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The influence of vibrating plates on poiseuille flow of a binary mixture
Int. f. Engng Sci‘ Vol. 26, No. 4, pp. 313-323, 1988
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Copyright @ 1988 Pergamon Press plc
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THE INFLUENCE OF VIBRATING PLATES ON POISEUILLE FLOW OF A BINARY MIXTURE Faculty of Mech~i~~
En~nee~ng, Department of Mechanics, Istanbul Technical Unive~ity, ~~rn~~su~~ Ist~bul, Turkey
Abstmct-A mixture of two incompressible Newtonian fluids is considered and the flow induced by steady vibrations of plates on poiseuille flow between two parallel plates, under a constant pressure gradient, is then investigated. Calculations are made for longitudinal vibrations of the plates. 1. INTRODUCTION In
recent years much work has been published on the continuum theory of mixtures. 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 [Z], uses only an equation of energy balance and entropy production inequality as basic postulates. Coincidentally, a paper by A, C. Eringen and 9. D. Ingram has been published which presents a unified approach for the derivation of basic conservation equations for a chemically reacting continuum [3]. Mixtures of fluids which are viscous in the sense that their constitutive equations contain gradients of the velocity fields of the constituents have been considered in the context of the continuum theory by Green and Naghdi [4], Craine et al. [53, Mills [6J, Bowen [7], Mtiller 181, Adkins [9], R. M. Bowen and Garcia [lo], Eringen and Ingram [ll] have investigated mixtures of chemically reacting fluids in addition there is a considerable literature in which general two-phase flows are viewed from a different ~e~oint, much of this work being concerned with suspensions of solid particles in fluids. Lastly we mention the time dependent problems which has been investigated by Craine 112, 131. In the work described here a binary mixture of incompressible, Newtonian fluids is considered. In Section 2 we use the governing field equations of the continuum theory of mixtures and the notations given by R. E. Craine [12, 131. The appropriate constitutive theory developed by the authors mentioned above is given in Section 2 [4-61. In Sections 3 and 4 of present paper, the flow induced by longitudinal vibrations of two infinite parallel plates (plane poiseuille flow), under a constant pressure gradient, is investigated and an exact solution is obtained. 2. BASIC THEORY In this section we summarize the theory, for more details the reader should consult Craine [12]. A mixture of two continua, in motion relative to each other, is considered. At an arbitrary time t it is assumed that at each place z 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. The velocities of the l- and, 2-constituents are denoted by Vfl) and V@ respectively, and we define DC*) a _=_+.j_,d
Dr
dt
&j
42-l)
where a takes the values 1, 2. The governing kinematic equations and the energy equation of a binary mixture in which the constituents have a common temperature T and do not interact chemically are
(2.2)
314
The quantities pm> Ua, CT(~),FCC), e_3,P and q are in turn rhe delE&.yy,the fnicrmal energy pei? unit mass, the partial stress, the body force per unit mass of the ath c~nst~t~e~t~ the difksive force vector, the heat supply per unit vo~~~~, and the heat flux vecteor[4]. The partial stresses satisfy O@l4” Cr_i$= 0
<2.6)
where suffixes enclosed in square brackets refer to components of the ant-isymmetric part of a second order tensor and circular brackets have a similar meaning in relation to the symmetric ~~a~s~~s-~~he~ ~n~~ua~~~ymay be written [S] , part. If s, denotes the partial entropy then t
The formulation of constitutive equations f~sr a mixture of Newtonian fluids is now considered, attention being restricted to the case in which both constituents are incompsressible. At some time tOr SI and S, are separated and have constant densities plOl pzo respectiveiy. The mixture is formed by adding the element: of ~7~aa _XA, of mass dq and volume d& to ehe element of S, at YA, of mass $mz and volume dV,. There is assumed to be no change of total volume when the mixt~~ Is formed, so that a~ element of the mixture at xi of mass dmr t dmz oceq&s a volume de/, + d&2; Such a mixture was considered by Mills [6]. We now write
