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Effect of couple stresses on pulsatile hydromagnetic poiseuille flow
FLUID DYNAMICS
RESEARCH Fluid Dynamics Research 15 (1995) 313 324
ELSEVIER
Effect of couple stresses on pulsatile hydromagnetic poiseuille flow Nabil T.M. EL-Dabe, Salwa M.G. EL-Mohandis Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Cairo, Egypt Received 11 March 1994; revised 13 June 1994
An analytical study of unsteady magneto-hydrodynamic flow of an incompressible electrically conducting fluid filling the space between two parallel plates is presented, taking into account the couple stresses and pulsation of the pressure gradient effect. The solution of the problem is obtained with the help of perturbation technique. Analytical expression is given for the velocity field, and the effectsof the various parameters entering into the problem are discussed with the help of graphs.
I. Introduction Couple stresses and the concept of internal spin are two main physical concepts that go into building theories of fluids with microstructure. Couple stresses are a consequence of assuming that the mechanical action of one part of a b o d y on another, across a surface, is equivalent to a force and a moment distribution. In classical nonpolar mechanics, moment distributions are not considered, and the mechanical action is assumed to be equivalent to a force distribution only. The laws of motion can then be used for defining the stress tensor which, necessarily, turns out to be symmetric. Thus, in nonpolar mechanics, the state of stress at a point is defined by a symmetric second order tensor which is a point function that has six independent components. However, in polar mechanics the mechanical action is assumed to be equivalent to both a force and a moment distribution. The state of stress is then measured by a stress tensor and a couple stress tensor. In general neither of these second order tensors is symmetric, so the state of stress at a point is measured by eighteen independent components. Thus, the concept of couple stresses results from a study of the mechanical interaction taking place across a surface and, conceptually, is not related to the kinematics of motion. O n the other hand, the concept of microstructure is a kinematic one. F o r classical fluids without microstructure, all the kinematic parameters are assumed to be determined once the velocity field is specified. Thus, if the velocity field is identically zero, then there is no motion, and the linear and angular m o m e n t a of all material elements must also be identically zero. However, even when the velocity field is zero, the angular m o m e n t u m may be visualized as being nonzero by "magnifying" 0169-5983/95/$4.00 © 1995 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. Amsterdam. All rights reserved SSD! 0 1 6 9 - 5 9 8 3 ( 9 4 ) 0 0 0 4 9 - 2
314
N. T.M. El-Dabe, S.M. G. El-Mohandis / Fluid Dynamics Research 15 (1995) 313-324
the continuum picture until the identities of individual particles can be seen. A particle may not have any velocity of translation, so that its macroscopic velocity and thus its linear m o m e n t u m is zero, but the particle may be spinning about an axis. This spin would give rise to an angular momentum. If the same phenomenon is assumed to hold true at the continuum level, then angular m o m e n t u m can exist even in the absence of linear momentum. This is not true in the classical theories of fluid mechanics. At the kinematic level, a specification of the velocity field is then not sufficient, and additional kinematic measures, independent of velocity field, must be introduced to describe this internal spin. Such a fluid is said to have microstructure. The concepts of couple stresses and microstructure are conceptually different. The first concept has its origins in the way mechanical interactions are modeled, while the second is essentially a kinematic one, and arises out of an attempt to describe point particles having structure. Whereas in a general theory of fluids with microstructure, couple stresses and internal spin may be present simultaneously, theories of fluids in which couple stresses are present but microstructure is absent are also possible. Similarly, microstructure may be considered in the absence of couple stresses. In this way the main consequences of each of these concepts may be studied before proceeding to the study of more general theories. Several different approaches may be used for formulating such theories. For example, a statistical mechanics model, which assumes noncentral forces between particles, is known to give rise to couple stresses. Thus a continuum theory for fluids with microstructure may be obtained from such a model. The concept of couple stresses may also be introduced purely on the basis of continuum argument. Microstructure can be introduced heuristically. Such theories may also be formulated by an averaging procedure in which the macroscopic variables are obtained by taking suitable averages over continuum domains. The most important effect of couple stresses is to introduce a size-dependent effect that is not predicted by the classical nonpolar theories. For example, in pipe Poiseuille flow, even after all the variables have been nondimensionalized in the usual way, the velocity profile is a function of the pipe radius. The simplest theory of couple stresses in fluids, in the absence of microstructure, is given in this manuscript. The possible effects of couple stresses will first be considered in isolation by assuming that the fluid has no microstructure at the kinematic level, so that kinematics of motion are fully determined by the velocity field. In this case the intrinsic angular m o m e n t u m and the kinetic energy of spin density are not taken into account, whereas couple stresses will be considered; the fluid will not have any microstructure. A list of equations that are of interest is useful. Once the intrinsic angular m o m e n t u m and the kinetic energy of spin density are taken to be zero, the various field equations governing the motion of a fluid will be given by the following. The continuity equation p" + pvr, r = O,
