Journal of King Saud University - Science (2011) 23, 341-347

King Saud University Journal of King Saud University -Science

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ORIGINAL ARTICLE

A note on the unsteady torsional sinusoidal flow of fractional viscoelastic fluid in an annular cylinder

A. Mahmood a'*, N.A. Khan b, I. Siddique a, A. Zada c, A.U. Awan c

a Department of Mathematics, COMSATS Institute of Information Technology, Lahore, Pakistan b Department of Mathematics, University of Karachi, Karachi 75270, Pakistan c Abdus Salam School of Mathematical Sciences, Lahore, Pakistan

Received 2 July 2010; accepted 24 July 2010 Available online 23 March 2011

KEYWORDS

Second grade fluid; Shear stress;

Torsional oscillatory flow; Fractional calculus

Abstract In this note, the velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of Laplace and Hankel transforms. Initially both cylinders and fluid are at rest and then the two cylinders suddenly start torsional oscillations around their common axis with simple harmonic motions having angular frequencies x1 and x2. The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, flows corresponding to the Newtonian, second grade and generalized second grade fluids are shown graphically by plotting velocity profiles.

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* Corresponding author. Tel.: +92 3214858413. E-mail address: amir4smsgc@gmail.com (A. Mahmood).

1018-3647 © 2011 King Saud University. Production and hosting by Elsevier B.V. All rights reserved.

Peer review under responsibility of King Saud University. doi:10.1016/j.jksus.2010.07.022

1. Introduction

Flows in the neighborhood of spinning or oscillating bodies are of interest to both academic workers sand industry. Among them, the flows between oscillating cylinders are some of the most important and interesting problems of motion. As early as 1886, Stokes established an exact solution for the rotational oscillations of an infinite rod immersed in a classical linearly viscous fluid. Casarella and Laura (1969) obtained an exact solution for the motion of the same fluid due to both longitudinal and torsional oscillations of the rod. Later,

Rajagopal (1983) found two simple but elegant solutions for the flow of a second grade fluid induced by the longitudinal and torsional oscillations of an infinite rod. These solutions have been already extended to Oldroyd-B fluids by Rajagopal and Bhatnagar (1995). Others interesting results have been recently obtained by Khan et al. (2005), Fetecau and Fetecau (2006) Mahmood et al. (2009), Vieru et al. (2007), Fetecau et al. (2008), Massoudi and Phuoc (2008), Khan et al. (2009), and Mahmood et al. (2010).

Recently, the fractional calculus has encountered much success in the description of viscoelasticity. Specifically, rheologi-cal constitutive equations with fractional derivatives play an important role in the description of the properties of polymer solutions and melts. The starting point of the fractional derivative model of non-Newtonian fluids is usually a classical differential equation which is modified by replacing the time derivative of an integer order by so-called Riemann-Liouville fractional differential operator. This generalization allows us to define precisely non-integer order integrals or derivatives (Podlubny, 1999).

It is important to mention here that a number of research papers in the literature (Fetecau et al., 2008; Massoudi and Phuoc, 2008; Khan et al., 2009; Mahmood et al., 2010) are devoted to the study of the flow of different viscoelastic fluids between two cylinders, when only one of them is oscillating and other is at rest. On the other hand, the exact solutions corresponding to the flow of these fluids between two cylinders, when both of them are oscillating along or around their common axis simultaneously, are very rare in literature. Recently, Mahmood et al. (2009, 2010) have studied the flow of fractional Maxwell and second grade fluids between two cylinders, when both of them are oscillating around, respectively, along their common axis. As far as the knowledge of authors is concerned, in the literature, no attempt has been made to study the flows of fractional second grade fluid due to torsional oscillations of two cylinders. Therefore, in this paper, we are interested into the torsional oscillatory motion of a generalized second grade fluid between two infinite coaxial circular cylinders when both of them are oscillating around their common axis with given constant angular frequencies m1 and w2. Velocity field and associated tangential stress of the motion are determined by using Laplace and Hankel transforms and are presented under integral and series forms in terms of the generalized G and R functions. It is worthy to point out that the solutions that have been obtained satisfy the governing differential equation and all imposed initial and boundary conditions as well. The solutions corresponding to the ordinary second grade fluid and those for Newtonian fluid, performing the same motion, are also determined as special cases of our general solutions. Furthermore, the respective solutions for the oscillatory motion between the cylinders, when one of them is at rest, can be obtained from our general solutions.

