details the reader should consult Craine [5]. A mixture of two continua, in motion relative to each other, is considered. 705
At an arbitrary time t it is assumed that at each place t in the region of space occupied by the mixture, two particles are situated, one belonging to each constituent. This is a basic assumption of the continuum theory of mixtures and its validity requires that the mixture appears homogeneous when viewed on the scale of the applied disturbance. If vca) denotes the velocity of the cvth constituent, the derivative D’“‘/Dt is defined by: W) a -=_+ul,& L>t dt
(2.1)
dXk’
where cy takes the values 1, 2. The governing kinematic equations and the energy equation for a binary mixture in which the constituents have a common temperature T and do not interact chemically are:
(2.4)
The quantities pn, U,, ucQ), F’“’ are in turn the density, internal energy per unit mass, partial stress and body force per unit mass of the arth constituent. In addition r. the heat supply per unit volume, and q, the heat flux vector, refer to the mixture as a whole, and o denotes the diffusive force vector 1151. The partial stresses satisfy: o@, = a&,
(2.6)
where suffixes enclosed in square brackets refer to components of the antisymmetric part of a second-order tensor and circular brackets have a similar meaning in relation to the symmetric part. If S, denotes the partial entropy then the Clausius-Duhem inequality may be written [16]: D”‘S
‘I
(2.7)
----%p,---
Dt
where f, and S2 are the entropies per unit mass of the l- and 2-constituents, 3. MECHANICAL
CONSTITUTIVE
respectively.
THEORY
The formulation of constitutive equations for a mixture of Newtonian fluids given by Craine [5] is now summarized. If both constituents are incompressible with densities p,,,, p2(, when separated, we have: (P2,t - P)d!‘k’ + (P - Pl&$
- &% = 0,
(3. I)
Ok = u!” - uy,
(3.2)
the quantities 5 and d(“) being defined by: Sk = P.!s>
(jj;‘=
yJ
and P=Pl+Pz?
p, =I PdPm p20
p2 = PdP
- PI -
PII,
’
- PI01
Pm - PII1
(3.3)
Introducing a partial Helmholtz free energy A, through A, = U, - Ts,,
then as the basic constitutive postulate we assume that the quantities A,,
(3.4) S,, w, q and of*) are
707
Letters in Applied and Engineering Sciences
arbitrary functions of p but depend linearly on 5, a, dtp) and r, where: n&“’ = &fl&k13
F, = i&’ - r@.
(3.5)
Adopting an analogous method to that outlined in Ref. [S], it is possible to deduce from the Clausius-Duhem inequality that the constitutive equations have the reduced forms:
(3.7) @I-=
iWi
-
?bgj;it
(3.8)
qj = - Tkoai,
(3.9)
@ = -P&j,, + 2yld$ + 2,usdj:’ f n,r,, &’ = -Pzfiik + 2~~d~~’+ 2pzd$z’ - h$,,
(3.10) (3.11)
where the coefficients o, ko, pl, . +. , p4 are functions of p, and satisfy the inequalities: iYa0,
CL1 20, i%!*:o,
(5%+
P4Y
6
(3.12)
4EllF62.
4. FLOW INDUCED BY HARMONIC MOTION OF A PRESSURE-GRADIENT ON THE POISEUILLE FLOW OF A BINARY MIXTURE In this section we shall consider a binary mixture of incompressible Newtonian fluids flowing through a uniform pipe of circular cross-section, with radius a, under the action of a simple harmonic motion of pressure-gradient in a direction parallel to its length. Let Vti) be the velocity of the fluids in a cylindrical polar coordinate system. It will be assumed that body forces are absent and that the motion of the mixed fluid takes place under isothermal conditions. The second of these requirements is strictly satisfied only if the heat created by viscous dissipation and diffusive resistance is extracted through agency of an appropriate negative heat supp1y.t The physical artificiality of this situation is of little account so long as the rate of dissipation of energy is “small” and on this argument rests the standard practice of regarding the energy equation as irrelevant in problems of “slow” viscous flow. Since the temperature T is no longer a variable, the coefficients 1-1,etc. appearing in the constitutive equations depend only on the total density p [5]. For simplicity the form of the pressure-gradients will be taken as a simple harmonic motion and written in complex form fA” and B” are the complex conjugates of A and B):
2 -_
A
*e’“‘,
(4-l)
where w = 21tf is the angular frequency in radlsec of the oscillatory motion, with f the frequency in Hz. The solution for this case is especially important because, with the aid of the Fourier series, any periodic function, such as the arterial pulse, can be represented 1171. We assume a flow field of the form u1 = u(r)@“, u1 = v(r)e’““‘:
d’) = [O, 0, u(rpq,
d2)= [O, 0, u(r)e”“‘],
P = p(r,
t),
(4.2)
where CIand v are functions of r. Then, the no-slip boundary conditions yield: n(‘)z:u(‘)=O
on
r=a,
v\‘) and vi2) must be finite when r = 0, .~S~~stitution of the solution (4.2) into the residual energy eq~~tiun (2.5) gives:
_.