Cauchy's first law of motion pai:
tri, r + Pfi,
Cauchy's second law of motion mri, r + pli + ei~st~ = O,
(1)
N.T.M. El-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
315
where p is the density of the fluid, vi are the velocity components, ai are the components of the acceleration, tj~ is the second order stress tensor, mji is the second order couple stress tensor, J] is the body force per unit volume, li is the body m o m e n t per unit volume, and eijs is the third order alternating pseudo tensor, which is equal to 1 if (i,j, s) is an even permutation of (1,2, 3), and is equal to - 1 if (i,j, s) is an odd permutation of (1, 2, 3), and is equal to zero if two or more of the indices i,j, s are equal. For the elastic case, Mindlin and Tierstem [1] obtained the constitutive equations for a linear perfectly elastic solid in the form t~., = - P~ + (2 + t~)v,.,~ + tw~.... erismis, 1 -~- pe~isl~ + erisespqtpq :
0,
which, on simplification, gives tA = -- ~e,i~(mi~. 1 1 + pls)
t A = - - ½ [ e , i s m s + 4~le~isco~, 11] + pe,isl~
as ~o,, 1 ----0 and eris(Ds
~
(J)ri,
hence
t A = - 2qe9,i, ll + ½e, i~(m,s + pls) tAr,, r =
- - 2 q ~ r i , r l 1 + ½eirs(pls), r
since the term e , ~ m ~ must be zero. Finally, since 2co~i = vi,~ - v~, i, t,~,, = - rlvi,,,1, + rlv,.,ill + ½ei~(pl~).,
and the equations of motion become Pai = -- P,i + ('~" -'~ ~t)(Vr, r),i -~ ~l)i, rr -- I~Vi, rrss + t~(Ur,,),iss + ½ei,~(pl~),~ + p f
which, in the notation of Gibbs, can be written as pa = - Vp + (2 + tt)V(V.v) + qVSVV-v + flV 2U - qV 4U ~- p f +
½V A(pI),
(2)
where a = Cv/Ct + v. Vr, p is the pressure, and 2, tt, and q are material constants. The dimensions of the material constants 2 and #, which do not have the same significance as in the nonpolar case, are those of viscosity, namely M / L T , whereas those of r/are those of momentum, namely M L / T . The ratio r//tL therefore has dimensions of length square, and this material constant will be denoted by L = (,1/~) 1/~.
For incompressible fluids V. v = O, then, if the body force f and the body m o m e n t I are absent, the equations of motion reduce to pa = -- VP + ~/V2u - - /']V4v.
The general expression for t~j is given by tij = ti~ -}- tiAj = -- P(~ij -k- ~(Ui, j + Vj, i) -- qVj, jr r - - a e i j r ( m r
-}- pl,).
This is the simplest theory that shows all the important features and effects of couple stresses and results in equations that are similar to the Navier-Stokes equations. The main effect of couple
316
N. T.M. El-Dabe, S.M. G. El-Mohandis / Fluid Dynamics Research 15 (1995) 313-324
l' y'
B~
y-h
y'--o
X
Fig. 1. Schematic of the fluid channel bound by two parallel plates.
stresses will be to introduce a length-dependent effect that is not present in the classical nonpolar theories. Many authors [1-5] have studied the effects of couple stresses on a flow past an infinite plate or past a sphere and through channels for constant or absent pressure gradient (Annapurna and Ramanaiah, 1976; Chaturani, 1978; Mindlin, 1973; Mindlin and Tierstea, 1962; Stokes, 1971), while in the absence of an effect of couple stress, Edwards et al. (1972), discussed the pulsating flow of non-Newtonian fluids in pipes. Also, oscillatory fluid flow through a porous medium channel bounded by two impermeable parallel plates has been studied by Khodadadi (1991). Some of the applications related to this idea are the flow of oil under ground where there is a natural magnetic field; and the motion of the blood through the arteries. The main idea of our work is the mathematical study of these phenomena, and out-purpose is to show the relation between the different parameters of the motion and the external forces, in order to investigate how to control the velocity of the fluid motion by changing these parameters and external forces. The aim of our work is to study the effects of couple stresses, pressure pulsation, and magnetic parameters on the velocity distribution of an incompressible, electrically conducting fluid flowing between two parallel fixed plates (see Fig. 1).
2. Mathematical analysis
We consider the unsteady hydromagnetic flow of a viscous, incompressible, and electrically conducting fluid induced by the pulsation of the pressure gradient. The plates are assumed to be electrically insulated. We also consider a Cartesian coordinate system Ox'y'z' in such a way that the x'z'-plane is taken on the lower plate and this y'-axis is normal to the plates. A uniform magnetic field B~ is acting along the y'-axis. The induced magnetic field is assumed to be negligible. Within the framework of these assumptions, the equations which govern the M H D flow in the absence of both body force f and body moment I are ~u'
1 0P'
/t ~2u'
11 ~4 u,
0./~ On,2u'
~t' -
p Ox ~ + p ~y,2
p ay,4
p,
(3)
The term - t l ~ 4 u ' / p O y '4 in this equation gives the effect of couple stresses, while -aB'o2u'/p ' signifies the electromagnetic force. All the physical quantities in the above equations have their usual meaning.
