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Appl. Math. Inf. Sci.6, No. 1, 29-33 (2012) 29

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Natural Sciences Publishing Cor.

Some couette flows of a Maxwell fluid with wall slip

condition

Dumitru Vieru1,2 and Azhar Ali Zafar2

1 Department of Theoratical Mechanics, Technical University of Iasi, Romania2 Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan

Received: Jul 8, 2011; Revised Oct. 4, 2011; Accepted Oct. 6, 2011

Published online: 1 January 2012

Abstract: Couette flows of a Maxwell fluid produced by the motion of a flat plate are analyzed under the slip condition at boundaries.

The bottom plate is assumed to be translated in its plane with a given velocity. The flow of the fluid is studied in the assumption that the

relative velocity between the fluid at the wall and the wall is proportional to the shear rate at the wall. Exact expressions for velocity and

shear stress are determined by means of a Laplace transform. The velocity fields corresponding to both slip and non slip conditions for

Maxwell and Newtonian fluids are obtained. Two particular cases, namely translation with constant velocity and sinusoidal oscillations

of the bottom plate, are studied. Results for Maxwell fluids are compared with those of Newtonian fluids in both cases with slip and

non slip conditions. Some properties of the flow are also presented.

Keywords: Maxwell fluid, couette flows, wall slip condition.

1. Introduction

Since 1867 J.C.Maxwell (1831-1879) observed that somefluids, such as air, exhibit both viscous and elastic behaviours.The constitutive relation, in modern notations, proposedby Maxwell for these fluids is given by [1-3]

S + (S LS SL

T

) =A, (1)

whereS is the extra stress tensor, L is the velocity gradi-ent, A= L+LT is the first Rivlin-Erickson tensor, ( 0)and(> 0) are the relaxation time and dynamic viscos-ity, respectively, and the superposed dot indicates the ma-terial time derivative. Maxwell fluids also can be consid-ered as a special case of a Jeffreys-Oldroyd B fluid, whichcontain both relaxation and retardation time coefficients[1]. Maxwells constitutive relation can be recovered fromthat corresponding to Jeffreys-Oldroyd B fluids by settingthe retardation time to be zero. The fluids described by(1) are referred to as viscoelastic fluids of Maxwell type,or simply Maxwell fluids. Several fluids, such as glyc-

erin, crude oils or some polymeric solutions, behave asMaxwell fluids. The reference [4] contains more examples

of this type of fluids. The Maxwell model has been the sub-ject of study for many researchers. The first exact solutionof Stokes first problem, also known as Rayleighs prob-lem, for Maxwell fluids was given by Tanner [5]. Othersolutions of Stokes first problem for Maxwell fluids, to-gether some interesting properties, have been obtained byJordan et al [6], Jordan and Puri [7] and, for Oldroyd B

fluid, by Christov [8]. The unsteady Couette flow of a Maxwellfluid between two infinite parallel plates was studied byDenn and Porteous [9] while, for second grade dipolarfluids, by Jordan [10] and Jordan and Puri [11]. Interest-ing subjects and solutions regarding the Couette or Stokesflows of non-Newtonian fluids can be found in reference[12-15]. In aforementioned papers the assumption that aliquid adheres to the solid boundary, so called nonslip bound-ary condition, was considered fulfilled. The nonslip bound-ary condition is one of the basic principles in which themechanics of the linearly viscous fluids was built. Manyexperiments are in favour of the nonslip boundary condi-tion for a large class of flows. An interesting discussionregarding the acceptance of the nonslip condition can be

found in [16]. Even if the nonslip condition has provedto be successful for a great variety of flows, it has been

Corresponding author: e-mail: dumitru [email protected]