2. Torsional oscillations between two cylinders

Among the many constitutive assumptions that have been employed to study non-Newtonian fluid behavior, one class that has gained support from both the experimentalists and the theoreticians is that of Rivlin-Ericksen fluids of second grade. The Cauchy stress tensor T for such fluids is given by

where —p is the pressure, I is the unit tensor, i is the coefficient of viscosity, a1 and a2 are the normal stress moduli and A1 and A2 are the kinematic tensors defined through

Ai = grad v +(grad v)T,

A2 = ddA + Ai (grad v) + (grad v)TAi.

In the above relations, v is the velocity, d/dt denotes the material time derivative and grad the gradient operator. Since the fluid is incompressible, it can undergo only isochoric motions and hence

div v = tr A1 = 0.

If this model is required to be compatible with thermodynamics, then the material moduli must meet the following restrictions:

l P 0, a1 P 0 and a1 + a2 = 0.

The sign of the material moduli a1 and a2 has been the subject of much controversy. A comprehensive discussion on the restrictions given in (4), as well as a critical review on the fluids of differential type, can be found in the extensive work of Dunn and Rajagopal (1995).

Generally, the constitutive equation of the generalized second grade fluids has the same form as (1), but A2 is defined by

A2 = D? A1 + A! (grad v) + (grad v)TAl5 (5)

where D? is the Riemann-Liouville fractional calculus operator of order ? with respect to t defined by

bf(A - .

C(1 - b) dt

(t — s) f(s)ds, 0 < b 6 1,

where C(-) is the Gamma function. When ? =1, Eq. (5) may be simplified as (2)2 while for a1 = 0 the constitutive relationship (1) describes the Rainer-Rivlin viscous fluid.

2.1. Mathematical formulation of the problem and governing equation

Suppose that an incompressible generalized second grade fluid is situated in the annular region between two infinite straight circular cylinders of radii R1 and R2(> R1) as shown in Fig. 1. At time t = 0, the fluid and cylinders are at rest. At time t = 0 + , inner and outer cylinders suddenly begin to oscillate around their common axis (r = 0) with the velocities W1sin(ra1t) and W2sin(w2t), where m1 is the frequency of velocity of inner cylinder and x2 is that of outer cylinder. Owing to the shear, the fluid between the cylinders is gradually moved, its velocity being of the form

v = v(r, t) = v(r, t)ee,

where eh is the unit vector along h-direction. For such flows the constraint of incompressibility is automatically satisfied. Introducing (7) into the constitutive equation, we find that

T = —pi + iA1 + a1 A2 + a2Ap

s(r, t) = (1 + a1Db) (JT — 1) v(r, t),

Figure 1 Flow geometry.

where s(r, t) = Srh(r, t) is the shear stress, which is different from zero. In the absence of body forces and assuming no pressure gradient in the flow direction, the balance of the linear momentum leads to the relevant equation

dv(r. t) ' dt

2 I sir. t).

Eliminating s(r, t) between Eqs. (8) and (9) we get the governing equation of our problem

-v(r, t) , (—2 1 - 1 \ , N

--T = (v + aD'\ -P +1 @r -1) v(r,t))

r 2 (Rl; R2), t > 0; (10)

where a = a1/p and v = i/q is the kinematic viscosity of the fluid (q being its constant density).

The appropriate initial and boundary conditions are

v(r, 0) = 0; r 2(Ri; R2), (11)

v(R1; t) = W1 sin(ra11), v(R2, t) = W2sin(x2t) for t > 0. (12)

2.2. Calculation of the velocity field

Applying the Laplace transform to Eqs. (10)-(12) and using the Laplace transform formula for sequential fractional derivatives (Podlubny, 1999), we obtain the ordinary differential equation

-2y(r; q) . 1 -y(r; q) 1 , q n

a 2 +--~---2 v(r; q) —ih~ v(r; q) = 0

-r2 r -r r2 aqb + v

r 2 (Ri, R2), (13)

where the image function v(r, q) of v(r, t) has to satisfy the conditions

-cd \ Wi^! W2œ2

v(R1.q)=—l—2. v(R2.q)=—l—2.

q2 + q2 + «2

In the following, let us denote by

Vn{q)= rv(r, q)Bi(rrn)dr; n = 1,2,3,..., ■JR1

the finite Hankel transforms of v(r, q), where rn are the positive roots of the transcendental equation B1(R1r) = 0 and

B1 (rrn) = J (rrn) Y1 (R2rn)- J Rrn) Y (rrn).