N.T.M. El-Dabe, S.M.G. EI-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
317
The boundary conditions corresponding to the problem under consideration are u'=0,
y'=0,
at
u'(1)=0,
at
(4) (5)
y' = h,
since the couple stresses vanish at both the plates which in turn, implies that
~2Ut at
Oy,z-O,
y'=0,
~2Ut ~y,2 -- 0, at y' =
(6)
h,
(7)
where u'(y, t) is the fluid velocity field and h is the distance between the two plates. By introducing the following dimensionless quantities: t' = th/Vo,
u' = Vou, x'=xh,
y'=yh,
L 2 = tl/P,
~0' = (Voo~)/h,
P' = p V 2 p ,
a 2 = h 2 / L 2,
Re = (Vop'h)/p',
(8)
M 2 = (B'o2ah2)/ff,
the equation of motion (3) and boundary conditions (4)-(7) become Ou _ Rea2 0t
u=0,
~gp a2 ~2u Rea2 ~ x + Oy 2-
u"=0
at
y=0,
Cq4U •y4
(9)
aZM2u,
u=0,
u"=0
at
y=l,
(10)
where prime denotes the differentiation with respect to y. For pulsation pressure gradient, let
hence Eq. (9) can be written as
c~u= Rea2 [ ( t-~x 3 P ) ( 6+3 P )~ox Rea2 & s
eiO~t]+ a2~2u ~4U -t~Y2
~3Y4
a2Meu.
(12)
The above equation can be solved by using the following perturbation technique: u = us + u o e i°'t
(13)
Substituting from (13) in (9) and (10) and equating the like terms on both sides, we get the following system of equations: d4us
2d2us
d4uo
2d2uo
dy----~ - a ~
dy---T - a ~
(~P'],
+ a2M2us = nea 2 \-~x Js
+
2/t3P'X aZ(M 2 + Reko)Uo = Rea ~-~x flo'
(14) (15)
N.T.M. El-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
318
subject to the b o u n d a r y c o n d i t i o n s i!
u,=uo=0,
u'~'=Uo=0,
fory=0
a n d 1.
(16)
T h e solutions of e q u a t i o n s (14) a n d (15) subject to the b o u n d a r y c o n d i t i o n s (16) give the velocity d i s t r i b u t i o n of the fluid u n d e r c o n s i d e r a t i o n : u =
.
Rep)
- N 9 ea'y + N8 e'2y + N7 e;'y + N 6 ea'y + ~ - 7
+ I - N15e&Y + N18e&Y + N17e)TY + N16e&Y +
s
RePo
] el,O,,
M 2 + Reio) A
where 21 = [ ( a 2 + (a 4 - -
4a2M2)X/2)] 1/2, 22 = - 21,
/~3 = [ ( a2 - - ( a4 - - 4a2M2)1/2)] 1/2,
24 = -- 23,
25 = [ ( a 2 .-..b(a 4 - - 4 a 2 ( M 2 Jr- Reifo))l/2)/2]l/2,
t~6 = - 25,
27 = [ ( a 2 + (a 4 - - 4 a 2 ( M 2 + Reico))X/2)/2] 1/2,
28 = - 27,
No = (e 2x --
e23)(/~2
--
)~2),
N1 = (e 2, _ e2,)(222 _ A,4) ,42
RePs
N2 = ~ T
(e
~,
N3 = (2 2 e a: - 22 e~')(22 e ~' - 22 e ~) - (22 e a~ - 22 e)")(212 e )' - 22 e<),
N4 = (22 e ~ ' ' - 2] e ~') (22 e < - 22 ea') - (22 e z" - 242 e x') (22 e < - 22 ea~), Ns-
R~22P~ ,_ ~ I_e<,22 [ i e).~ - 2 2 e ~ ) - ( 2 2 e N5 No
N2 N3 N6 = N1N3 - N4No' -
x ~ - 22 eX~)],
N2 N4 - N1 Ns N7 = N1N3 __ NON,,'
1 ( RePs(e'h - 1 ) ) N8 - e~.2 _ e~ N,r(e ":~' -- e ~) + N6(e ~' - e ~'') + M 2 RcP~
N9 = N6 + N7 + N8 + M-----T , Na~ = (e a' - e~")(22 - 22),
N12 - M 2
RePo (e~.~ _ 1)22, + R~i~o
N , o = (e ~ - e;7)(262 - 2~),
-- 1)22),
(17)
N.T.M. El-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
319
N,3 = (2~e ~" - 26Zea')(2~ e ~~ - 22e ~) -(2Zse a' - 27ZeX~)(22e ~ ' - 26ze~'), N14 = (2~e 2' - 2sz e~")(2s2 e ~° - 22e ~') - (22e)" - 2sZe~')(22 e a' - 22 e)°),
N15 - M2 + Reie) N16
=
NlsN10 - Na2N13 , NIIN13 -- N l g N 1 0
(
N17 =
N l z N 1 4 -- NasN11 , N1aN13 -- N 1 4 N 1 o
~
R°P°
N18 = e A --1 e,l~ Na7(e~' -- e)~) + N~6(e~~ -- ea") + M 2 + R~io)
(e;,-1))
RoPo N19 = N16 + N17 + N18 + M 2 +
Reie)"
The non-dimensional discharge between the plates per unit depth is given by Q, where
Q= ff u(y,t)dy [ --
Ns (e'~2-1) . NT(e~3 - 1) + N-~6(e;~, - 1) + M-:-:v.~ lReP~-] N-~9(e;'l- 1)+ ~-2 +-~-3 -j A1
[
+
A4
N17 (e)~ -- l ) + -7-g16 (e)~"-- l ) + N19 ) N18 a 1) + -7--- 2--~-(e~,-1)+ 2---~(e ~ ,% ,ts
_RePs
] eio9'.
MZ+R¢icoJ
(18)
Also, the wall shear is given in dimensionless form as: du
To° ~ d-Y~ y = l
= [ _ N921e;" + Ns22e~ ~ + NvJ[3e~ + N6J[4e,~,] + [ - - N9)~5 e "~ + N18)~6 e~~ + N17J~Te~'~ + N1628 e;~] e i~t.