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30 Dumitru Vieru et al : Some couette flows of a Maxwell fluid with wall slip condition

found to be inadequate in several situations, such as prob-lems involving multiple interfaces, flows in micro chan-nels or in wavy tubes, flows of polymeric liquids or flowsof rarefid fluids. Many years ago, Navier [17] proposed aslip boundary condition wherein the relative velocity (theslip velocity) depended linearly on the shear stress. A largenumber of models have been proposed for describing theslip that occurs at solid boundaries. Many of them can befound in the reference [18]. One of the early studies ofthe slip at the boundary was under taken by Monney [19].Recently, several papers regarding flows of Newtonian ornon-Newtonian fluids with slip at the boundary have beenpublished. Khalid and Vafai [20] were studied the effect ofthe slip condition on Stokes and Coutte flows due to an os-cillating wall; Vieru and Rauf [21] analyzed Stokes flowsof a Maxwell fluid with wall slip condition; the Coutte

flow of a third grade fluid with rotating frame and slipcondition was studied by Abelman et al [22]. Many in-teresting and useful results regarding solutions for flowsof non-Newtonian fluids with slip effects are in references[23-25].

In this study, Couette flows of a Maxwell fluid pro-duced by the motion of a flat plate are analyzed under theassumption of the slip boundary conditions between theplates and the fluid. The motion of the bottom plate is arectilinear translation in its plane with velocity uw(t) =Uof(t), while the upper plate is at rest. Exact expressionsfor velocity and shear stress are determined by means ofa Laplace transform for Maxwell and Newtonian fluids.

Expressions of the relative velocity are determined, andthe solutions corresponding to flows with nonslip at theboundary are also presented. Two particular cases, namelythe translation of the bottom plate with a constant veloc-ity and sinusoidal oscillations are studied. In each case,the expression of the velocity is written as a sum betweenthe permanent solution and the transient solution. Forlarge values of time t the transient solution tends to zeroand the fluid flows according to the permanent solution.Some relevant properties of the velocity and comparisonsbetween solutions with slip and nonslip at the boundariesare presented.

2. Problem formulation and solution

Consider an infinite solid plane wall situated in the (x,z)-plane of Cartesian coordinate system with the positiveyaxis in the upward direction. The second infinite solidplane wall occupies the planey = h > 0. Let an incom-pressible, homogeneous Maxwell fluid fill the slab y (0, h). Initially, the fluid and plates are at rest. At the mo-ment t= 0+, the fluid is set in motion by the bottom plate,which begins to translate along the xaxis with the veloc-ityuw(t) =Uof(t), whereUo >0 is a constant andf(t)is a piecewise continuous function defined on [0,) andf(0) = 0. Also, we suppose that the Laplace transform of

the functionf(.)exists. In the case of parallel flow alongthe xaxis, the velocity vector is v = (u(y, t), 0, 0)while the constitutive relation and the governing equationare given by [7, 8, 21]

+

t =

u

y, (y, t) (0, h) (0,), (2)

+

t =

u

y, (y, t) (0, h) (0,), (3)

where(y, t) = Sxy(y, t) is one of the nonzero compo-nent of the extra stress tensor and is the constant densityof the fluid. In this paper, we consider the existence of slipat the walls and assume that the relative velocity between

the velocity of the fluid at the wall and wall is proportionalto the shear rate at the wall [20,21]. The boundary condi-tions due to wall slip as well as the initial conditions are

u(0, t) u(0, t)y

=Uof(t), t >0, (4)

u(h, t) + u(h, t)

y = 0 , t >0, (5)

u(y, 0) = 0, u(y,0)t = 0,(y, 0) = 0, y [0, h], (6)

whereis the slip coefficient. We introduce the followingnon-dimensionalization:

t = tT , y = yh , u

= uUo , =

(hUo/T),

= T , =

h

(7)

T >0 being a characteristic time. Equations (2-6), in non-dimensional form are (dropping the * notation)

+

t =

1

R

u

y, y (0, 1) (0,) (8)

2

ut2

+ ut

= 1R

2

uy2

, y (0, 1) (0,) (9)

u(0, t) u(0, t)y

=g(t), t >0 (10)

u(1, t) + u(1, t)

y = 0 , t >0 (11)

u(y, 0) = 0, u(y,0)t = 0,(y, 0) = 0, y [0, 1] (12)

whereR = h2

Tis the Reynolds number,= is the kine-matic viscosity,g(t)is given byf(T t).