(4.3)
. . ._._, _.._
M. !jINASI G66i$
708
On substituting equations (4.2) into equations (2.2) we find that: P = p(r).
(4.4)
Using (4.1) (4.2) and (4.4) the constitutive equations (3.7-3.11) are greatly simpli~ed and it follows that the kinematic equations with the physical components of the stress in a cylindrical polar coordinate system r, 8, z become: (4.5) (4.6) (4.7) (4.8)
It is easily shown from (4.7) that h is independent
of 0. The most general form of A is: (4.10)
with the use of (3.3), elimination of dA/dr between (4.5) and (4.6) gives, after manipulation (for discussion of this result see Craine [S]),
some
(4.11) since, in generai p + prcl, p # p2() and d2(~A)fd~2 f0 we must have dpldr = 0, whence p = pII, a constant. From (4.5) and (4.6) it follows that hr = h,,), also a constant. Since p has been proved to be constant and isothermal conditions have been assumed, all the coefficients in equations (4.8) and (4.9) are constant. It is convenient at this point to introduce dimensionless variabies and material constants defined by:
A* = A*faV,
B* = B*/aV,
(4.12)
we find the dimensionless forms:
replacing (4.13) and (4.14) by the sum of (4.13) and (4.14) the surviving pair of governing equations can be written as [lo]:
(4.15)
(4.16)
709
Letters in Appliedand EngineeringSciences
Now let 9 = R,)(tZF)be a solution of the modified Bessel’s differential equation of order zero, so that 7 satisfies: (4.18)
$++=0.
If we use (4.18) it follows that the homogeneous differential equations formed from (4.15) and (4.16) may be satisfied by solution of the form [lo]: (4.19)
ij = DR&P),
r; = CR&),
where C and D are complex constants, proving that:
(4.20)
D(r2- A')-F2C =O.
C(Q2 - AZ)- E2D = 0, Equations (4.20) are consistent if:
(4.21)
A4+P,A2+/12=0, where fit={-(-
111 P,+jiz+k+P4)--~~
(
82
=
(-W2f12
PlP2+F2Pl+-2
+
s
Bn,
in)//%,
(4.22) Solution of biquadratic equation (4.21) for A with roots [5]: +s,
+it, +-p,
+@f-4&)1/2},
A2= s2 + it2= i (--p, - (/3f - 4/32)“2},
(4.23)
s, and t, being real constants. The complex numbers s, f it, have unique square roots with negative real parts, say -1, + im,, where: I, = 2-'"{(s2,+ t",)'" +s*}"2,
we denote the values of k2 given by (4.21) by A: and &$, so that the homogeneous ii and b are given by: G = C,,i,,((--I,
solutions for
+ im,)?} + im,)?} + C,2Kl,{-(-11 +
8=
(4.24)
m, = -t*/21,,
D,,l,,{(-1, -Iim,)?} + D,,&,{-(-1,
C,,l,,{(--l, + im2)F} + C,,&{-(-I,+ irQF>,
+ im,)?} + Dz,{(-12+im2)F}
+ D22Ko{-(-12+im2)~},
(4.25)
Daiare related to C, by the conditions [lo, 121: C,i( Q’ - A”,)- E'Dmi= 0,
D&t'- A",)F=C,,50,
(4.26)
where (Y= I, 2; i = 1, 2 ((Y not summed). In order to obtain a solution which is finite when F = 0, it is necessary to take Cbz = Cz2 = D,,= Dz2= 0.The complete solutions for ii and 0 are
M. SINAS
710
Cibt.Ii;iis
given by: B=C,,4,{(-f,
+im,)r}
+c*,4~{(-~~+~~*)~}
ti = D,,4,{(-1,
+ im,)P} + &I,,{(-12
+
f im,)F} +
(A* + P)
+ ijQ&*
j3,p2n2 - in
’
(A* + B*) + ip,nB* p,p2n2 - in
(4.27)
*
The complete solutions contain four constants C&iand Da;. This may easily be determined impose the boundary conditions: G=jj=O ii
and
on
if we
f=G,
ti must be finite with F = 0.
(4.28)
With appIying the boundary conditions (4.28), these equations have the unique solution: c
=
(A* + b* + i&d*). (A:
A$).
in). I,,{(-I,
(p,p,n’-
@*+ l?* + i&&i*).
c,,=D
-
(Q’-
v: - G) * (P,P2 _
(A* + B* + @,nE*)
D,,=-
-
A:)
+ im,)d}
’
Ai) - (A* + l?* + @,nL?*) . E2
n2
in) . I,,{(-l,
-
+ im,)E}
’ F2
’ ( t2 - #If> - (A*&+ + i&d*).