When S, and Sz are separated we have
Using (2.8) and (2.10) it follows that
For the densities of SIs,,,!&in the rni~t~r~ we write
So that the incompressibility
condition (2.11) now becomes
where y is a ~orn~os~t~~~ factor. The density p of the mixture is given by P = P1-k Pz and we have
The influence of vibrating plates on poiseuille flow of a binary mixture
315
(2.16)
(pzo - P)d~’ + (P - PlO)d~’ - ~iUj= 0
the quantities 5 and d(“) being defined by Ei
=
dZi”)
P,i
=
(2.17)
“&)
partial Helimholtz free energies A, are introduced,
For convenience
defined by the relations
A a = U, - TS,
(2.18)
As the basic constitutive postulate we assume that the quantities A,, S,, co, q and a(“) are arbitrary functions of p and T but depend linearly on 5, b, a, d’“) and I’, where J.j$;’ = #
bi = T,i,
JYik= j.@ - wp
Id+
a.I = vi(l)- $2)
(2.19)
Adopting an analogous method to that outlined in [12], it is possible to deduce from the Clausius-Duhem inequality that the constitutive equations have the reduced forms A, = A&
‘A =A@,
7%
(P$++
-(Pzo-P)
Pl’
T),
Oi 4i =
= -
s, = UP,
=
__dA dT
(2.20) (2.21)
P2=(P-P1o)(P2$$-")
aai - Agi + ybi
(2.22)
T(koai + k,bi)
(2.23)
Us’ = (-Pl+
n,d~)+ a3d~‘)6, + 2pldik
Us’ = (-Pz +
n,d~) + &df))6,
where the coefficients satisfy the inequalities
s
T),
+ Z!/igdik+ n,rik
(2.24)
+ 2/44dik+ 2cczdik - a5rik
(2.25)
E, ill, . . . , d5, y, ko, k3, pl, . . . , y, are functions
of p and T, and
Q 2 0,
k3 3 0,
,420,
,&~001
(Y,+!h)2s‘hh
(2.26)
2 ;1,+-/L,ao, 3
(2.27) (2.28)
3. FLOW
INDUCED
BY STEADY LONGITUDINAL TWO PARALLEL PLATES
VIBRATIONS
OF
In this section we consider a binary mixture of incompressible Newtonian fluids, to be contained between infinite parallel rigid plates normal to the y-axis of a rectangular Cartesian coordinate system X and to flow under the action of constant pressure gradients PI, P2 in the direction of the x-axis. Simultaneously the two plates are subjected to a sinusoidal vibration in their own planes, of equal amplitudes with angular frequency w, parallel to the x-direction. For convenience set x1 = x, x2 = y, x3 = z, and let the plates be located at y = T/z. We assume that body forces are absent, and that the temperature remains constant throughout the mixture (for discussion of this latter assumption see Craine [12]). Since the temperature may now be dropped from the list of variables in the problem, all the coefficients pl etc. appearing in the constitutive equations (2.24) and (2.25) depend only on the density of the mixture p. We assume a flow field of the form v(‘) = (u,(y,t),
0, 0) ,
vC2’ = (%(Y,
t), 0, 0)
P = P(Y,
t)
(3.1)
in which u,(y, t) and v,(y, t) are u,(y, t) = z+,(y) + L sin cot+ M cos ot
(3.2)
v,(y, t) = v,(y)
(3.3)
P = P(Y,
+ N sin ot + G cos wt t>
(3.4)
The iniluence of vibrating plates on poiseuille flow of a binary mixture
we have p # plo, p # pzo and d2(pA)/dp2 # 0. It must the mixture p is a constant. Since p has been proved have been assumed, all the coefficients in eqns (3.17) From (3.17), (3.18), (3.19) and (3.20) it follows constant A is
A=
_-=
a& dx
have dpldy = 0, from here the density of to be constant and isothermal conditions and (3.18) are constant. that A and fI(y) are constant, where
34 dx --=A
Pzo-P
317
P-P10
di2
(3.24)
3.x
On substituting eqns (3.2) and (3.3) into eqns (3.17) and (3.18) we find that (3.25) (3.26) (3.27) (3.28) (3.29) (3.30)
introducing the dimensionless
quantities,
(3.31)
where ,U is a typical viscosity dimensionless forms,
in the problem,
eqns (3.25)-(3.30)
take on the following
Replacing (3.33) by the sum of (3.32) and (3.33), the equations can be written
with the use of the boundary (3.26) gives,
conditions
G&F@ = 0, “Ls,(Th) = 0 on f = T&, integrating
Substituition of (3.40) into eqn (3.38) gives a linear differential linear equation, for solution &, and DOwe obtain
equation.