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Appl. Math. Inf. Sci.6, No. 1, 29-33 (2012) / www.naturalspublishing.com/Journals.asp 31

2.1. Velocity field

By applying the temporal Laplace transform, L{

.}

[26],to Eqs. (9)-(11) and using the initial conditions (12)1,2 weobtain the following set of equations:

2u(y, q)

y2 R(q2 + q)u(y, q) = 0 (13)

u(0, q) u(0, q)y

=G(q) =L{q(t)} (14)

u(1, q) + u(1, q)

y = 0 (15)

whereqis the Laplace transform parameter and u(y, q) =L

{u(y, t)

}. The solution of differential equation (13) with

the boundary conditions (14) and (15) is given by

u(y, q) =G(q)G1(y, q) (16)

where

G1(y, q) =

= sh[(1y)

Rw(q)]+

Rw(q)ch[(1y)

Rw(q)]

[1+2Rw(q)]sh[

Rw(q)]+2

Rw(q)ch[

Rw(q)]

(17)

and

w(q) = (q2 + q) =[(q+ 1

2)2 ( 1

2)2]. (18)

In order to determine the inverse Laplace transform of func-tionG1(y, q), we consider the auxiliary function

F1(y, q) =sh[(1 y)Rq] + Rqch[(1 y)Rq]

(1 + 2Rq)sh(

Rq) + 2

Rqch(

Rq)(19)

Observing that the singular points ofF1(y, q) are simplepoles located at

qn = pn2

R , n= 1, 2,... (20)

wherepn= 0are the real roots of the equation

tan(pn) = 2pn

2pn2 1 , (21)

we invert functionF1(y, q)

by using the residue theoremto evaluate the Laplace inversion integral [26]. Such that,after simplifying, we obtain

f1(y, t) =L1{F1(y, q)} =

=

n=1Res[F1(y, q)e

qt, qn] =

=

n=1An(y)exp(pn

2

R t)

(22)

where

An(y) = sin(1y)pn+pncos(1y)pn

(+1)R sinpn R2pn(1+22pn2) cospn=

= 2pnRsin(ypn)+pncos(ypn)

(1+2)+2pn2 .

(23)

By comparing (17) and (19) we observe that G1(y, q) =F1[y, w(q)] and, using the inverse Laplace transform for

composed functions (see (A1) and (A2) from the AppendixA), we obtain

g1(y, t) =L1{G1(y, q)} =

0

f1(y, s)h(s, t)ds (24)

where

h(s, t) =L1{esw(q)} == t

2es2t4

k=0

(s)k(k+1)!(2k+1)!

0

z2k+1J2(2

zt)dz (25)

andJ()is the Bessel function of first kind and order .Replacing (22) and (25) into (24) we find that

g1(y, t) =

0

[

n=1

An(y)e pn2s

R ][ t2

es2t4

k=0

(s)k(k+1)!(2k+1)!

0

z2k+1J2(2

zt)dz]ds=

= t2

e t2

n=1

An(y)

k=0

()k(k+1)!(2k+1)!

0

z2k+1J2(2

zt)dz

0

ske(pn

2

R 1

4)sds=

= t2

e t2

n=1

An(y)

0J2(2

zt)

k=0()k(k+1)(k+1)!(2k+1)!

z2k+1

bnk+1dz,

(26)

wherebn = pn

2

R 14 >0 andis the Gamma function.By using (A3) from the Appendix A we obtain a new

expression of the functiong1(y, t), namely

g1(y, t) =

= 2te t

2

n=1

An(y)0

1z sin

2(

z2

bn)J2(2

zt)dz. (27)

Now, using the properties of the Bessel functions [27] wecan show that

g1(y, t) =L1[G1(y, q)] =

=e t2

n=1

An(y)

bnsin(tbn

) (28)

Finally, we obtain:a. The velocity field corresponding to the flow of a Maxwellfluid with slip at the boundary is given by

uMs (y, t) = (g g1)(t) =t0

g(t s)g1(y, s)ds=

=

n=1

An(y)bn

t0

g(t s)e s2 sin(s

bn)ds.