21 (Af
. E2
A:) - (A* + l?* + iP,nl?*)
(Q’-
21
* (p,p2nZ
-
(A* + B* + ip,nB*) . (t’(A: - k$(p,jj,n’
in) . I,,{ ( -12 -t imz)d}
’
Al) - (A* + l?* + iP2nA*). F2 - in) - &{ (-f,
+ im,)ci}
(4.29)
’
where p,p2n2 - in # 0,
n:-a;$:o,
(4.30)
PO# 0,
s,, s2, t, and t2 being real constants, we may calculate from equations (4.22), (4.23) and (4.24) (for discussion of this latter calculation see Gogus [7]). Substitution of solution (4.29) into the equation (4.27) gives: A* + S” + iji,nA*
1 + -. Q2 - Af 4,{(-l2 + im2)F} -p.Q’ - A$ I,d(--l, + im,)F) Af- h2, I,,{(--I, +im,)G} A: - A’, 4,{(-I2 + im,)~} E2
I,,{(--/,
im,)F}
+
+ kt - As { ],,{(--I, + h&i)
I,,{(-I2
+
- -
PIP2n
* --In *
A” + B” + ip,nE*
im2)F>
- I,,{(--I2 + im&i} I( .
p,&n2 -in
I*
ij =t l + t* - AT Z,,{(-f, + im2)7} r2 - AZ I,,{(-I, + im,)F) + A* + B* + @,nB* --. -. A::- A: I,,{(-1,+im,)d} A: - AZ I,,{(--I, +im,)5} I I 1. c - . fw2nZ - En
+ F2 4,{(-f, + im,)F} 4,{(--f, + im)r) A$&{(-f,
A:-
+im,)G}
(4.31)
-4,{(--~2+im2)~~
’
C,=P2, &=ii4- - 2 5- - 0 and A* = fi” results reduce to the familiar case of a single incompressible Newtonian fluid.
when PI=P2, iht=P20r
5. APPROXIMATE
Low- and high-frequency (4.31). In the low-frequency given by:
SOLUTION
FOR
LOW
AND
HIGH
FREQUENCIES
approximations to 6 and V can be derived from the expressions limit n << I the first approximations to U and V are identical, being n
-.-=(A*+Bf)
>
@,+p2+i&+F4
in
n >
I
’ {I
+0(n)}.
6
1)
711
Letters in Applied and Engineering Sciences
The velocities of the two fluids therefore coincide and the mixture behaves as a single fluid with average properties. Some difficulty is experienced in obtaining the high-frequency limit n >> 1. Consider first the special case of a mixture for which j& = p4 = & = 0. In this case the partial stresses are symmetric and each one depends only on the velocity gradient of that constituent, no error-terms being included. The expressions (4.31) for ii and fi are then simplified, reducing to: “’ (l+i) F I( > 77
4,
A-* ip,n
(( - )I’* I,, np, i
PI
(l+i) 777
_ I
_ 1
‘.
(5.2)
The velocities of the individual constituents are now seen to be different, and in this limit each fluid moves independently form the other. When 0, = p2, pi = p2, pi0 = p2(,, j& = p4 = & = 0 and A* = t?*, the results reduce to the familiar case of a single incompressible Newtonian fluid [8,18, 191: (i) n << 1. In this case (4.13) and (4.14) imply ti - V = O(n), and so ii = 0 to first approximation. (ii) n >> 1. All the terms in (4.13) and (4.14) except ii - V are O(n) and therefore much larger than ij - 0. In the leading approximation ii - fi can hence be neglected [18]. The approximate solutions (5.1) and (5.2) are concerned with complex quantities and modified Bessel functions of order zero. Real and imaginary parts of a Bessel function of order zero are given by [19,20]: @vJ = ber( vT) + i bei( vf) &(@a) ber(vZ) + i bei(vci)’
(5.3)
where @=$(l+i).
6.
NUMERICAL
RESULTS
(5.4)
AND DISCUSSION
Numerical results are obtained for the variation with distance of velocities of the constituent fluids for fixed dimensionless times (d = mn/6; m = 0, 1, 312, 2, 3, 4, 6) and certain fixed frequencies of vibration of pressure-gradient. In all of the curves shown, certain parameters are fixed (p,=l, p2=1, 4, F3=1/2, j&=1/2, &=-l/2, p,=5/6, c5=1, A*=B*) and the remaining parameters take given values [6-8, 18, 191. Moreover, it is approximated by: I,,(X) = 1 + c + g
for small values of X.
The variation of the velocities of the fluid caused by the simple harmonic motion of pressure-gradient on poiseuille flow in a uniform circular pipe is investigated for n = 1 (Figs 1, 4), n = 10 (Figs 2, 5), n = 50 (Figs 3, 6) all at nf = mn/6. As it may be expected, the velocity difference is more marked in the different harmonic components (see Figs l-4, Figs 2-5, Figs 3-6). As the frequency increases there is a parabolic profile at any time, for example, in Fig. 2.
712
Fig. 1. Variation of fi, and U, with distance; n = 1, A; = -1, A; ~0.
90
60
4s
30
Fig. 2. Variation of ti, and 0, with distance;n=lO,A;=-l,A$=O.
60
Fig. 3. Variation of P, and CI,with distance; n = 50, Ar = -1, & = 0.
