Strange Beautiful (Ravi)i

7/29/2019 Strange Beautiful (Ravi)i

1/16

2

which can be shown to be a series of trigonometric sine waves. Harmonies becomemathematical ratios which can occur in everything.Music is a primary illustration of the ability of individuals to unknowingly appreciate geometryand mathematics, even when theyre not serious thinkers. Music is basically about ratios,frequencies , and timing . There is also a strong geometrical connection, in that, if one takesthe unique right triangle , and strings a continuous fine wire to each of the three points of thetriangle, it is then possible to tune one of the sides to a particular note, and have the other

two sides be in a tuned harmony. The three sides of the triangle form a series of tones thatare equivalent to the first three strings of a tuned guitar. Every musical pulse consists ofnumerous sine-wave tones. Even a square wave is made up of a large number of oddharmonics, and thus by extrapolation, a truly infinite pulse would consist of allpossible puretones. The manner in which musicians examine a spectrum of musical harmonies is, in fact,exactly the same procedure mathematicians call a Fourier Transform. The sum of all musicalfrequencies thus constitutes the whole of the universe.Music is the art of organizing tones to produce a coherent sequence of sounds intended toelicit an aesthetic response in a listener. Music can incite passion, belligerence, serenity, fear,or sadness. Interestingly one can play the national anthem on guitar using the number andtiming technique. Usually a guitar consists of 21 frets (the metallic strips on the neck of theguitar). If the six horizontal lines represents the six strings of guitar and the numberrepresents the position on the fret board (the neck of guitar) then following the below patternone can play the national anthem on guitar.

e|-------------------------|----------------------|--------------------------|B|-----------------7-------|---10-----7-----------|------------------7--9h10-|G|-----------7--9-----7--9-|-------9-----7--------|------------7--9----------|D|----7--11----------------|----------------11--9-|--9--11--9----------------|A|-------------------------|----------------------|--------------------------|E|-------------------------|----------------------|--------------------------|

e|----------------|----------------------|-----------------------------------|B|--9--7----------|----------------------|--10--12--15--13--12--9s10---------|G|--------9--7--6-|--9--7--6--7--6-------|-----------------------------------|D|----------------|-----------------9--7-|-----------------------------------|A|----------------|----------------------|-----------------------------------|E|----------------|----------------------|-----------------------------------|

e|--9--10--12--14--15--17--18-|--9--10--12-14-15--18--19--18--16--18--16--15-|B|----------------------------|----------------------------------------------|G|----------------------------|----------------------------------------------|D|----------------------------|----------------------------------------------|

A|----------------------------|----------------------------------------------|E|----------------------------|----------------------------------------------|

e|-----------------------------|BI--9--7--6--7--6--------------|G|-----------------8--6--------|D|-----------------------------|A|-----------------------------|E|-----------------------------|

e|-------------------------|----------------------|--------------------------|B|-----------------7-------|---10-----7-----------|------------------7--9h10-|G|-----------7--9-----7--9-|-------9-----7--------|------------7--9----------|D|----7--11----------------|----------------11--9-|--9--11--9----------------|A|-------------------------|----------------------|--------------------------|E|-------------------------|----------------------|--------------------------|

e|----------------|----------------------|B|--9--7----------|----------------------|

G|--------9--7--6-|--9--7--6--7--6-------|D|----------------|-----------------9--7-|A|----------------|----------------------|E|----------------|----------------------|

Well the temporal lobes of the brain, just behind the ears, act as the music center. Formusicians, who had begun their training before the age of 7, they actually increased the sizeof their brains -- specifically the corpus callosum -the trunk line which connects the brainsright and left hemispheres. This neural path increase may also explain why the bettermusicians are not only technically adept (the left brains partiality to cognition), but can playwith emotion (the right brains forte). Even more striking is the fact that mental imagery or