(29)

b. For the flow of a Maxwell fluid with a nonslip boundarycondition, that is= 0, the functionAn(y)given by Eq.(23) becomes

A1n(y) = 2n

R sin(ny), n= 1, 2, ..... (30)

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and the velocity field has the expression

uM(y, t) = 2

R

n=1

n sin(ny)Rn

t0

g(t s)e s2 sin(scn)ds,(31)

wherecn = n22

R 14 >0.c. The solution in the transform domain corresponding tothe flow of a viscous fluid with slip at the wall, that is for= 0and= 0, is given by

u(y, t) =G(q)F1(y, q) (32)

and the(y, t)-domain solution is

UNs (y, t) =t0

g(t s)f1(y, s)ds==

n=1An(y)

t0

g(t s)exp(pn2R s)ds,(33)

whereAn(y)is given by Eq. (23). For f(t) = sin(t) orf(t) = cos(t), the velocity field given by Eq.(33) wasdetermined in equivalent forms by Khalid and Vafai [20].d. For flows of viscous Newtonian fluids with a nonslipboundary condition, that is = 0 and = 0, the transformdomain solution is

uN

(y, q) =G(q)G2

(y, q), (34)

where

G2(y, q) = sh[(1 y)Rq]

sh[

Rq] , (35)

and the(y, t)-domain solution is given by

uN(y, t) = 2

R

n=1

n sin(ny)

t

0

g(t s)en22

R sds.(36)

The relative velocity between the fluid at the bottom wall

and wall itself for Maxwell fluids is

uMrel(t) =uMs (0+, t) g(t) == 2R

n=1

pn2

bn{2pn2+(1+2)}

t0

g(t s)e s2 sin(s

bn)ds g(t)

(37)

and for a viscous Newtonian fluid is given by

uNrel(t) =uN s(0+, t) g(t) == 2R

n=1

pn2

2pn2+(1+2)

t0

g(t s)e pn2

R sds g(t).(38)

2.2. Shear Stress

In order to determine the shear stress (y, t) we use Eqs.(8),(16) and (28). Applying the Laplace transform to Eq.(8) with the initial condition (12)3, we obtain

(y, q) = 1

RG(q)G3(y, q) (39)

where

G3(y, q) = 1

q+ 1/

G1(y, q)

y . (40)

The inverse Laplace transform of functionG3(y, q)is

g3(y, t) =t

0e

ts

g1(y,s)y ds=

=

n=1

Bn(y)bn

et

t0

e s2 sin(s

bn)ds

(41)

where

Bn(y) = dAn(y)

dy =

= 2pn2

Rcos(ypn)pnsin(ypn)

(1+2)2pn2 .(42)

Eq.(41) can be written in the simple form

g3(y, t) =

n=1

Bn(y)bn

2e

t2

1+4bn

[2bne t2 + sin(t

bn) 2

bncos(t

bn)]

(43)

The(y, t)-domain solution for the shear stress is given by

(y, t) = 1

R(g g3)(t) = 1

R

t

0

g(t s)g3(y, s)ds.(44)

3. Some particular cases of the motion of the

plate

In this section we consider two functions correspondingto the motion of the bottom plate, namelyf(t) = H(t)

and f(t) = sin(t) , with > 0 being a constant. Wechoose the characteristic timeT = 1 , for the dimension-less variables and functions given by Eq.(7), and obtaing(t) =H(t)andg(t) = sin t, respectively.

3.1. Solution for the translation of the bottom

plate with a constant velocity

The motion of the bottom plate is given by the functiong(t) =H(t)and the velocity u(y, t)is obtained from Eqs.(29), (31), (33) and (36) with g (t s) = 1 . The veloc-ity fields corresponding to this type of the motion have the

following expressions:

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a. Maxwell fluids with slip at the boundary:

uMs (y, t) =H(t){

4

n=1

An(y)

1+4bn2

n=1

An(y)e

t2

bn(1+4bn)

[sin(t

bn) + 2

bncos(t

bn)]},

(45)

which, by using (A4) from the Appendix A, can be writtenin the simpler form

uMs (y, t) =H(t){1+y1+2e t2

n=1

sin(ypn)+pncos(ypn)pn(1+2+2pn2)

[2 cos(t

bn) +

1bn

sin(t

bn)]}.