(19)
3. Results and discussion The effect of the properties of the parameters of the problem on the velocity, discharge, and skin-friction is illustrated graphically t h r o u g h Figs. 2-9. It is found that the phase of the velocity decreases with increasing time t (see Fig. 2). Fig. 3 shows that the velocity increases with the increase of the amplitude of the pulsation P0. F r o m Fig. 4 we show that the velocity decreases with increasing the magnetic p a r a m e t e r M. Also, Fig. (5) illustrates that the effect of the couple stresses is to increase the velocity with increasing stress. The discharge Q will decrease with increased magnetic p a r a m e t e r M a n d increases with couple stresses (see Figs. 6 and7). It is clear from Figs. 8 a n d 9 that the skin-friction r,o increases with an increase in the magnetic p a r a m e t e r M while it decreases with increase in both the amplitude of the pulsation Po and couple stresses a.
320
N. T,M. El-Dabe, S.M.G. EI-Mohandis/Fluid Dynamics Research 15 (1995) 313-324 I
0.06
0.05
0.04
0.03
0.02
0.01
1
Fig. 2. V e l o c i t y d i s t r i b u t i o n
u
I 5
10
lJ 210
15
215
30 --~y
p l o t t e d v e r s u s p o s i t i o n for M = 5, a = 0.5, w = 1, Ps = 5, P0 = 5, Re = 2.
i
i
i
=
,
=
0.03
0.025
0.02
0.015
0.01
0.005
I
I
i
I
0.2
0.4
0.6
0,8
~ Y
Fig. 3. V e l o c i t y d i s t r i b u t i o n p l o t t e d v e r s u s p o s i t i o n for M = 5, a = 0.5, w = 1, Ps = 5, t = •/2, Re = 2.
N.T.M. EI-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
321
0.025
0.02
0.015
0.01
0.005
i
0.2
/
0.4
i
I
0.6
0.8
--.~y
Fig. 4. V e l o c i t y d i s t r i b u t i o n p l o t t e d versus p o s i t i o n for Po = 5, a = 0.5, w = 1, P~ = 5, t = rt/2, Re = 2.
0.35
,
,
_
,
U
, =
t o.a 0.25
0.2
0.15
0.1
0.05
i
i
i
i
0.2
0.4
0.6
0.8
--~y
Fig. 5. V e l o c i t y d i s t r i b u t i o n p l o t t e d versus p o s i t i o n for M = 5, P0 = 5, w = 1, P~ = 5, t = ~/2,
R e =
2.
322
--~M
N. T.M. El-Dabe, S.M. G. El-Mohandis / Fluid Dynamics Research 15 (1995) 313-324
2
Q
4
6
8
[
I
10
12
I
1
14
t 0.019 0,018
0,017 0.016 0.015 0,014
0,013 I
I
Fig. 6. D i s c h a r g e
Q p l o t t e d v e r s u s M for t = 0.5, a = 0.5, w = 1, P~ = 5, Po = 20, Re = 2.
I
Q
1
I
I
i
i
I
li0
i
I
0.5
0.4
0.31
~a=5
0.2
0.1
I
4
1
6
8
112
ll4 ~ M
Fig. 7. D i s c h a r g e
Q p l o t t e d v e r s u s M for t = 0.5, w = 1, Ps = 5, Po = 5, Re = 2.
N.T.M. El-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
323
~ M -r w
6
8
10
12
14
i
i
i
i
i
,
,
,
Fig. 8. Skin-friction zw plotted versus M for t = 0.5, w = 1, Ps = 5, 1°o = 5, Re = 2.
"--,,'-M 2
4
6
8
10
12
14
I
I
l
I
I
T~ -0.2
-0.3
-0.4
-0.5
J
Fig. 9. Skin-friction rw plotted versus M for t = 0.5, w = 1, P~ = 5, a = 0.5, Re = 2.
References A n n a p u r n a , N. and G. R a m a n a i a h (1976) Effect of couple stresses on the u n s t e a d y D r a i n a g e of a M i c r o - P o l a r fluid on a flat surface, Jpn. J. Appl. Phys. 15, 2441 2444. Chaturani, P. (1978), Viscosity of poiseuille flow of a couple stress fluid with applications to b l o o d flow, Biorheology 15, 119-128.
324
N.T.M. E1-Dabe, S.M.G. El-Mohandis /Fluid Dynamics Research 15 (1995) 313-324
Edwards, M.F., D.A. Nelist and W.L. Wilkinson (1972) Pulsating flow of non-Newtonian fluids in pipes, Chemical Engineering Science 27, 545-553. Khodadadi, J.M. (1991) Oscillatory fluid flow through a porous medium channel bounded by two impermeable parallel plates, J. Fluids Engineerin 9 133, 509-511. Mindlin, R.D. (1973) Influence of couple stresses in linear elasticity, Exp. Mech. 20, 1-7. Mindlin, R.D. and H.F. Tierstea (1962) Effects of couple stresses in linear elasticity, Arch. Ration. Mech. Anal. 11, 415-448. Stokes, V.K. (1971), Effects of couple stresses in fluids on the creeping flow past a sphere, Phys. Fluids 14, 1580-1582.