2/16

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

c 2012 NSP

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, Pakistann

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 velo-

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

ditions 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)

where S 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


3/16

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 positive yaxis in the upward direction. The second infinite solidplane wall occupies the plane y = 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-ity uw(t) = Uof(t), where Uo > 0 is a constant and f(t)is a piecewise continuous function defined on [0,) andf(0) = 0. Also, we suppose that the Laplace transform of

the function f(.) exists. In the case of parallel flow alongthe x axis, 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)

where is 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)

where R = h2

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


4/16

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)

where qis the Laplace transform parameter and u(y, q) =L{

u(y, t)}

. The solution of differential equation (13) withthe 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-

tion G1(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 of F1(y, q) are simplepoles located at

qn = pn2

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

where pn = 0 are the real roots of the equation

tan(pn) =2pn

2pn2

1

, (21)

we invert function F1(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=1

Res[F1(y, q)eqt, qn] =

=

n=1An(y)exp(pn

2

R t)(22)

where

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

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

= 2pnR

sin(ypn)+pn cos(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)

and J() 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 ][ t2es2t4

k=0

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

0

z2k+1J2(2

zt)dz]ds =

= t2

et2

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

et2

n=1

An(y)

0J2(2

zt)

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

z2k+1

bnk+1dz,

(26)

where bn =pn

2

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

expression of the function g1(y, t) , namely

g1(y, t) =2t

e

t2

n=1

An(y)

0

1

zsin2(

z

2

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)bn

sin(t

bn

)(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 function An(y) given by Eq.(23) becomes

A1n(y) =2n

Rsin(ny), n = 1, 2, ..... (30)


5/16


and the velocity field has the expression

uM(y, t) =2R

n=1

n sin(ny)Rn

t0

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

where cn =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 = 0 and = 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)

where An(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)

t0

g(t s)en22

Rsds.(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 function G3(y, q) is

g3(y, t) =t

0e

ts

g1(y,s)y ds =

=

n=1

Bn(y)bn

et

t0

es2 sin(s

bn )ds

(41)

where

Bn(y) =dAn(y)

dy =

= 2pn2

Rcos(ypn)pn sin(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(tbn ) 2

bn cos(t

bn )]

(43)

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

(y, t) =1

R(g g3)(t) = 1

R

t0

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, namely f(t) = H(t)

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


6/16


a. Maxwell fluids with slip at the boundary:

uM s

(y, t) = H(t){

4

n=1

An(y)

1+4bn 2

n=1

An(y)e

t2

bn(1+4bn)[sin(t

bn ) + 2

bn cos(t

bn )]}

,(45)

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

uM s(y, t) = H(t){1+y1+2 et2

n=1

sin(ypn)+pn cos(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+2 2

n=1sin(ypn)+pn cos(ypn)

pn(1+2+2pn2)e

pn2

Rt}. (49)

c. Newtonian fluids with nonslip at the boundary:

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

n=1

sin(ny)

ne

n22

Rt. (50)

The relative velocity is given by

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

n=1

pnpn(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

Rt],

(52)

for Newtonian fluids. By using the illustrations generatedwith the software Mathcad, we discuss some relevant phys-ical aspects of the flow. Also, the roots pn , 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 rootspn are presented in theTable 1 from the Appendix A. In the figures, we use =

9.15255

103m2

/s

, = 0.55s , = 1.050kg/m

3 corre-

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 is R = 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 tfor y {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 variable y, the velocity cor-responding to a Maxwell fluid with slip at the boundaryis zero for a short time, after that, is increasing and tends

towards 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 thetime t they tend to the stationary solutions ustNs = u

stMs

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

velocity u(y, t) corresponding to Maxwell and Newtonianfluids for both slip and non slip conditions. The velocitywas plotted versus y for t {0.5, 1.5, 2.5} . For small val-ues of time t the Newtonian fluid with slip at the wall is

slower 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 time t.

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(ts) is replaced by sin(ts).a. Integrating by parts into Eq. (29), this yields after somesimplifications, the velocity field corresponding to the flowof a Maxwell fluid with slip at the wall

uMs (y, t) = Q1(y)sin t + Q2(y)cos t+

+et2

n=1

[Q3n(y)cos(tbn )++Q4n(y)sin(t

bn )],

(53)


7/16


Figure 1 Plot of u(y, t) versus t for both cases with slip andnonslip condition, Fig (b) Plot ofu(y, t) versus y for both caseswith slip and nonslip condition.

where

Q1(y) = 2

n=1

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

Qn(y), (54)

Q2(y) = 2

n=1

Rpn

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

Q3n(y) =2pnR

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

Q4n(y) =pnbn

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

Qn(y), (57)

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

Qn(y) =sin(ypn) + pn cos(ypn)

1 + 2+ 2pn2, (58)

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

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

and the transient solution uM st(y, t) = uMs (y, t)uM sp(ywhich can be neglected for large values of the time t. Byusing the residue theorem to evaluate the inverse Laplacetransform of function u(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) = 1M[M1P1(y) + M2P2(y)]sin t++ 1

M[M1P2(y) + M2P1(y)]cos t,

(60)


8/16


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

Newtonian fluids.