(46)

For large values of the time t the velocity field given byEq. (46) tends to the stationary solution

ustMs (y, t) =1 + y

1 + 2 (47)

and for t 0+ uMs (y, t) tends to zero. Maxwell fluidswith nonslip at the boundary:

uM(y, t) =H(t){1 y e t2

n=1

sin(ny)n

[2 cos(tcn) + 1cn sin(t

cn)]}.

(48)

b. Newtonian fluids with slip at the boundary:

uN s(y, t) =H(t){1+y1+22

n=1

sin(ypn)+pncos(ypn)pn(1+2+2pn2)

epn

2

R t}. (49)

c. Newtonian fluids with nonslip at the boundary:

uN(y, t) =H(t){1 y 2

n=1

sin(ny)

n e

n22

R t. (50)

The relative velocity is given by

uMrel(t) =H(t){ 1+2e t2

n=1pn

pn(1+2+2pn2)

[2 cos(t

bn) +

1bn

sin(t

bn)]},

(51)

for the Maxwell fluid, respectively,

uNrel(t) =H(t)[ 1+22

n=1

pnpn(1+2+2pn2)

epn

2

R t],

(52)

for Newtonian fluids. By using the illustrations generatedwith the software Mathcad, we discuss some relevant phys-ical aspects of the flow. Also, the rootspn , n = 1, 2,....,of Eq.(21) are determined by means of the software root(f(x), x , a , b) from Mathcad. For the dimensionless slipcoefficient, {0.4, 0.7} the rootspnare presented in theTable 1 from the Appendix A. In the figures, we use =

9.15255103m2s , = 0.55s , = 1.050kgm3corre-sponding to Maxwell fluid 1%P M M A in DEM(Poly(methyl-metha crylate) in diethyl malonate) [4], and h = 0.2m,Uo = 0.6m/s, = 0.14m. The Reynolds number cor-responding to aforementioned values isR = 3.4962934,the dimensionless slip coefficient is = 0.7 and the di-mensionless relaxation time is = 0.44. Also, we use thefollowing abbreviations for dimensionless velocities:uM s,the velocity for Maxwell fluid with slip at the wall; uM, thevelocity for Maxwell fluid with nonslip at the wall; uN s,the velocity for Newtonian fluid with slip at the wall; uN,the velocity for Newtonian fluid with nonslip at the wall. InFig. 1 we show the dimensionless velocity, u(y, t), versus tfory {0.05, 0.25, 0.85} . For comparison, we have plot-ted the functions corresponding to Maxwell and Newto-nian fluids with both slip and nonslip boundary conditions.

For a fixed value of the spatial variabley, the velocity cor-responding to a Maxwell fluid with slip at the boundaryis zero for a short time, after that, is increasing and tendstowards the stationary solution ustMs given by Eq. (47).The velocity corresponding to Maxwell fluid with nonslipcondition has a non uniform variation at the beginning ofthe motion after that approaches to the stationary solu-tion ustM(y, t) = 1y. For Newtonian fluids, the velocityis larger in the case of a nonslip than in the case of slip atthe boundary near the bottom plate. For large values of thetimet they tend to the stationary solutions ustNs = u

stMs

, respectively ustN = ustM . Fig. 2 shows the diagrams of

velocityu(y, t)corresponding to Maxwell and Newtonian

fluids for both slip and non slip conditions. The velocitywas plotted versus yfor t {0.5, 1.5, 2.5} . For small val-ues of timet the Newtonian fluid with slip at the wall isslower than the Maxwell fluid with slip condition near thelower plate and faster near upper plate. For increasing tthe Newtonian fluid with slip condition is slower than theMaxwell fluid throughout domain of flow. In Fig.3 we haveplotted the relative velocity corresponding to Maxwell andNewtonian fluids versus t for two values of the dimension-less slip coefficient . In absolute value, the relative ve-locity decreases with increasing values ofand flatten outfor large values of timet.