180
PA 180
120
60 45 30 0 5,
Fig. 4. Variation of E, and D, with distance; n = 1, /if = -1, AI = 1.
Fig. 5. Variation of 0, and D, with distance; n = 10, A: = -1, A: = 1 I80 180 120 90 y
A \ 60
60
Fig. 6. Variation of ii, and D, with distance; n = 50, A; = -1, Al = 1. 713
714
M. SINAS
G6GoS
In the numerical results, the radius of the pipe was taken sufficiently small and the harmonics were taken as A* = B* = A: - (A: in the calculations). As can be seen from Fig. 1, for n = 1 and A: = -1, A: = 0, the mixture behaves as a single continuous medium at low frequencies. But, as shown in Fig. 4, for the AT = -1 and A: = 1 values of the harmonic components, the two continuous media flow independently at low frequencies. It is also seen from Figs 3 and 6 that high-frequency decreases the difference in velocity resulting from the harmonic components being different. Pumping systems, heat and mass transfer and blood circulation in living creatures can be shown as the most important applications related to pulsative flows at present.
REFERENCES [l] C. TRUESDELLand R. TOUPIN, The Classical Field Theories, Handbuck der Physik (Edited by S. FLUDGE), Band 11 l/l Springer-Verlag, Berlin (1960). [2] A. E. GREEN and P. M. NAGHDI, Inr. J. Engng Sci. 3,231 (1965). [3] A. C. ERINGENand J. D. INGRAM, lnt. /. Engng Sci. 3, 197 (1965). [4] R. J. ATKINand R. E. CRAINE, Q. J, Me&. AppL Math. 29,209 (1976). [5] R. E. CRAINE, Inr. /. Engng Sci. 9, 1177 (1971). (61 R. E. CRAINE,/. Appl. Math. Phys (ZAMP) 24, 365 (1973). [7] M. S. GOGUS, Inr. J. Engng Sci. M, 313 (1988). [8] M. !j. GqG$$ Bull. Tech. Llniu. Istanbul 40, 789 (1987). [9] M. 9. GqGv$, Int. J. Engng Sci. 29, 1651 (1991). [lo] M. S. GGGUS, Int. J. Engng Sci. 30, 141 (1992). [ll] M. S. GOGUS, Int. J. Engng Sci. 30, 665 (1992). [12] N. MILLS, Inr. J. Engng Sci. 4, 97 (1966). 1131 A. E. GREEN and P. M. NAGHDI, Q. /. Mech.Math. 22, 427 (1969). [14] R. E. CRAINE. A. E. GREEN and P. M. NAGHDI, Q. J. Mech. Appl. Math. 23, 171 (1970). [15] R. M. BOWEN and D. T. GARCIA, Inf. /. Engng Sci. 5, 289 (1967). [16] R. M. BOWEN, J. Chem. Phys. 49, 1625 (1968). [ 17) D. A. MCDONALD, Blood Flow in Arferies. Edward Arnold, London (1960). 1181 J. D. INGRAM and A. C. ERINGEN, In!. J. Engng Sci. 5, 289 (1967). (191 J. Y. KAZAKIA and R. S. RIVLIN, Rheol. Acta 17, 210 (1978). 120) J. Y. KAZAKIA and R. S. RIVLIN, Rheol. Acta 18,224 (1979). (Received and accepted 20 April 1993)
APPENDIX The rate at which the mixture
flows through
the pipe is given by: Q = Q ,e’“‘,
(Al)
Q,=2+(P,ri+&it)df
(A2)
Thus we have:
g=fi
[
(AC,, + P2D11NK-~I +im,Pl+(PlC2, +~2021)11[(-12+im,)81+
d. (A* + B*) 2
1+ip,&n . [ j5,&n2-in
II.ein’, (A3)
Pergaaon
LETTERS IN APPLIED
AND ENGINEERING
THE VELOCITY PROFILE IN PULSATILE BINARY MIXTURE
SCIENCES
FLOW OF A
M. SINASI GbG;ijS Division of Mechanics, Faculty of Mechanical Engineering, Istanbul Technical University, Giimiigsuyu, Istanbul, Turkey Absiract-A mixture of two incompressible Newtonian fluids is considered, and the flow induced by the simple harmonic motion of pressure-gradient on poiseuille flow in a uniform circular pipe is then investigated. Explicit results are obtained in the limiting cases in which a suitable non-dimensionalized frequency is very small or every large. The flow is not steady, and thus, it is to be anticipated that the velocity profile will not be of the same parabolic form that is found in steady laminar flow.
1.