RESEARCH Fluid Dynamics Research 15 (1995) 313 324
ELSEVIER
Effect of couple stresses on pulsatile hydromagnetic poiseuille flow Nabil T.M. EL-Dabe, Salwa M.G. EL-Mohandis Department of Mathematics, Faculty of Education, Ain Shams University, Heliopolis, Cairo, Egypt Received 11 March 1994; revised 13 June 1994
An analytical study of unsteady magneto-hydrodynamic flow of an incompressible electrically conducting fluid filling the space between two parallel plates is presented, taking into account the couple stresses and pulsation of the pressure gradient effect. The solution of the problem is obtained with the help of perturbation technique. Analytical expression is given for the velocity field, and the effectsof the various parameters entering into the problem are discussed with the help of graphs.
I. Introduction Couple stresses and the concept of internal spin are two main physical concepts that go into building theories of fluids with microstructure. Couple stresses are a consequence of assuming that the mechanical action of one part of a b o d y on another, across a surface, is equivalent to a force and a moment distribution. In classical nonpolar mechanics, moment distributions are not considered, and the mechanical action is assumed to be equivalent to a force distribution only. The laws of motion can then be used for defining the stress tensor which, necessarily, turns out to be symmetric. Thus, in nonpolar mechanics, the state of stress at a point is defined by a symmetric second order tensor which is a point function that has six independent components. However, in polar mechanics the mechanical action is assumed to be equivalent to both a force and a moment distribution. The state of stress is then measured by a stress tensor and a couple stress tensor. In general neither of these second order tensors is symmetric, so the state of stress at a point is measured by eighteen independent components. Thus, the concept of couple stresses results from a study of the mechanical interaction taking place across a surface and, conceptually, is not related to the kinematics of motion. O n the other hand, the concept of microstructure is a kinematic one. F o r classical fluids without microstructure, all the kinematic parameters are assumed to be determined once the velocity field is specified. Thus, if the velocity field is identically zero, then there is no motion, and the linear and angular m o m e n t a of all material elements must also be identically zero. However, even when the velocity field is zero, the angular m o m e n t u m may be visualized as being nonzero by "magnifying" 0169-5983/95/$4.00 © 1995 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. Amsterdam. All rights reserved SSD! 0 1 6 9 - 5 9 8 3 ( 9 4 ) 0 0 0 4 9 - 2
314
N. T.M. El-Dabe, S.M. G. El-Mohandis / Fluid Dynamics Research 15 (1995) 313-324
the continuum picture until the identities of individual particles can be seen. A particle may not have any velocity of translation, so that its macroscopic velocity and thus its linear m o m e n t u m is zero, but the particle may be spinning about an axis. This spin would give rise to an angular momentum. If the same phenomenon is assumed to hold true at the continuum level, then angular m o m e n t u m can exist even in the absence of linear momentum. This is not true in the classical theories of fluid mechanics. At the kinematic level, a specification of the velocity field is then not sufficient, and additional kinematic measures, independent of velocity field, must be introduced to describe this internal spin. Such a fluid is said to have microstructure. The concepts of couple stresses and microstructure are conceptually different. The first concept has its origins in the way mechanical interactions are modeled, while the second is essentially a kinematic one, and arises out of an attempt to describe point particles having structure. Whereas in a general theory of fluids with microstructure, couple stresses and internal spin may be present simultaneously, theories of fluids in which couple stresses are present but microstructure is absent are also possible. Similarly, microstructure may be considered in the absence of couple stresses. In this way the main consequences of each of these concepts may be studied before proceeding to the study of more general theories. Several different approaches may be used for formulating such theories. For example, a statistical mechanics model, which assumes noncentral forces between particles, is known to give rise to couple stresses. Thus a continuum theory for fluids with microstructure may be obtained from such a model. The concept of couple stresses may also be introduced purely on the basis of continuum argument. Microstructure can be introduced heuristically. Such theories may also be formulated by an averaging procedure in which the macroscopic variables are obtained by taking suitable averages over continuum domains. The most important effect of couple stresses is to introduce a size-dependent effect that is not predicted by the classical nonpolar theories. For example, in pipe Poiseuille flow, even after all the variables have been nondimensionalized in the usual way, the velocity profile is a function of the pipe radius. The simplest theory of couple stresses in fluids, in the absence of microstructure, is given in this manuscript. The possible effects of couple stresses will first be considered in isolation by assuming that the fluid has no microstructure at the kinematic level, so that kinematics of motion are fully determined by the velocity field. In this case the intrinsic angular m o m e n t u m and the kinetic energy of spin density are not taken into account, whereas couple stresses will be considered; the fluid will not have any microstructure. A list of equations that are of interest is useful. Once the intrinsic angular m o m e n t u m and the kinetic energy of spin density are taken to be zero, the various field equations governing the motion of a fluid will be given by the following. The continuity equation p" + pvr, r = O,