3.2. Solution for the sinusoidal oscillations of

the bottom plate

In this section we consider for the motion of the bottomplate the function g(t) = sin t. The velocity fields cor-responding to this type of motion are given by Eqs. (29),(31), (33) and (36) in which g(t

s)is replaced bysin(t

s).a. Integrating by parts into Eq. (29), this yields after some

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Figure 1 Plot ofu(y, t) versus t for both cases with slip andnonslip condition.

simplifications, the velocity field corresponding to the flowof a Maxwell fluid with slip at the wall

uM s(y, t) =Q1(y)sin t + Q2(y)cos t+

+ e t2

n=1

[Q3n(y)cos(t

bn)+

+ Q4n(y)sin(t

bn)],

(53)

where

Q1(y) = 2

n=1

pn(pn2 R)(pn2 R)2 + R2

Qn(y), (54)

Figure 2 Plot ofu(y, t) versus y for both cases with slip andnonslip condition.

Q2(y) = 2

n=1

Rpn

(pn2 R)2 + R2Qn(y), (55)

Q3n(y) = 2pnR

(pn2 R)2 + R2Qn(y), (56)

Q4n(y) = pn

bn

R 2(pn2 R)(pn2 R)2 + R2

Qn(y), (57)

Qn(y) =sin(ypn) + pncos(ypn)

1 + 2+ 2pn2 , (58)

The velocity field given by Eq. (53) is the sum between the

permanent solutionuMsp (y, t) =Q1(y)sin t + Q2(y)cos t (59)

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Figure 3 Plot of the relative velocity versus t for Maxwell and

Newtonian fluids.

and the transient solution uMst (y, t) =uMs (y, t)uMsp (y, t)which can be neglected for large values of the time t. Byusing the residue theorem to evaluate the inverse Laplacetransform of functionu(y, q) given by Eq. (16), with G(q) =L{sin t} = 1q2+1 , we obtain for the permanent solutionuMsp (y, t)an equivalent expression, namely

uMsp (y, t) = 1

M

[M1P1(y) + M2P2(y)] sin t++ 1M[M1P2(y) + M2P1(y)]cos t, (60)

where

M1= [(1 2R)sh(1

R)+

+21

Rch(1

R)] cos(2

R) [2Rch(1

R)+

+ 22

Rsh(1

R)] sin(2

R),

(61)

M2= [(1 2R)ch(1

R)+

+21

Rsh(1

R)] sin(2

R)+

+[2Rsh(1R) + 22Rch(1R)] cos(2

R),

(62)

M=M12 + M2

2, (63)

P1(y) =sh[1

R(1 y)]{cos[2

R(1 y)]2

R sin[2

R(1 y)]}+

+ 1

Rch[1

R(1 y)] cos[2

R(1 y)],(64)

P2(y) =ch[1

R(1

y)]

{sin[2

R(1

y)]+

+2R cos[2R(1 y)]}++1

Rsh[1

R(1 y)]sin[2

R(1 y)],

(65)

and 1,2=

(

2 + 1 )/2. b. The velocity field cor-responding to Maxwell fluids with nonslip at the boundary

is given by

uM(y, t) = 2 sin t

n=1

(n)(n22R) sin(ny)(n22R)2+R2

2cos t

n=1

R(n) sin(ny)(n22R)2+R2 +

+ 2e t2 {

n=1

R(n) sin(ny)(n22R)2+R2 cos(t

cn)+

+ (n)[R2(n22R)] sin(ny)

cn[(n22R)2+R2] sin(t

cn)}.

(66)

The permanent solution corresponding to this type of themotion, can be written in the equivalent form

uM p (y, t) =P3(y)sin t + P4(y)cos t, (67)

where

P3(y) = {sh[1

R(1 y)] cos[2

R(1 y)]sh(1

R)cos(2

R)+

+ ch[1

R(1 y)]sin[2

R(1 y)]ch(1

R)sin(2

R)} 1

sh2(1

R)+sin2(2

R),

(68)

P4(y) = {ch[1

R(1 y)] sin[2

R(1 y)]sh(1

R)cos(2

R)+

+ sh[1

R(1 y)] cos[2

R(1 y)]ch(1

R)sin(2

R)} 1

sh2(1

R)+sin2(2

R).