eqn
With the soMow
oi
where,
(3.43)
The last four of eqns (3.34)-(3.37) the notation
may be rewritten
Ri,=b-dti,
as two complex equations by introducing
R,=A+iC
(3.44)
Thus (3.45)
319
The influence of vibrating plates on poiseuille flow of a binary mixture
Replacing (3.46) by the sum of (3.45) and (3.46) the governing equations can be written. (3.47) [&
+ ,i&)D2 - i&n]&
where D denotes the differential operator
+ [(p2 + ,i&)D2 - ip2n]Z?, = 0
(3.48)
d/d7 and put
WI = [(fi2 + ,i&)D2 - i&n]Y,
Z&= [-(PI + ,&)D2 + @&I’
(3.49)
the eqn (3.48) is satisfied identically and (3.47) becomes (D4 + P1D2 + /?$I’ = 0
(3.50)
where
-(ih+P2+P3+rii4)-in
P1p2+~2~1+~~5) (
ie2
=el+
A=
(3.51)
PO
p =
-P&n2
+ in =e3+
2
ie,
Bo
seeking a solution of (3.50) of the form W = B exp AJ leads to a bi-quadratic with roots
equation for il
?L;= &+ itI = ; [ - ,& + (f?f - @$“] (3.52) a; = s, + it2= ; [--PI - (pf - 4p2)‘“] S, and ta being real constants. The complex numbers S, + it, have unique square roots with negative real part, say -1, + im,, where m, = -t,/21,
1, = 2-‘“[(S2, + @‘2 + S,]l”,
(3.53)
the solution for Y is Y = B exp(-1,
+ im,)jj + C exp(-l2
+ E exp[-(-1,
+ im2)jj+ D exp[-(-1,
+ im,)jj]
+ im2)y]
(3.54)
where B, C, D and E are complex constants, and it follows from (3.49) that l?, = B[(P2 + P&f - iP2n] * exp[(-1, + im,)j] + C[(p2 + @,)A,”- io2n] - exp[(-1, + D[(jQ2+ j13)A.f- ip2n] * exp[-(-lI + im&] + E[(j12 + ,~I,)dz- iP2n] - exp[-(-1, & = B[-(PI + ,k,)kf + i&n]exp[(-ll
+ im&] + C(-(PI + ,&)A$+ ip2n]exp[(-1,
f D[-(j& +&&If + ip,n]exp[-(-lI + E[-(pl
+ im,)y] + im2)9] -I-im2)Y]
+ im&j] + a4)Ag+ iP2n]exp[ - (-l2 + im2)y] (3.55)
Applying the boundary conditions (3.5) and (3.6) to eqns (3.55) gives jil = sin nZ,
R, = 1,
z= 1,
M=O,
on
J=FR
0, = sin nZ,
a2=1,
N=l,
G=O,
on
jj=T&
and
(3.56)
in (C - E)[exp(-
12
where
we must lame B = D,
Thus,
setting
It fQilOW§ frsm (3.55)
wkere
The influence of vibrating plates on poiseuilie flow of a binary mixture
i&,12”)
qm
iqf2l-o
321
q (gl
Fig. 1. Variation of fluid velocities. a = S/6, pz = 1.4, n = 1. sl,
s2, tl
and t2 being real constants, we calculate from eqns (3.51) and (3.52) that
[~(rl+r~)]liz]
,,=&+
S2=~[-e,-[~(r1+r,)lin]
r
IJZ
t*=;i-e2-t- &-r*+KJ) II t2=;[-ez
- [;
(-r* +
(3.64)
rq]
where rl =
ef - ef - 4e3
r2 =
e, . e2
-
2~
(3.65)
r3= (T?+ 4ry
G
’
,a 2
u,
(2lT)
“l,rl,
i7( 11, 2
Fig. 2. Variation of fluid velocities. p1 = 5/6, pz = 5, n = 1.