INTRODUCTION
Recently there have been two alternative approaches to the theory of non-linear diffusion and flow of mixture of continuous media. In the first theory, given by Truesdell and Toupin [l], equations of mass, momentum, moment of momentum and energy balance are basic postulates for each ingredient in the mixture. The second theory, given by Green and Naghdi [2], uses only an equation of energy balance and an entropy production inequality as basic postulates. Coincidentally, a paper by Eringen and Ingram has been published which presents a unified approach for the derivation of basic conservation equations for a chemically reacting continuum [3]. Several other papers also allow for the interchange of mass between the constituents of a mixture. In all cases there are similarities with the formulation suggested in Atkin and Craine [4] but differences occur, usually in the thermodynamic arguments adopted. There is now considerable agreement amongst the various authors on the basic balance equations for mixtures with either single or multiple temperatures, and a number of problems have been solved. These solutions arise from applications to a wide range of mixtures and, in all the cases so far considered, the non-linear theories can be readily simplified to the appropriate classicat ones. Lastly we mention the time-dependent problems which have been investigated by Craine [S, 61 and Gogus (7-111. In the work described here a binary mixture of incompressible Newtonian fluids is considered. In Sections 2 and 3 we use the governing field equations of the continuum theory of mixtures and notations given by Craine [5,6] and Mills [12]. The appropriate constitutive theory developed by the authors mentioned above is given in Section 3 [13,14]. in Sections 4, 5 and 6 of the present paper, the flow induced by the simple harmonic motion of pressure-gradient on poiseuille flow in a uniform circular pipe is investigated and an exact solution is obtained, but explicit results are obtainable only in the low- and high-frequency limits. Throughout in this work, it will be assumed that body forces are absent and that the motion of the mixed fluid takes place under isothermal conditions.
2.
BASIC
THEORY
The governing equations are summarized in this section, for more details the reader should consult Craine [5]. A mixture of two continua, in motion relative to each other, is considered. 705
At an arbitrary time t it is assumed that at each place t in the region of space occupied by the mixture, two particles are situated, one belonging to each constituent. This is a basic assumption of the continuum theory of mixtures and its validity requires that the mixture appears homogeneous when viewed on the scale of the applied disturbance. If vca) denotes the velocity of the cvth constituent, the derivative D’“‘/Dt is defined by: W) a -=_+ul,& L>t dt
(2.1)
dXk’
where cy takes the values 1, 2. The governing kinematic equations and the energy equation for a binary mixture in which the constituents have a common temperature T and do not interact chemically are:
(2.4)
The quantities pn, U,, ucQ), F’“’ are in turn the density, internal energy per unit mass, partial stress and body force per unit mass of the arth constituent. In addition r. the heat supply per unit volume, and q, the heat flux vector, refer to the mixture as a whole, and o denotes the diffusive force vector 1151. The partial stresses satisfy: o@, = a&,
(2.6)
where suffixes enclosed in square brackets refer to components of the antisymmetric part of a second-order tensor and circular brackets have a similar meaning in relation to the symmetric part. If S, denotes the partial entropy then the Clausius-Duhem inequality may be written [16]: D”‘S
‘I
(2.7)
----%p,---
Dt
where f, and S2 are the entropies per unit mass of the l- and 2-constituents, 3. MECHANICAL
CONSTITUTIVE
respectively.
THEORY
The formulation of constitutive equations for a mixture of Newtonian fluids given by Craine [5] is now summarized. If both constituents are incompressible with densities p,,,, p2(, when separated, we have: (P2,t - P)d!‘k’ + (P - Pl&$
- &% = 0,
(3. I)
Ok = u!” - uy,
(3.2)
the quantities 5 and d(“) being defined by: Sk = P.!s>
(jj;‘=
yJ
and P=Pl+Pz?
p, =I PdPm p20
p2 = PdP
- PI -
PII,
’
- PI01
Pm - PII1
(3.3)
Introducing a partial Helmholtz free energy A, through A, = U, - Ts,,
then as the basic constitutive postulate we assume that the quantities A,,
(3.4) S,, w, q and of*) are
707
Letters in Applied and Engineering Sciences
arbitrary functions of p but depend linearly on 5, a, dtp) and r, where: n&“’ = &fl&k13
F, = i&’ - r@.
(3.5)
Adopting an analogous method to that outlined in Ref. [S], it is possible to deduce from the Clausius-Duhem inequality that the constitutive equations have the reduced forms:
(3.7) @I-=
iWi
-
?bgj;it
(3.8)
qj = - Tkoai,
(3.9)
@ = -P&j,, + 2yld$ + 2,usdj:’ f n,r,, &’ = -Pzfiik + 2~~d~~’+ 2pzd$z’ - h$,,
(3.10) (3.11)
where the coefficients o, ko, pl, . +. , p4 are functions of p, and satisfy the inequalities: iYa0,
CL1 20, i%!*:o,
(5%+
P4Y
6
(3.12)
4EllF62.