(69)

c. The velocity field corresponding to the flows of Newto-

nian fluids with slip at the wall has expression

uNs (y, t) = 2 sin t

n=1

p3nQn(y)p4n+R

2

2cos t

n=1

RpnQn(y)p4n+R

2 + 2

n=1

RpnQn(y)p4n+R

2 e p

2nR

t,(70)

whereQn(y) is given by Eq. (58). The permanent solu-tion corresponding to this type of flows can be written inthe equivalent form

UNsp (y, t) = 1D [D1E1(y) + D2E2(y)]sin t+

+ 1D [D1E2(y) D2E1(y)] cos t, (71)

where

D1= sh(

R2) cos(

R2) 2Rch(

R2) sin(

R2)+

+

2R[ch(

R2

) cos(

R2

) sh(

R2

) sin(

R2

)],(72)

D2= ch(

R2) sin(

R2) +

2Rsh(

R2) cos(

R2)+

+

2R[sh(

R2) sin(

R2) + ch(

R2) cos(

R2)]

,(73)

D= D12 + D2

2, (74)

E1(y) =sh[

R2

(1 y)] cos[

R2

(1 y)]+

+R

2 {ch[R

2(1 y)] cos[R

2(1 y)] sh[

R2(1 y)] sin[

R2

(1 y)]},(75)

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E2(y) =ch[

R2

(1 y)] sin[

R2

(1 y)]++

R2{sh[

R2

(1 y)]sin[

R2

(1 y)]+

+ch[R

2(1 y)] cos[R2(1 y)]}.(76)

d. The Couette flow of a Newtonian fluid with nonslipboundary condition is characterized by the velocity field

uN(y, t) = 2 sin t

n=1

(n)3 sin(ny)

(n)4+R2

2cos t

n=1

R(n) sin(ny)(n)4+R2

+

+ 2

n=1

R(n) sin(ny)(n)4+R2

en22

R t.

(77)

The permanent solution corresponding to the expression(77) can be written in the following form

uNp (y, t) == sin t

sh2(

R2)+sin2(

R2)

{sh(

R2

) cos(

R2)sh[

R2

(1 y)] cos[

R2

(1 y)]++ch(

R2

) sin(

R2

)ch[

R2

(1 y)]sin[

R2

(1 y)]}++ cos t

sh2(

R2)+sin2(

R2)

{sh(

R2) cos(

R2)ch[

R2(1 y)]sin[

R2(1 y)]

ch(

R2

) sin(

R2

)sh[

R2

(1 y)] cos[

R2

(1 y)]}.

(78)

Some important properties of flow due to sinusoidal oscil-

lations of the bottom plate are presented using illustrationsfrom Figs. 4-6.

Figure 4 Plot of theu(y, t)versust for both cases with slip andnonslip condition.

Figure 5 Plot of theu(y, t)versusy for both cases with slip andnonslip condition.

Figure 6 Plot of the relative velocity versus t for Maxwell and

Newtonian fluids.

In Fig. 4 we plotted the velocity u(y, t) and the per-manent solutions corresponding to Maxwell and Newto-nian fluids with both slip and non slip boundary condi-tions. These functions were presented versus t for y {0.05, 0.25, 0.85} and, it is evident that the difference be-tween the velocity u(y, t) and the permanent velocityis significant only for small values of the time . We seethat in the considered case, after the moment t = 4 forMaxwell fluid with slip at the wall, respectively t = 6forthe Maxwell fluid with non slip condition the transient ve-locitiesutMs (y, t) =uMs (y, t) uMsp (y, t),utM(y, t) =

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uM(y, t)uMp (y, t)can be neglected. For Newtonian flu-ids t = 6 in the case of the flow with slip at the wallandt = 4 in the case of nonslip at the wall. After thesemoments, the fluids flow according to the permanent so-lution. Fig. 5 contains diagrams of velocity u(y, t), ver-susy for six different values of time, t. The curves cor-responding to the slip and nonslip boundary conditions,for Maxwell and Newtonian fluids were considered. At thesmall values of time the Maxwell fluid has not a monotonousflow. After the valuet = 1, the absolute values of the ve-locity corresponding to both types of Maxwell fluids in-crease for increasing y. The absolute values of the velocitycorresponding to both cases of Newtonian fluids increasefor increasing y and for all values of the time t . In Fig.6 we have plotted the relative velocity corresponding toMaxwell and Newtonian fluids versus t for two values of

the dimensionless slip coefficient. The relative velocity,in absolute terms, is an increasing function of.