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(1

R) + 22

Rch(1

R)]

cos(2R),(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)]++2

R cos[2

R(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 boundaryis given by

uM(y, t) = 2sin t

n=1

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

2cos t

n=1

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

+ 2et2 {

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 the

motion, can be written in the equivalent formuM p (y, t) = P3(y)sin t + P4(y)cos t, (67)

where

P3(y) =

{sh[1

R(1

y)]

cos[2R(1 y)]sh(1R)cos(2R)++ ch[1

R(1 y)]sin[2

R(1 y)]

ch(1

R)sin(2

R)} 1sh2(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+R2

2cos t

n=1

RpnQn(y)p4n+R

2 + 2

n=1

RpnQn(y)p4n+R

2 e p

2nR

t,(70)

where Qn(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)]++

R2{ch[

R2

(1 y)] cos[

R2

(1 y)]

sh[R2 (1

y)]sin[R2 (1 y)]

},

(75)

E2(y) = ch[

R2

(1 y)] sin[

R2

(1 y)]++

R2{sh[

R2 (1 y)]sin[

R2 (1 y)]+

+ ch[

R2 (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

Rt.

(77)


9/16


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 illustrations

from Figs. 4-6.

Figure 4 Plot of the u(y, t) versus t for both cases with slip andnonslip condition.

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 = 6 forthe Maxwell fluid with non slip condition the transient ve-locities utMs (y, t) = uMs (y, t) uMsp (y, t), utM(y, t) =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 walland t = 4 in the case of nonslip at the wall. After these

Figure 5 Plot of the u(y, t) versus y for both cases with slip andnonslip condition.

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

Newtonian fluids.

moments, the fluids flow according to the permanent so-lution. Fig. 5 contains diagrams of velocity u(y, t), ver-sus y 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 monotonouflow. After the value t = 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.


10/16


6 we have plotted the relative velocity corresponding toMaxwell and Newtonian fluids versus t for two values ofthe 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 the

bottom 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 thewall 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 the

constant 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 time t 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. Thedifference between the velocity u(y, t) and the permanentsolution is significant only for the small values of the timet (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 calculations

and to generate the diagrams presented herein and the rootsof Eq. (21) (See Table 1 from Appendix A).

pn = 0.4 = 0.7

p1

1.8615134 1.513246

p2 4.2127514 3.8518918

p3 6.9717948 6.7031418p4 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 37.7747121

p14 40.9626254 40.9105149

p15 44.0955657 44.04714

p16 47.2296565 47.184424

p17 50.3646768 50.3222441p18 53.5004611 53.4605064

p19 56.636892 56.5991374

p20 59.7738602 59.7380791

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)+pn cos(pny)pn(1+2+2pn2)

=

= 1+y1+2

Table 1. Roots of Eq. (21)

Acknowledgement

The authors acknowledge the financial support ....., projectNo. ...... The author is grateful to the anonymous refereefor a careful checking of the details and for helpful com-ments that improved this paper.

References

[1] R.B.Bird, et al., Dynamics of Polymeric Liquids: vol 1,

Fluid Mechanics, (Wiley, New York, 1987).

[2] G.Bohme, Non- Newtonian Fluid Mechanics, (North-

Holland, New York, 1987).

[3] D.D. Joseph, Fluid Dynamics of Visco elastic liquids,

(Springer, New York, 1990).


11/16


[4] O. Riccius, D.D. Joseph, M. Arney, Shear-Wave speeds and

elastic moduli for different liquids. Part. 3. Experiments-

update. Rheol. Acta 26 (1987) 96-99.

[5] R.I. Tanner, Note on the Rayleigh problem for a visco-elastic fluid, Z. Angew. Math. Phys. 13 (1962) 573-580.