Cauchy's first law of motion pai:
tri, r + Pfi,
Cauchy's second law of motion mri, r + pli + ei~st~ = O,
(1)
N.T.M. El-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
315
where p is the density of the fluid, vi are the velocity components, ai are the components of the acceleration, tj~ is the second order stress tensor, mji is the second order couple stress tensor, J] is the body force per unit volume, li is the body m o m e n t per unit volume, and eijs is the third order alternating pseudo tensor, which is equal to 1 if (i,j, s) is an even permutation of (1,2, 3), and is equal to - 1 if (i,j, s) is an odd permutation of (1, 2, 3), and is equal to zero if two or more of the indices i,j, s are equal. For the elastic case, Mindlin and Tierstem [1] obtained the constitutive equations for a linear perfectly elastic solid in the form t~., = - P~ + (2 + t~)v,.,~ + tw~.... erismis, 1 -~- pe~isl~ + erisespqtpq :
0,
which, on simplification, gives tA = -- ~e,i~(mi~. 1 1 + pls)
t A = - - ½ [ e , i s m s + 4~le~isco~, 11] + pe,isl~
as ~o,, 1 ----0 and eris(Ds
~
(J)ri,
hence
t A = - 2qe9,i, ll + ½e, i~(m,s + pls) tAr,, r =
- - 2 q ~ r i , r l 1 + ½eirs(pls), r
since the term e , ~ m ~ must be zero. Finally, since 2co~i = vi,~ - v~, i, t,~,, = - rlvi,,,1, + rlv,.,ill + ½ei~(pl~).,
and the equations of motion become Pai = -- P,i + ('~" -'~ ~t)(Vr, r),i -~ ~l)i, rr -- I~Vi, rrss + t~(Ur,,),iss + ½ei,~(pl~),~ + p f
which, in the notation of Gibbs, can be written as pa = - Vp + (2 + tt)V(V.v) + qVSVV-v + flV 2U - qV 4U ~- p f +
½V A(pI),
(2)
where a = Cv/Ct + v. Vr, p is the pressure, and 2, tt, and q are material constants. The dimensions of the material constants 2 and #, which do not have the same significance as in the nonpolar case, are those of viscosity, namely M / L T , whereas those of r/are those of momentum, namely M L / T . The ratio r//tL therefore has dimensions of length square, and this material constant will be denoted by L = (,1/~) 1/~.
For incompressible fluids V. v = O, then, if the body force f and the body m o m e n t I are absent, the equations of motion reduce to pa = -- VP + ~/V2u - - /']V4v.
The general expression for t~j is given by tij = ti~ -}- tiAj = -- P(~ij -k- ~(Ui, j + Vj, i) -- qVj, jr r - - a e i j r ( m r
-}- pl,).
This is the simplest theory that shows all the important features and effects of couple stresses and results in equations that are similar to the Navier-Stokes equations. The main effect of couple
316
N. T.M. El-Dabe, S.M. G. El-Mohandis / Fluid Dynamics Research 15 (1995) 313-324
l' y'
B~
y-h
y'--o
X
Fig. 1. Schematic of the fluid channel bound by two parallel plates.
stresses will be to introduce a length-dependent effect that is not present in the classical nonpolar theories. Many authors [1-5] have studied the effects of couple stresses on a flow past an infinite plate or past a sphere and through channels for constant or absent pressure gradient (Annapurna and Ramanaiah, 1976; Chaturani, 1978; Mindlin, 1973; Mindlin and Tierstea, 1962; Stokes, 1971), while in the absence of an effect of couple stress, Edwards et al. (1972), discussed the pulsating flow of non-Newtonian fluids in pipes. Also, oscillatory fluid flow through a porous medium channel bounded by two impermeable parallel plates has been studied by Khodadadi (1991). Some of the applications related to this idea are the flow of oil under ground where there is a natural magnetic field; and the motion of the blood through the arteries. The main idea of our work is the mathematical study of these phenomena, and out-purpose is to show the relation between the different parameters of the motion and the external forces, in order to investigate how to control the velocity of the fluid motion by changing these parameters and external forces. The aim of our work is to study the effects of couple stresses, pressure pulsation, and magnetic parameters on the velocity distribution of an incompressible, electrically conducting fluid flowing between two parallel fixed plates (see Fig. 1).
2. Mathematical analysis
We consider the unsteady hydromagnetic flow of a viscous, incompressible, and electrically conducting fluid induced by the pulsation of the pressure gradient. The plates are assumed to be electrically insulated. We also consider a Cartesian coordinate system Ox'y'z' in such a way that the x'z'-plane is taken on the lower plate and this y'-axis is normal to the plates. A uniform magnetic field B~ is acting along the y'-axis. The induced magnetic field is assumed to be negligible. Within the framework of these assumptions, the equations which govern the M H D flow in the absence of both body force f and body moment I are ~u'
1 0P'
/t ~2u'
11 ~4 u,
0./~ On,2u'
~t' -
p Ox ~ + p ~y,2
p ay,4
p,
(3)
The term - t l ~ 4 u ' / p O y '4 in this equation gives the effect of couple stresses, while -aB'o2u'/p ' signifies the electromagnetic force. All the physical quantities in the above equations have their usual meaning.