4. FLOW INDUCED BY HARMONIC MOTION OF A PRESSURE-GRADIENT ON THE POISEUILLE FLOW OF A BINARY MIXTURE In this section we shall consider a binary mixture of incompressible Newtonian fluids flowing through a uniform pipe of circular cross-section, with radius a, under the action of a simple harmonic motion of pressure-gradient in a direction parallel to its length. Let Vti) be the velocity of the fluids in a cylindrical polar coordinate system. It will be assumed that body forces are absent and that the motion of the mixed fluid takes place under isothermal conditions. The second of these requirements is strictly satisfied only if the heat created by viscous dissipation and diffusive resistance is extracted through agency of an appropriate negative heat supp1y.t The physical artificiality of this situation is of little account so long as the rate of dissipation of energy is “small” and on this argument rests the standard practice of regarding the energy equation as irrelevant in problems of “slow” viscous flow. Since the temperature T is no longer a variable, the coefficients 1-1,etc. appearing in the constitutive equations depend only on the total density p [5]. For simplicity the form of the pressure-gradients will be taken as a simple harmonic motion and written in complex form fA” and B” are the complex conjugates of A and B):
2 -_
A
*e’“‘,
(4-l)
where w = 21tf is the angular frequency in radlsec of the oscillatory motion, with f the frequency in Hz. The solution for this case is especially important because, with the aid of the Fourier series, any periodic function, such as the arterial pulse, can be represented 1171. We assume a flow field of the form u1 = u(r)@“, u1 = v(r)e’““‘:
d’) = [O, 0, u(rpq,
d2)= [O, 0, u(r)e”“‘],
P = p(r,
t),
(4.2)
where CIand v are functions of r. Then, the no-slip boundary conditions yield: n(‘)z:u(‘)=O
on
r=a,
v\‘) and vi2) must be finite when r = 0, .~S~~stitution of the solution (4.2) into the residual energy eq~~tiun (2.5) gives:
_.
(4.3)
. . ._._, _.._
M. !jINASI G66i$
708
On substituting equations (4.2) into equations (2.2) we find that: P = p(r).
(4.4)
Using (4.1) (4.2) and (4.4) the constitutive equations (3.7-3.11) are greatly simpli~ed and it follows that the kinematic equations with the physical components of the stress in a cylindrical polar coordinate system r, 8, z become: (4.5) (4.6) (4.7) (4.8)
It is easily shown from (4.7) that h is independent
of 0. The most general form of A is: (4.10)
with the use of (3.3), elimination of dA/dr between (4.5) and (4.6) gives, after manipulation (for discussion of this result see Craine [S]),
some
(4.11) since, in generai p + prcl, p # p2() and d2(~A)fd~2 f0 we must have dpldr = 0, whence p = pII, a constant. From (4.5) and (4.6) it follows that hr = h,,), also a constant. Since p has been proved to be constant and isothermal conditions have been assumed, all the coefficients in equations (4.8) and (4.9) are constant. It is convenient at this point to introduce dimensionless variabies and material constants defined by:
A* = A*faV,
B* = B*/aV,
(4.12)
we find the dimensionless forms:
replacing (4.13) and (4.14) by the sum of (4.13) and (4.14) the surviving pair of governing equations can be written as [lo]:
(4.15)
(4.16)
709
Letters in Appliedand EngineeringSciences
Now let 9 = R,)(tZF)be a solution of the modified Bessel’s differential equation of order zero, so that 7 satisfies: (4.18)
$++=0.
If we use (4.18) it follows that the homogeneous differential equations formed from (4.15) and (4.16) may be satisfied by solution of the form [lo]: (4.19)
ij = DR&P),
r; = CR&),
where C and D are complex constants, proving that:
(4.20)
D(r2- A')-F2C =O.
C(Q2 - AZ)- E2D = 0, Equations (4.20) are consistent if:
(4.21)
A4+P,A2+/12=0, where fit={-(-
111 P,+jiz+k+P4)--~~
(
82
=
(-W2f12
PlP2+F2Pl+-2
+
s
Bn,
in)//%,
(4.22) Solution of biquadratic equation (4.21) for A with roots [5]: +s,
+it, +-p,
+@f-4&)1/2},
A2= s2 + it2= i (--p, - (/3f - 4/32)“2},
(4.23)
s, and t, being real constants. The complex numbers s, f it, have unique square roots with negative real parts, say -1, + im,, where: I, = 2-'"{(s2,+ t",)'" +s*}"2,
we denote the values of k2 given by (4.21) by A: and &$, so that the homogeneous ii and b are given by: G = C,,i,,((--I,
solutions for
+ im,)?} + im,)?} + C,2Kl,{-(-11 +
8=
(4.24)
m, = -t*/21,,
D,,l,,{(-1, -Iim,)?} + D,,&,{-(-1,
C,,l,,{(--l, + im2)F} + C,,&{-(-I,+ irQF>,
+ im,)?} + Dz,{(-12+im2)F}
+ D22Ko{-(-12+im2)~},
(4.25)
Daiare related to C, by the conditions [lo, 121: C,i( Q’ - A”,)- E'Dmi= 0,
D&t'- A",)F=C,,50,
(4.26)
where (Y= I, 2; i = 1, 2 ((Y not summed). In order to obtain a solution which is finite when F = 0, it is necessary to take Cbz = Cz2 = D,,= Dz2= 0.The complete solutions for ii and 0 are
M. SINAS
710
Cibt.Ii;iis
given by: B=C,,4,{(-f,
+im,)r}
+c*,4~{(-~~+~~*)~}
ti = D,,4,{(-1,
+ im,)P} + &I,,{(-12
+
f im,)F} +
(A* + P)
+ ijQ&*
j3,p2n2 - in
’
(A* + B*) + ip,nB* p,p2n2 - in
(4.27)
*
The complete solutions contain four constants C&iand Da;. This may easily be determined impose the boundary conditions: G=jj=O ii
and
on
if we
f=G,
ti must be finite with F = 0.