322
Fig. 3. Variation of fluid veiocities. p1 = 516, p2 = 1.4, PI= 100.
and it follows that
The numerical results are divided into two groups, The first group gives the variation of the velocities of the fluid with frequency at a fixed dimensionless time (f = mz/Zn, ~PX = I, 2, 3, 4; pz= 1, 100) and at a fixed distance from the plates. In the second set a fixed frequency of oscillation of the plates is assumed and the variation of the velocities with distance from the plates are found, again at a fixed dimensionless time. We use some numericai results which are E. Craine [I?]. With a varied selection of values for given to complete the discussion by
Fig. 4. Variation of fluid velocities. pI = 5/6, p2 = 5, n = 180.
The influence of vibrating plates on poiseuille flow of a binary mixture
323
these quantities computer programmes were run. Though in all the curve shown certain parameters are fixed (aI = 1, p3 + +1/2, ,& = +1/2, x5 = -l/2, PI = -1, pz = -1). Variation of fluid velocities is investigated for dimensionless plate vibration frequency ranging 12= 1 (Figs 1 and 2) and y1= 100 (Figs 3 and 4), taking viscosity coefficient ranging I.$= 1, 4 and 5, nS= mn/2 m(l,2, 3,4). Comparison of Figs 1 and 2 show that flow characteristics do not show a distinctive variation in Fig. 1 compared to Fig. 2. Comparing Figs 3 and 4, one concludes that figure exhibits nearly single fluid characteristics compared to Fig. 4.
REFERENCES [l] C. TRUESDELL and R. TOUPIN, The Classical Field Theories, Hundbuch der Physik, (Ed. S. FLOGGE), Band 111/l. Springer, Berlin (1960). [2] A. E. GREEN and P. M. NAGHDI, ht. J. EngrzgSci. 3,231 (1965). [3] A. C. ERINGEN and J. D. INGRAM, Znt. J. Engng Sci. 3, 197 (1965). [4] A. E. GREEN and P. M. NAGHDT, Q. J. Mech. uppl. Math. 22, 427 (1969). [5] R. E. CRAINE, A. E. GREEN and P. M. NAGHDI, Q. J. Mech. appl. Math. 23, 171 (1970). [6] N. MILLS, Znt. J. Engng Sci. 4, 97 (1966). [7] R. M: BOWEN, J. Chem. Phys. 49, 162.5 (1968). [S] I. MULLER, Arch. Rntl. Mech. Anal. 28, 1 (1968). [9] R. J. ATKIN, Q. J. Mech. appl. Math. 21, 171 (1968). [lo] R. M. BOWEN and D. J. GARCIA, Znt. J. Engng Sci. 8,63 (1970). [ll] J. D. INGRAM and A. C. ERINGEN, ht. J. Engng Sci. 5, 289 (1967). [12] R. E. CRAINE, Oscillations of a plate in a Binary mixture of incompressible Newtonian fluids. Znt. J. Engng. Sci. 9, 1177 (1971). [13] t9Fi)CRAINE
An oscillating plate in a binary mixture of hemihedral fluids. J. Appl. Math. Phys. (ZAMP)
[14] J. Y. IkAZAKIA and R. S. RIVLIN, Rheol. Acta 17,210-226 (1978). [15] J. Y. KAZAKIA and R. S. RIVLIN, Rheol. Acta 18, 224-255 (1979). (Received 21 April 1987)
24
~2~7225/88
$3.00+ 0.00
Copyright @ 1988 Pergamon Press plc
Printed in Great Britain. AU rights reserved
THE INFLUENCE OF VIBRATING PLATES ON POISEUILLE FLOW OF A BINARY MIXTURE Faculty of Mech~i~~
En~nee~ng, Department of Mechanics, Istanbul Technical Unive~ity, ~~rn~~su~~ Ist~bul, Turkey
Abstmct-A mixture of two incompressible Newtonian fluids is considered and the flow induced by steady vibrations of plates on poiseuille flow between two parallel plates, under a constant pressure gradient, is then investigated. Calculations are made for longitudinal vibrations of the plates. 1. INTRODUCTION In