[6] P.M. Jordan, Ashok Puri, G. Boros, On a new exact solu-

tion to Stokes first problem for Maxwell fluids, Int. J. Non-

linear Mech.,39(2004)1371-1377.

[7] P.M. Jordan, A. Puri, Revisiting Stokes first problem for

Maxwell fluids, Q.J.Mech..Appl. Math. 58(2)(2005)213-

227.

[8] I.C. Christov, Stokes first problem for some non-Newtonian

fluids: Results and Mistakes, Mech. Res. Commun.

37(2010) 717-723.

[9] M.M. Denn, K.C. Porteous, Elastic effects in flow of visco-

elastic liquids, Chem. Eng. J. 2(1971) 280-286.

[10] P.M.Jordan, A note on start-up, plane Couette flow involving

second- grade fluids, Math. Prob. Eng. 5 (2005), 539-545.

[11] P.M. Jordan, P.Puri, Exact solutions for the unsteady plane

Coutte flow of a dipolar fluid, Proc. Roy. Soc. London Ser.

A 458 (2002) No. 2021, 1245-1272.

[12] Zhang Jin Xue, Jun Xiang Ni, Exact solutions of Stokes

first problem for heated generalized Burgers fluid via

porous half-space, Non-linear Analysis: Real World Appl.,

9(4) (2008) 1628-1637.

[13] H.A. Attia, Unsteady hydro magnetic Couette flow of dusty

fluid with temperature dependent viscosity and thermal

conductivity under exponential decaying pressure gradient,

Comm. Non-Lin. Sci. Num. Simul. 13(6) (2008) 1077-1088.

[14] S. Asghar, T. Hayat, P. D. Ariel, Unsteady Couette flows in a

second grade fluid with variable material properties, Comm.

Non-Lin. Sci. Num. Simul. 14(1) (2009) 154- 159.

[15] M. Danish, DS. Kumar, Exact analytical solutions for the

Poiseuille and Couette- Poiseuille flow of third grade fluid

between parallel plates, Comm. Non-Lin. Sci. Num. Simul.

17(3), (2012) 1089-1097.

[16] M.A. Day, The non-slip boundary condition in fluid me-

chanics, Erkenntnis 33,(1990) 285-296.

[17] C.L.M.H. Navier, Sur les lois du movement des fluids. Mem.

Acad. R. Sa: Inst. Fr. 6.(1827) 389-440.

[18] I.J.Rao, K.Rajagopal, The effect of the slip boundary con-

dition on the flow of fluids in a channel, Acta Mech. 135

(1999) 113-126.

[19] M.Mooney , Explicit formula for slip and fluidity, J. Rheol.

2(1931) 210-222.

[20] A.R.A. Khaled, K. Vafai, The effect of the slip condition on

Stokes and Couette flows due to an oscillating wall: exact

solutions, Int. J. Non-Lin. Mech. 39(2004) 795-809.

[21] D. Vieru, A. Rauf, Stokes flows of a Maxwell fluid with

wall slip condition, Can. J. Phys. 89(2011) 1-12.

[22] S.Abelman, E. Momoniat, T. Hayat, Non-Linear Analysis:

Real World Appl. 10(6)(2009) 3329-3334.

[23] L. Zhenga, Y. Liu, X. Zhang, Slip effects on MHD flow of a

generalized Oldroyd B fluid with fractional derivative, Non-

Lin. Anal.: Real World Appl. 12(2) (2012) 513-523.

[24] R.Ellahi, T. Hayat, F.M. Mahomed, S. Asghar, Effects of

slip on the non-linear flows of a third grade fluid, Non-Lin.

Anal.: Real World Appl. 11(1)(2010) 139-146.

[25] T.Hayat, M. Khan, M. Ayub, The effect of the slip condi-

tion on flows of an Oldroyd 6-constant fluid, J. Comp. Appl.

Math. 202(2007) 402-413.

[26] H.S. Carslaw, J.C. Jaeger, Operational methods in Applied

Mathematics, (2nd edition, Dover, New York 1963).

[27] G.N.Watson, A treatise on the theory of Bessel functions,

(Cambridge University Press, 1995).