(4.28)
With appIying the boundary conditions (4.28), these equations have the unique solution: c
=
(A* + b* + i&d*). (A:
A$).
in). I,,{(-I,
(p,p,n’-
@*+ l?* + i&&i*).
c,,=D
-
(Q’-
v: - G) * (P,P2 _
(A* + B* + @,nE*)
D,,=-
-
A:)
+ im,)d}
’
Ai) - (A* + l?* + @,nL?*) . E2
n2
in) . I,,{(-l,
-
+ im,)E}
’ F2
’ ( t2 - #If> - (A*&+ + i&d*).
21 (Af
. E2
A:) - (A* + l?* + iP,nl?*)
(Q’-
21
* (p,p2nZ
-
(A* + B* + ip,nB*) . (t’(A: - k$(p,jj,n’
in) . I,,{ ( -12 -t imz)d}
’
Al) - (A* + l?* + iP2nA*). F2 - in) - &{ (-f,
+ im,)ci}
(4.29)
’
where p,p2n2 - in # 0,
n:-a;$:o,
(4.30)
PO# 0,
s,, s2, t, and t2 being real constants, we may calculate from equations (4.22), (4.23) and (4.24) (for discussion of this latter calculation see Gogus [7]). Substitution of solution (4.29) into the equation (4.27) gives: A* + S” + iji,nA*
1 + -. Q2 - Af 4,{(-l2 + im2)F} -p.Q’ - A$ I,d(--l, + im,)F) Af- h2, I,,{(--I, +im,)G} A: - A’, 4,{(-I2 + im,)~} E2
I,,{(--/,
im,)F}
+
+ kt - As { ],,{(--I, + h&i)
I,,{(-I2
+
- -
PIP2n
* --In *
A” + B” + ip,nE*
im2)F>
- I,,{(--I2 + im&i} I( .
p,&n2 -in
I*
ij =t l + t* - AT Z,,{(-f, + im2)7} r2 - AZ I,,{(-I, + im,)F) + A* + B* + @,nB* --. -. A::- A: I,,{(-1,+im,)d} A: - AZ I,,{(--I, +im,)5} I I 1. c - . fw2nZ - En
+ F2 4,{(-f, + im,)F} 4,{(--f, + im)r) A$&{(-f,
A:-
+im,)G}
(4.31)
-4,{(--~2+im2)~~
’
C,=P2, &=ii4- - 2 5- - 0 and A* = fi” results reduce to the familiar case of a single incompressible Newtonian fluid.
when PI=P2, iht=P20r
5. APPROXIMATE
Low- and high-frequency (4.31). In the low-frequency given by:
SOLUTION
FOR
LOW
AND
HIGH
FREQUENCIES
approximations to 6 and V can be derived from the expressions limit n << I the first approximations to U and V are identical, being n
-.-=(A*+Bf)
>
@,+p2+i&+F4
in
n >
I
’ {I
+0(n)}.
6
1)
711
Letters in Applied and Engineering Sciences
The velocities of the two fluids therefore coincide and the mixture behaves as a single fluid with average properties. Some difficulty is experienced in obtaining the high-frequency limit n >> 1. Consider first the special case of a mixture for which j& = p4 = & = 0. In this case the partial stresses are symmetric and each one depends only on the velocity gradient of that constituent, no error-terms being included. The expressions (4.31) for ii and fi are then simplified, reducing to: “’ (l+i) F I( > 77
4,
A-* ip,n
(( - )I’* I,, np, i
PI
(l+i) 777
_ I
_ 1
‘.
(5.2)
The velocities of the individual constituents are now seen to be different, and in this limit each fluid moves independently form the other. When 0, = p2, pi = p2, pi0 = p2(,, j& = p4 = & = 0 and A* = t?*, the results reduce to the familiar case of a single incompressible Newtonian fluid [8,18, 191: (i) n << 1. In this case (4.13) and (4.14) imply ti - V = O(n), and so ii = 0 to first approximation. (ii) n >> 1. All the terms in (4.13) and (4.14) except ii - V are O(n) and therefore much larger than ij - 0. In the leading approximation ii - fi can hence be neglected [18]. The approximate solutions (5.1) and (5.2) are concerned with complex quantities and modified Bessel functions of order zero. Real and imaginary parts of a Bessel function of order zero are given by [19,20]: @vJ = ber( vT) + i bei( vf) &(@a) ber(vZ) + i bei(vci)’
(5.3)
where @=$(l+i).
6.