recent years much work has been published on the continuum theory of mixtures. 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 [Z], uses only an equation of energy balance and entropy production inequality as basic postulates. Coincidentally, a paper by A, C. Eringen and 9. D. Ingram has been published which presents a unified approach for the derivation of basic conservation equations for a chemically reacting continuum [3]. Mixtures of fluids which are viscous in the sense that their constitutive equations contain gradients of the velocity fields of the constituents have been considered in the context of the continuum theory by Green and Naghdi [4], Craine et al. [53, Mills [6J, Bowen [7], Mtiller 181, Adkins [9], R. M. Bowen and Garcia [lo], Eringen and Ingram [ll] have investigated mixtures of chemically reacting fluids in addition there is a considerable literature in which general two-phase flows are viewed from a different ~e~oint, much of this work being concerned with suspensions of solid particles in fluids. Lastly we mention the time dependent problems which has been investigated by Craine 112, 131. In the work described here a binary mixture of incompressible, Newtonian fluids is considered. In Section 2 we use the governing field equations of the continuum theory of mixtures and the notations given by R. E. Craine [12, 131. The appropriate constitutive theory developed by the authors mentioned above is given in Section 2 [4-61. In Sections 3 and 4 of present paper, the flow induced by longitudinal vibrations of two infinite parallel plates (plane poiseuille flow), under a constant pressure gradient, is investigated and an exact solution is obtained. 2. BASIC THEORY In this section we summarize the theory, for more details the reader should consult Craine [12]. A mixture of two continua, in motion relative to each other, is considered. At an arbitrary time t it is assumed that at each place z 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. The velocities of the l- and, 2-constituents are denoted by Vfl) and V@ respectively, and we define DC*) a _=_+.j_,d
Dr
dt
&j
42-l)
where a takes the values 1, 2. The governing kinematic equations and the energy equation of a binary mixture in which the constituents have a common temperature T and do not interact chemically are
(2.2)
314
The quantities pm> Ua, CT(~),FCC), e_3,P and q are in turn rhe delE&.yy,the fnicrmal energy pei? unit mass, the partial stress, the body force per unit mass of the ath c~nst~t~e~t~ the difksive force vector, the heat supply per unit vo~~~~, and the heat flux vecteor[4]. The partial stresses satisfy O@l4” Cr_i$= 0
<2.6)
where suffixes enclosed in square brackets refer to components of the ant-isymmetric part of a second order tensor and circular brackets have a similar meaning in relation to the symmetric ~~a~s~~s-~~he~ ~n~~ua~~~ymay be written [S] , part. If s, denotes the partial entropy then t
The formulation of constitutive equations f~sr a mixture of Newtonian fluids is now considered, attention being restricted to the case in which both constituents are incompsressible. At some time tOr SI and S, are separated and have constant densities plOl pzo respectiveiy. The mixture is formed by adding the element: of ~7~aa _XA, of mass dq and volume d& to ehe element of S, at YA, of mass $mz and volume dV,. There is assumed to be no change of total volume when the mixt~~ Is formed, so that a~ element of the mixture at xi of mass dmr t dmz oceq&s a volume de/, + d&2; Such a mixture was considered by Mills [6]. We now write
When S, and Sz are separated we have
Using (2.8) and (2.10) it follows that
For the densities of SIs,,,!&in the rni~t~r~ we write
So that the incompressibility
condition (2.11) now becomes
where y is a ~orn~os~t~~~ factor. The density p of the mixture is given by P = P1-k Pz and we have