N.T.M. El-Dabe, S.M.G. EI-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
317
The boundary conditions corresponding to the problem under consideration are u'=0,
y'=0,
at
u'(1)=0,
at
(4) (5)
y' = h,
since the couple stresses vanish at both the plates which in turn, implies that
~2Ut at
Oy,z-O,
y'=0,
~2Ut ~y,2 -- 0, at y' =
(6)
h,
(7)
where u'(y, t) is the fluid velocity field and h is the distance between the two plates. By introducing the following dimensionless quantities: t' = th/Vo,
u' = Vou, x'=xh,
y'=yh,
L 2 = tl/P,
~0' = (Voo~)/h,
P' = p V 2 p ,
a 2 = h 2 / L 2,
Re = (Vop'h)/p',
(8)
M 2 = (B'o2ah2)/ff,
the equation of motion (3) and boundary conditions (4)-(7) become Ou _ Rea2 0t
u=0,
~gp a2 ~2u Rea2 ~ x + Oy 2-
u"=0
at
y=0,
Cq4U •y4
(9)
aZM2u,
u=0,
u"=0
at
y=l,
(10)
where prime denotes the differentiation with respect to y. For pulsation pressure gradient, let
hence Eq. (9) can be written as
c~u= Rea2 [ ( t-~x 3 P ) ( 6+3 P )~ox Rea2 & s
eiO~t]+ a2~2u ~4U -t~Y2
~3Y4
a2Meu.
(12)
The above equation can be solved by using the following perturbation technique: u = us + u o e i°'t
(13)
Substituting from (13) in (9) and (10) and equating the like terms on both sides, we get the following system of equations: d4us
2d2us
d4uo
2d2uo
dy----~ - a ~
dy---T - a ~
(~P'],
+ a2M2us = nea 2 \-~x Js
+
2/t3P'X aZ(M 2 + Reko)Uo = Rea ~-~x flo'
(14) (15)
N.T.M. El-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
318
subject to the b o u n d a r y c o n d i t i o n s i!
u,=uo=0,
u'~'=Uo=0,
fory=0
a n d 1.
(16)
T h e solutions of e q u a t i o n s (14) a n d (15) subject to the b o u n d a r y c o n d i t i o n s (16) give the velocity d i s t r i b u t i o n of the fluid u n d e r c o n s i d e r a t i o n : u =
.
Rep)
- N 9 ea'y + N8 e'2y + N7 e;'y + N 6 ea'y + ~ - 7
+ I - N15e&Y + N18e&Y + N17e)TY + N16e&Y +
s
RePo
] el,O,,
M 2 + Reio) A
where 21 = [ ( a 2 + (a 4 - -
4a2M2)X/2)] 1/2, 22 = - 21,
/~3 = [ ( a2 - - ( a4 - - 4a2M2)1/2)] 1/2,
24 = -- 23,
25 = [ ( a 2 .-..b(a 4 - - 4 a 2 ( M 2 Jr- Reifo))l/2)/2]l/2,
t~6 = - 25,
27 = [ ( a 2 + (a 4 - - 4 a 2 ( M 2 + Reico))X/2)/2] 1/2,
28 = - 27,
No = (e 2x --
e23)(/~2
--
)~2),
N1 = (e 2, _ e2,)(222 _ A,4) ,42
RePs
N2 = ~ T
(e
~,
N3 = (2 2 e a: - 22 e~')(22 e ~' - 22 e ~) - (22 e a~ - 22 e)")(212 e )' - 22 e<),
N4 = (22 e ~ ' ' - 2] e ~') (22 e < - 22 ea') - (22 e z" - 242 e x') (22 e < - 22 ea~), Ns-
R~22P~ ,_ ~ I_e<,22 [ i e).~ - 2 2 e ~ ) - ( 2 2 e N5 No
N2 N3 N6 = N1N3 - N4No' -
x ~ - 22 eX~)],
N2 N4 - N1 Ns N7 = N1N3 __ NON,,'
1 ( RePs(e'h - 1 ) ) N8 - e~.2 _ e~ N,r(e ":~' -- e ~) + N6(e ~' - e ~'') + M 2 RcP~
N9 = N6 + N7 + N8 + M-----T , Na~ = (e a' - e~")(22 - 22),
N12 - M 2
RePo (e~.~ _ 1)22, + R~i~o
N , o = (e ~ - e;7)(262 - 2~),
-- 1)22),
(17)
N.T.M. El-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
319
N,3 = (2~e ~" - 26Zea')(2~ e ~~ - 22e ~) -(2Zse a' - 27ZeX~)(22e ~ ' - 26ze~'), N14 = (2~e 2' - 2sz e~")(2s2 e ~° - 22e ~') - (22e)" - 2sZe~')(22 e a' - 22 e)°),
N15 - M2 + Reie) N16
=
NlsN10 - Na2N13 , NIIN13 -- N l g N 1 0
(
N17 =
N l z N 1 4 -- NasN11 , N1aN13 -- N 1 4 N 1 o
~
R°P°
N18 = e A --1 e,l~ Na7(e~' -- e)~) + N~6(e~~ -- ea") + M 2 + R~io)
(e;,-1))
RoPo N19 = N16 + N17 + N18 + M 2 +
Reie)"
The non-dimensional discharge between the plates per unit depth is given by Q, where
Q= ff u(y,t)dy [ --
Ns (e'~2-1) . NT(e~3 - 1) + N-~6(e;~, - 1) + M-:-:v.~ lReP~-] N-~9(e;'l- 1)+ ~-2 +-~-3 -j A1
[
+
A4
N17 (e)~ -- l ) + -7-g16 (e)~"-- l ) + N19 ) N18 a 1) + -7--- 2--~-(e~,-1)+ 2---~(e ~ ,% ,ts
_RePs
] eio9'.
MZ+R¢icoJ
(18)
Also, the wall shear is given in dimensionless form as: du
To° ~ d-Y~ y = l
= [ _ N921e;" + Ns22e~ ~ + NvJ[3e~ + N6J[4e,~,] + [ - - N9)~5 e "~ + N18)~6 e~~ + N17J~Te~'~ + N1628 e;~] e i~t.