4. Conclusion

Couette flows of a Maxwell Fluid were analyzed under slipconditions between the fluid and walls. The motion of thebottom plate was assumed to be a rectilinear translation inits plane while, the upper plate is at rest. Two particularcases, namely translation with constant velocity and sinu-soidal oscillations of the bottom plate, were considered.The relative velocity between the fluid at the wall and the

wall was assumed to be proportional to the shear rate atthe wall. The exact expressions for the velocity u(y, t)andshear stress, have been determined by means of Laplacetransform. For a complete study and for comparisons, wepresented velocity fields corresponding to both flows (withslip and nonslip conditions) for Maxwell and Newtonianfluids. The expressions of the relative velocity have alsobeen determined. If the bottom plate translates with theconstant velocity then the velocity fields corresponding tothe four types of the flows were written as sums betweenthe stationary solutions and transient solutions. For largevalues of the time t the transient solutions can be neglectedand the fluid flows according to the stationary solutions.

For Maxwell fluids the velocity is zero a short period afterthe staring of the motion. After this period the values ofthe velocity increase for increasing timet and tend to thevalues of the stationary solutions. For Newtonian fluids thevelocities are increasing functions oft. The velocity corre-sponding to the flow with slip condition is smaller than thevelocity for the flow with non slip condition for both typesof fluids (see Fig. 1 and 2). The relative velocity, in abso-lute value, is a decreasing function of the slip coefficient (see Fig. 3). For sinusoidal oscillations of the plate the ex-pressions of the velocities corresponding to the four typesof flows were written as the sums between the permanentsolutions and transient solutions. In each case two equiv-alent forms of the permanent solution were presented. The

difference between the velocity u(y, t)and the permanentsolution is significant only for the small values of the time

t(see Fig. 4).For large values of the time t the fluids flowaccording to the permanent solutions. The velocity fieldu(y, t)versus y, in absolute terms, is a decreasing function(see Fig. 5), and the relative velocity, in absolute terms isan increasing function of the slip coefficient . The soft-ware Mathcad 14.0 was used for numerical calculationsand to generate the diagrams presented herein and the rootsof Eq. (21) (See Table 1 from Appendix A).

5. Appendix A

A1. L1{F(q)} =f(t), L1{F[w(q)]} =

=0

f(x)g(x, t)dx, g(x, t) =L1{exw(q)}

A2. L1{qbeab} == 1b

n=0

an

(n+1)![b(n+1)]

0

xb(n+1)Jo(2

xt)dx, b >0

A3.

k=0

()kz2k+1(k+1)(2k+1)!bnk+1

= 2z [1 cos(z

bn

)] = 4z sin2( z

2

bn)

A4. 4

k=0

An(y)1+4bn

=2

n=1

sin(pny)+pncos(pny)pn(1+2+2pn2)

=

= 1+y1+2

Table 1 Table 1. Roots of Eq. (21)

pn = 0.4 = 0.7

p1 1.8615134 1.513246

p2 4.2127514 3.8518918

p3 6.9717948 6.7031418

p4 9.9185957 9.7167336

p5 12.9478517 12.7888567

p6 16.0176262 15.887318

p7 19.1097267 18.9996515

p8 22.2152756 22.1201338

p9 25.3295014 25.2457934

p10 28.449632 28.3749412

p11 31.5739545 31.5065484

p12 34.7013567 34.6399135

p13 37.8310854 37.7747121p14 40.9626254 40.9105149

p15 44.0955657 44.04714

p16 47.2296565 47.184424

p17 50.3646768 50.3222441

p18 53.5004611 53.4605064

p19 56.636892 56.5991374

p20 59.7738602 59.7380791

Acknowledgement

The second author is highly thankful and grateful to Ab-dus Salam School of Mathematical Sciences, GC Univer-

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sity, Lahore, Pakistan and Higher Education Commission,Pakistan for supporting and facilitating this research work.The authors are grateful to the anonymous referee for acareful checking of the details and for helpful comments.

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