The influence of vibrating plates on poiseuille flow of a binary mixture
315
(2.16)
(pzo - P)d~’ + (P - PlO)d~’ - ~iUj= 0
the quantities 5 and d(“) being defined by Ei
=
dZi”)
P,i
=
(2.17)
“&)
partial Helimholtz free energies A, are introduced,
For convenience
defined by the relations
A a = U, - TS,
(2.18)
As the basic constitutive postulate we assume that the quantities A,, S,, co, q and a(“) are arbitrary functions of p and T but depend linearly on 5, b, a, d’“) and I’, where J.j$;’ = #
bi = T,i,
JYik= j.@ - wp
Id+
a.I = vi(l)- $2)
(2.19)
Adopting an analogous method to that outlined in [12], it is possible to deduce from the Clausius-Duhem inequality that the constitutive equations have the reduced forms A, = A&
‘A =A@,
7%
(P$++
-(Pzo-P)
Pl’
T),
Oi 4i =
= -
s, = UP,
=
__dA dT
(2.20) (2.21)
P2=(P-P1o)(P2$$-")
aai - Agi + ybi
(2.22)
T(koai + k,bi)
(2.23)
Us’ = (-Pl+
n,d~)+ a3d~‘)6, + 2pldik
Us’ = (-Pz +
n,d~) + &df))6,
where the coefficients satisfy the inequalities
s
T),
+ Z!/igdik+ n,rik
(2.24)
+ 2/44dik+ 2cczdik - a5rik
(2.25)
E, ill, . . . , d5, y, ko, k3, pl, . . . , y, are functions
of p and T, and
Q 2 0,
k3 3 0,
,420,
,&~001
(Y,+!h)2s‘hh
(2.26)
2 ;1,+-/L,ao, 3
(2.27) (2.28)
3. FLOW
INDUCED
BY STEADY LONGITUDINAL TWO PARALLEL PLATES
VIBRATIONS
OF
In this section we consider a binary mixture of incompressible Newtonian fluids, to be contained between infinite parallel rigid plates normal to the y-axis of a rectangular Cartesian coordinate system X and to flow under the action of constant pressure gradients PI, P2 in the direction of the x-axis. Simultaneously the two plates are subjected to a sinusoidal vibration in their own planes, of equal amplitudes with angular frequency w, parallel to the x-direction. For convenience set x1 = x, x2 = y, x3 = z, and let the plates be located at y = T/z. We assume that body forces are absent, and that the temperature remains constant throughout the mixture (for discussion of this latter assumption see Craine [12]). Since the temperature may now be dropped from the list of variables in the problem, all the coefficients pl etc. appearing in the constitutive equations (2.24) and (2.25) depend only on the density of the mixture p. We assume a flow field of the form v(‘) = (u,(y,t),
0, 0) ,
vC2’ = (%(Y,
t), 0, 0)
P = P(Y,
t)
(3.1)
in which u,(y, t) and v,(y, t) are u,(y, t) = z+,(y) + L sin cot+ M cos ot
(3.2)
v,(y, t) = v,(y)
(3.3)
P = P(Y,
+ N sin ot + G cos wt t>
(3.4)
The iniluence of vibrating plates on poiseuille flow of a binary mixture
we have p # plo, p # pzo and d2(pA)/dp2 # 0. It must the mixture p is a constant. Since p has been proved have been assumed, all the coefficients in eqns (3.17) From (3.17), (3.18), (3.19) and (3.20) it follows constant A is
A=
_-=
a& dx
have dpldy = 0, from here the density of to be constant and isothermal conditions and (3.18) are constant. that A and fI(y) are constant, where
34 dx --=A
Pzo-P
317
P-P10
di2
(3.24)
3.x
On substituting eqns (3.2) and (3.3) into eqns (3.17) and (3.18) we find that (3.25) (3.26) (3.27) (3.28) (3.29) (3.30)
introducing the dimensionless
quantities,
(3.31)
where ,U is a typical viscosity dimensionless forms,
in the problem,
eqns (3.25)-(3.30)
take on the following
Replacing (3.33) by the sum of (3.32) and (3.33), the equations can be written
with the use of the boundary (3.26) gives,
conditions
G&F@ = 0, “Ls,(Th) = 0 on f = T&, integrating
Substituition of (3.40) into eqn (3.38) gives a linear differential linear equation, for solution &, and DOwe obtain
equation.