(19)
3. Results and discussion The effect of the properties of the parameters of the problem on the velocity, discharge, and skin-friction is illustrated graphically t h r o u g h Figs. 2-9. It is found that the phase of the velocity decreases with increasing time t (see Fig. 2). Fig. 3 shows that the velocity increases with the increase of the amplitude of the pulsation P0. F r o m Fig. 4 we show that the velocity decreases with increasing the magnetic p a r a m e t e r M. Also, Fig. (5) illustrates that the effect of the couple stresses is to increase the velocity with increasing stress. The discharge Q will decrease with increased magnetic p a r a m e t e r M a n d increases with couple stresses (see Figs. 6 and7). It is clear from Figs. 8 a n d 9 that the skin-friction r,o increases with an increase in the magnetic p a r a m e t e r M while it decreases with increase in both the amplitude of the pulsation Po and couple stresses a.
320
N. T,M. El-Dabe, S.M.G. EI-Mohandis/Fluid Dynamics Research 15 (1995) 313-324 I
0.06
0.05
0.04
0.03
0.02
0.01
1
Fig. 2. V e l o c i t y d i s t r i b u t i o n
u
I 5
10
lJ 210
15
215
30 --~y
p l o t t e d v e r s u s p o s i t i o n for M = 5, a = 0.5, w = 1, Ps = 5, P0 = 5, Re = 2.
i
i
i
=
,
=
0.03
0.025
0.02
0.015
0.01
0.005
I
I
i
I
0.2
0.4
0.6
0,8
~ Y
Fig. 3. V e l o c i t y d i s t r i b u t i o n p l o t t e d v e r s u s p o s i t i o n for M = 5, a = 0.5, w = 1, Ps = 5, t = •/2, Re = 2.
N.T.M. EI-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
321
0.025
0.02
0.015
0.01
0.005
i
0.2
/
0.4
i
I
0.6
0.8
--.~y
Fig. 4. V e l o c i t y d i s t r i b u t i o n p l o t t e d versus p o s i t i o n for Po = 5, a = 0.5, w = 1, P~ = 5, t = rt/2, Re = 2.
0.35
,
,
_
,
U
, =
t o.a 0.25
0.2
0.15
0.1
0.05
i
i
i
i
0.2
0.4
0.6
0.8
--~y
Fig. 5. V e l o c i t y d i s t r i b u t i o n p l o t t e d versus p o s i t i o n for M = 5, P0 = 5, w = 1, P~ = 5, t = ~/2,
R e =
2.
322
--~M
N. T.M. El-Dabe, S.M. G. El-Mohandis / Fluid Dynamics Research 15 (1995) 313-324
2
Q
4
6
8
[
I
10
12
I
1
14
t 0.019 0,018
0,017 0.016 0.015 0,014
0,013 I
I
Fig. 6. D i s c h a r g e
Q p l o t t e d v e r s u s M for t = 0.5, a = 0.5, w = 1, P~ = 5, Po = 20, Re = 2.
I
Q
1
I
I
i
i
I
li0
i
I
0.5
0.4
0.31
~a=5
0.2
0.1
I
4
1
6
8
112
ll4 ~ M
Fig. 7. D i s c h a r g e
Q p l o t t e d v e r s u s M for t = 0.5, w = 1, Ps = 5, Po = 5, Re = 2.
N.T.M. El-Dabe, S.M.G. El-Mohandis/Fluid Dynamics Research 15 (1995) 313-324
323
~ M -r w
6
8
10
12
14
i
i
i
i
i
,
,
,
Fig. 8. Skin-friction zw plotted versus M for t = 0.5, w = 1, Ps = 5, 1°o = 5, Re = 2.
"--,,'-M 2
4
6
8
10
12
14
I
I
l
I
I
T~ -0.2
-0.3
-0.4
-0.5
J
Fig. 9. Skin-friction rw plotted versus M for t = 0.5, w = 1, P~ = 5, a = 0.5, Re = 2.
References A n n a p u r n a , N. and G. R a m a n a i a h (1976) Effect of couple stresses on the u n s t e a d y D r a i n a g e of a M i c r o - P o l a r fluid on a flat surface, Jpn. J. Appl. Phys. 15, 2441 2444. Chaturani, P. (1978), Viscosity of poiseuille flow of a couple stress fluid with applications to b l o o d flow, Biorheology 15, 119-128.
324
N.T.M. E1-Dabe, S.M.G. El-Mohandis /Fluid Dynamics Research 15 (1995) 313-324
Edwards, M.F., D.A. Nelist and W.L. Wilkinson (1972) Pulsating flow of non-Newtonian fluids in pipes, Chemical Engineering Science 27, 545-553. Khodadadi, J.M. (1991) Oscillatory fluid flow through a porous medium channel bounded by two impermeable parallel plates, J. Fluids Engineerin 9 133, 509-511. Mindlin, R.D. (1973) Influence of couple stresses in linear elasticity, Exp. Mech. 20, 1-7. Mindlin, R.D. and H.F. Tierstea (1962) Effects of couple stresses in linear elasticity, Arch. Ration. Mech. Anal. 11, 415-448. Stokes, V.K. (1971), Effects of couple stresses in fluids on the creeping flow past a sphere, Phys. Fluids 14, 1580-1582.