NUMERICAL
RESULTS
(5.4)
AND DISCUSSION
Numerical results are obtained for the variation with distance of velocities of the constituent fluids for fixed dimensionless times (d = mn/6; m = 0, 1, 312, 2, 3, 4, 6) and certain fixed frequencies of vibration of pressure-gradient. In all of the curves shown, certain parameters are fixed (p,=l, p2=1, 4, F3=1/2, j&=1/2, &=-l/2, p,=5/6, c5=1, A*=B*) and the remaining parameters take given values [6-8, 18, 191. Moreover, it is approximated by: I,,(X) = 1 + c + g
for small values of X.
The variation of the velocities of the fluid caused by the simple harmonic motion of pressure-gradient on poiseuille flow in a uniform circular pipe is investigated for n = 1 (Figs 1, 4), n = 10 (Figs 2, 5), n = 50 (Figs 3, 6) all at nf = mn/6. As it may be expected, the velocity difference is more marked in the different harmonic components (see Figs l-4, Figs 2-5, Figs 3-6). As the frequency increases there is a parabolic profile at any time, for example, in Fig. 2.
712
Fig. 1. Variation of fi, and U, with distance; n = 1, A; = -1, A; ~0.
90
60
4s
30
Fig. 2. Variation of ti, and 0, with distance;n=lO,A;=-l,A$=O.
60
Fig. 3. Variation of P, and CI,with distance; n = 50, Ar = -1, & = 0.
180
PA 180
120
60 45 30 0 5,
Fig. 4. Variation of E, and D, with distance; n = 1, /if = -1, AI = 1.
Fig. 5. Variation of 0, and D, with distance; n = 10, A: = -1, A: = 1 I80 180 120 90 y
A \ 60
60
Fig. 6. Variation of ii, and D, with distance; n = 50, A; = -1, Al = 1. 713
714
M. SINAS
G6GoS
In the numerical results, the radius of the pipe was taken sufficiently small and the harmonics were taken as A* = B* = A: - (A: in the calculations). As can be seen from Fig. 1, for n = 1 and A: = -1, A: = 0, the mixture behaves as a single continuous medium at low frequencies. But, as shown in Fig. 4, for the AT = -1 and A: = 1 values of the harmonic components, the two continuous media flow independently at low frequencies. It is also seen from Figs 3 and 6 that high-frequency decreases the difference in velocity resulting from the harmonic components being different. Pumping systems, heat and mass transfer and blood circulation in living creatures can be shown as the most important applications related to pulsative flows at present.
REFERENCES [l] C. TRUESDELLand R. TOUPIN, The Classical Field Theories, Handbuck der Physik (Edited by S. FLUDGE), Band 11 l/l Springer-Verlag, Berlin (1960). [2] A. E. GREEN and P. M. NAGHDI, Inr. J. Engng Sci. 3,231 (1965). [3] A. C. ERINGENand J. D. INGRAM, lnt. /. Engng Sci. 3, 197 (1965). [4] R. J. ATKINand R. E. CRAINE, Q. J, Me&. AppL Math. 29,209 (1976). [5] R. E. CRAINE, Inr. /. Engng Sci. 9, 1177 (1971). (61 R. E. CRAINE,/. Appl. Math. Phys (ZAMP) 24, 365 (1973). [7] M. S. GOGUS, Inr. J. Engng Sci. M, 313 (1988). [8] M. !j. GqG$$ Bull. Tech. Llniu. Istanbul 40, 789 (1987). [9] M. 9. GqGv$, Int. J. Engng Sci. 29, 1651 (1991). [lo] M. S. GGGUS, Int. J. Engng Sci. 30, 141 (1992). [ll] M. S. GOGUS, Int. J. Engng Sci. 30, 665 (1992). [12] N. MILLS, Inr. J. Engng Sci. 4, 97 (1966). 1131 A. E. GREEN and P. M. NAGHDI, Q. /. Mech.Math. 22, 427 (1969). [14] R. E. CRAINE. A. E. GREEN and P. M. NAGHDI, Q. J. Mech. Appl. Math. 23, 171 (1970). [15] R. M. BOWEN and D. T. GARCIA, Inf. /. Engng Sci. 5, 289 (1967). [16] R. M. BOWEN, J. Chem. Phys. 49, 1625 (1968). [ 17) D. A. MCDONALD, Blood Flow in Arferies. Edward Arnold, London (1960). 1181 J. D. INGRAM and A. C. ERINGEN, In!. J. Engng Sci. 5, 289 (1967). (191 J. Y. KAZAKIA and R. S. RIVLIN, Rheol. Acta 17, 210 (1978). 120) J. Y. KAZAKIA and R. S. RIVLIN, Rheol. Acta 18,224 (1979). (Received and accepted 20 April 1993)
APPENDIX The rate at which the mixture
flows through
the pipe is given by: Q = Q ,e’“‘,
(Al)
Q,=2+(P,ri+&it)df
(A2)
Thus we have:
g=fi
[
(AC,, + P2D11NK-~I +im,Pl+(PlC2, +~2021)11[(-12+im,)81+
d. (A* + B*) 2
1+ip,&n . [ j5,&n2-in
II.ein’, (A3)