eqn
With the soMow
oi
where,
(3.43)
The last four of eqns (3.34)-(3.37) the notation
may be rewritten
Ri,=b-dti,
as two complex equations by introducing
R,=A+iC
(3.44)
Thus (3.45)
319
The influence of vibrating plates on poiseuille flow of a binary mixture
Replacing (3.46) by the sum of (3.45) and (3.46) the governing equations can be written. (3.47) [&
+ ,i&)D2 - i&n]&
where D denotes the differential operator
+ [(p2 + ,i&)D2 - ip2n]Z?, = 0
(3.48)
d/d7 and put
WI = [(fi2 + ,i&)D2 - i&n]Y,
Z&= [-(PI + ,&)D2 + @&I’
(3.49)
the eqn (3.48) is satisfied identically and (3.47) becomes (D4 + P1D2 + /?$I’ = 0
(3.50)
where
-(ih+P2+P3+rii4)-in
P1p2+~2~1+~~5) (
ie2
=el+
A=
(3.51)
PO
p =
-P&n2
+ in =e3+
2
ie,
Bo
seeking a solution of (3.50) of the form W = B exp AJ leads to a bi-quadratic with roots
equation for il
?L;= &+ itI = ; [ - ,& + (f?f - @$“] (3.52) a; = s, + it2= ; [--PI - (pf - 4p2)‘“] S, and ta being real constants. The complex numbers S, + it, have unique square roots with negative real part, say -1, + im,, where m, = -t,/21,
1, = 2-‘“[(S2, + @‘2 + S,]l”,
(3.53)
the solution for Y is Y = B exp(-1,
+ im,)jj + C exp(-l2
+ E exp[-(-1,
+ im2)jj+ D exp[-(-1,
+ im,)jj]
+ im2)y]
(3.54)
where B, C, D and E are complex constants, and it follows from (3.49) that l?, = B[(P2 + P&f - iP2n] * exp[(-1, + im,)j] + C[(p2 + @,)A,”- io2n] - exp[(-1, + D[(jQ2+ j13)A.f- ip2n] * exp[-(-lI + im&] + E[(j12 + ,~I,)dz- iP2n] - exp[-(-1, & = B[-(PI + ,k,)kf + i&n]exp[(-ll
+ im&] + C(-(PI + ,&)A$+ ip2n]exp[(-1,
f D[-(j& +&&If + ip,n]exp[-(-lI + E[-(pl
+ im,)y] + im2)9] -I-im2)Y]
+ im&j] + a4)Ag+ iP2n]exp[ - (-l2 + im2)y] (3.55)
Applying the boundary conditions (3.5) and (3.6) to eqns (3.55) gives jil = sin nZ,
R, = 1,
z= 1,
M=O,
on
J=FR
0, = sin nZ,
a2=1,
N=l,
G=O,
on
jj=T&
and
(3.56)
in (C - E)[exp(-
12
where
we must lame B = D,
Thus,
setting
It fQilOW§ frsm (3.55)
wkere
The influence of vibrating plates on poiseuilie flow of a binary mixture
i&,12”)
qm
iqf2l-o
321
q (gl
Fig. 1. Variation of fluid velocities. a = S/6, pz = 1.4, n = 1. sl,
s2, tl
and t2 being real constants, we calculate from eqns (3.51) and (3.52) that
[~(rl+r~)]liz]
,,=&+
S2=~[-e,-[~(r1+r,)lin]
r
IJZ
t*=;i-e2-t- &-r*+KJ) II t2=;[-ez
- [;
(-r* +
(3.64)
rq]
where rl =
ef - ef - 4e3
r2 =
e, . e2
-
2~
(3.65)
r3= (T?+ 4ry
G
’
,a 2
u,
(2lT)
“l,rl,
i7( 11, 2
Fig. 2. Variation of fluid velocities. p1 = 5/6, pz = 5, n = 1.