12/16


First Author is a leading world-known figure in math-ematics and is presently employed as HEC Foreign Profes-sor at CIIT, Islamabad. She obtained her PhD from WalesUniversity (UK). She has been awarded by the Presidentof Pakistan: Presidents Award for pride of performanceon August 14, 2010. She introduced a new technique, nowcalled as Noor Integral Operator which proved to be aninnovation in the field of geometric function theory andhas brought new dimensions in the realm of research inthis area. She is an active researcher coupled with the vast(40 years) teaching experience in various countries of theworld in diversified environments. She has been personallyinstrumental in establishing PhD/ MS programs at CIIT.She has been an invited speaker of number of conferencesand has published more than 340 (Three hundred and forty) research articles in reputed international journals of math-

ematical and engineering sciences.

Second Author is a leading world-known figure in math-ematics and is presently employed as HEC Foreign Profes-sor at CIIT, Islamabad. She obtained her PhD from WalesUniversity (UK). She has been awarded by the Presidentof Pakistan: Presidents Award for pride of performanceon August 14, 2010. She introduced a new technique, nowcalled as Noor Integral Operator which proved to be aninnovation in the field of geometric function theory andhas brought new dimensions in the realm of research inthis area. She is an active researcher coupled with the vast

(40 years) teaching experience in various countries of theworld in diversified environments. She has been personallyinstrumental in establishing PhD/ MS programs at CIIT.She has been an invited speaker of number of conferencesand has published more than 340 (Three hundred and forty) research articles in reputed international journals of math-ematical and engineering sciences.


13/16


14/16

3

mental rehearsals can activate the same regions of the brain as actual practice, and similarlyaffect the synapses!The numerous references to proportion and ratio in music and mathematics deserve a bitmore consideration. In a survey, infants were found to smile when music was played, whichconsisted of perfect fourths and perfect fifths -- i.e. chords or sequences which are separatedby either five half steps (as between C and F) or seven half steps (as between C and G),respectively. Babies, however, did notlike tritones, where two notes were separated by six

half steps (e.g. C and F sharp).Its hard for anyone to say what music looks like, but a new mathematical approach seesclassical music as cone-shaped and jazz as pyramid-like.The connections between math and music are many, from the unproven Mozart effect to themusic of the spheres -the ancient belief that proportions in the movements of the planetscould be viewed as a form of music. Now scientists have created a mathematical system forunderstanding music.Clifton Callender of Florida State University, Ian Quinn of Yale University and Dmitri Tymoczkoof Princeton University have presented their "geometrical music theory".The team designed a geometrical technique for mapping out music in coordinate space. Formusic made of chords containing two notes, all musical possibilities take the shape of a Mobiusstrip, which basically looks like a twisted rubber band. It is found that the shape of possibilitiesusing three-note chords is a three-dimensional cone, where types of chords, such as majorchords and minor chords, are unique points on the cone. The space of four-note chords is

what mathematicians would call a "cone over the real projective plane," which resembles apyramid in our 3-D universe. Any piece of music can be mapped in these spaces.You can use these geometrical spaces to provide ways of visualizing musical pieces, Thesespaces gives a much better and comprehensive picture of the space of all possible chords.It's probably no coincidence that math and music are so deeply linked,when music doesn't have words, it doesn't necessarily resemble anything in the real world.Traditionally, paintings always looked like things; poetry and literature were talking aboutthings. But music is coming closer to pure truth.The new techniques reveal fascinating differences between rock and classical music, and evenbetween Paul McCartney and John Lennon.One of the really exciting things about this research is that it allows seeing commonalitiesamong a much wider range of musicians.By looking at the mathematical essence behind the work of various musicians and musicalstyles, the scientists can better understand how they relate to one another.

Over the course of the 18th and 19th centurys people start exploring a wider variety ofgeometrical spaces. There's a general push toward increasing complexity and sophistication.They move from the three-dimensional cone to the four-dimensional space.While analyzing the math behind music can provide many insights. There is no way thatgeometry is going to help you become a great composer. Understanding the geometry willhelp to become a mediocre composer much more quickly, but composing is an artisticachievement. There's no royal road to becoming a great musician. Mathematics is not takingthe mystery away from music.


15/16

4

This is an image of the space of three-note chordtypes,the orange spheresrepresent the major and minor chords.

The space of two-note chords, as it is embedded in the infinite-dimensional spacecontaining chords with any number of notes. The two-note chords form a Mobius strip.


16/16

5

Documents

Strange Beautiful (Ravi)i