322
Fig. 3. Variation of fluid veiocities. p1 = 516, p2 = 1.4, PI= 100.
and it follows that
The numerical results are divided into two groups, The first group gives the variation of the velocities of the fluid with frequency at a fixed dimensionless time (f = mz/Zn, ~PX = I, 2, 3, 4; pz= 1, 100) and at a fixed distance from the plates. In the second set a fixed frequency of oscillation of the plates is assumed and the variation of the velocities with distance from the plates are found, again at a fixed dimensionless time. We use some numericai results which are E. Craine [I?]. With a varied selection of values for given to complete the discussion by
Fig. 4. Variation of fluid velocities. pI = 5/6, p2 = 5, n = 180.
The influence of vibrating plates on poiseuille flow of a binary mixture
323
these quantities computer programmes were run. Though in all the curve shown certain parameters are fixed (aI = 1, p3 + +1/2, ,& = +1/2, x5 = -l/2, PI = -1, pz = -1). Variation of fluid velocities is investigated for dimensionless plate vibration frequency ranging 12= 1 (Figs 1 and 2) and y1= 100 (Figs 3 and 4), taking viscosity coefficient ranging I.$= 1, 4 and 5, nS= mn/2 m(l,2, 3,4). Comparison of Figs 1 and 2 show that flow characteristics do not show a distinctive variation in Fig. 1 compared to Fig. 2. Comparing Figs 3 and 4, one concludes that figure exhibits nearly single fluid characteristics compared to Fig. 4.
REFERENCES [l] C. TRUESDELL and R. TOUPIN, The Classical Field Theories, Hundbuch der Physik, (Ed. S. FLOGGE), Band 111/l. Springer, Berlin (1960). [2] A. E. GREEN and P. M. NAGHDI, ht. J. EngrzgSci. 3,231 (1965). [3] A. C. ERINGEN and J. D. INGRAM, Znt. J. Engng Sci. 3, 197 (1965). [4] A. E. GREEN and P. M. NAGHDT, Q. J. Mech. uppl. Math. 22, 427 (1969). [5] R. E. CRAINE, A. E. GREEN and P. M. NAGHDI, Q. J. Mech. appl. Math. 23, 171 (1970). [6] N. MILLS, Znt. J. Engng Sci. 4, 97 (1966). [7] R. M: BOWEN, J. Chem. Phys. 49, 162.5 (1968). [S] I. MULLER, Arch. Rntl. Mech. Anal. 28, 1 (1968). [9] R. J. ATKIN, Q. J. Mech. appl. Math. 21, 171 (1968). [lo] R. M. BOWEN and D. J. GARCIA, Znt. J. Engng Sci. 8,63 (1970). [ll] J. D. INGRAM and A. C. ERINGEN, ht. J. Engng Sci. 5, 289 (1967). [12] R. E. CRAINE, Oscillations of a plate in a Binary mixture of incompressible Newtonian fluids. Znt. J. Engng. Sci. 9, 1177 (1971). [13] t9Fi)CRAINE
An oscillating plate in a binary mixture of hemihedral fluids. J. Appl. Math. Phys. (ZAMP)
[14] J. Y. IkAZAKIA and R. S. RIVLIN, Rheol. Acta 17,210-226 (1978). [15] J. Y. KAZAKIA and R. S. RIVLIN, Rheol. Acta 18, 224-255 (1979). (Received 21 April 1987)
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