In the above relation, J^-) and Y1(-) are Bessel functions of order one of the first and second kind. Applying the finite Hankel transform to Eq. (13) and taking into account the conditions (14), we find that

2W1m1 Ji(R2rn ) 2

- rn vn(q)

nWnœn

p(q2 + x2) p(q2 + «1) Ji(Rir,

aqß + v or equivalently,

(q) = 0.

v„(q) =

nWnan(aqß + v)

p(q2 + «2) {ar2nqb + q + vr2n)

2Wxrn 1 (aqß + v) JiRrn - p(q2 + x2)(arnqb + q + vrI) J^R^

In order to determine v(r,q), we firstly write vn(q) under the suitable form

n(q) = -

2W2 œ2

rV n2 4- «

2Wimi J1R rn

(q2 + x2) m^(q2 + «2) j1(R1 rn)

2 Wnœnq

mr2n(q2 + «2 )(ar2nqß + q + vr2)

_2W1«1q_J1(R2 rn

nrKq2 + a])(arlqß + q + vrl) J1 (R1 rn

and use the inverse Hankel transform formula (Sneddon, 1955)

_n 1 „2 t2

v(r. q)

rj j\(R1rn)B1(rrn)

2 ^ J2(R1rn)— J^^rn)

and (A1) from Appendix A. Furthermore, in order to avoid the burdensome calculations of residues and contour integrals, we apply the discrete inversion Laplace transform method, writing

a^q + q + vrl qb [vrlq—ß + (q1—ß + orD]

vr2n)kq—ßk—ß = (q1—ß + a^)k+1

(— q1

and use Eq. (A2), where (Lorenzo and Hartley, 1999)

~ (c)jd t{j+c)a—b—1 G^M. 0 = jL r(j + m(j + c)a — b]

is the generalized G function and (c) is the Pochhammer polynomial (Lorenzo and Hartley, 1999).

Finally, Eqs. (19)-(21), (A5) and (22) give the velocity field

W1R1{R1 - r2) sin(x!i) + W2R2{r2 - r2) sin(x2i)

v(r. t) =

(R2 — R\)r

■1 -I, 2 s k Jl(Rir„)Bi(rr„) "H h( Wn> J2(Rlrn) — Ji(R2rn)

x \w2X2Jl(Rlr„> J^ cos(«2(t — s>>Gl ß. ßk ß.k+i(—arl, s)ds

— W^OJ^J^(R2rn>J^ cos(œ1(t — т>>Gl ß. ßk ß^(—ar2n, s)d%^ . (23)

2.3. Calculation of the shear stress

Applying the Laplace transform to Eq. (8), we find that

s(r, q) = (l + a1q?)(@r —1) f(r> q);

@ 1\ . x 2R1R2 /r1w2m2 r2w1m1

— 7f(r'q) =

(r? — r>2 V q2 + « q2 + + p E XX(—vr2 )kx J1 (R1r«)[(2/r)B1 (rr») — rn'B1(rrn)}

W2M2/1(R1r„)

J?(R1rB)—J?(R2rB)

q (k + 1).(—ar?)

-W1W1/1 (R2r„)

q2 + w2 ;1qk+j(1—?)+1 q (k + 1).( —ar?)

q2 + w2 j!qk+j(1—?)+1

has been obtained from (23) and (A8), where in the above relation

B1(rr„) = Jfi(rrn) Y1 (R2 r„) — J1(R2rn)Y0(rrn).

Introducing (25) into (24), applying again the discrete inversion Laplace transform method to the obtained result and using (A3) and (A5), where

D . , . ^ C(t — d)(j+1)a—b—1 Rab(c;d;° = § n.H-1)a—b]

is R function (Lorenzo and Hartley, 1999), we find for the shear stress the expression

r(r, t) =

(R—R?)r2

|^l(R1 W2 sin(w21)— R2 W1 sin( w11))

R1W2w2R2>?(—0, t)—R2W1W1R2>?(—w?, 0, t)}

^ k J1 (R1rn)[(1/r)B1 (rrn)— rnB1 (rrn)] + "¿1 ^ J1(R1rn) —J1(R2rn)

W2w2J1(R1rn^cos(w2(t — s))

G1—?,—?k—?,k+1 (—arn, s)ds + a1 G1—?,—?k—1,k+1 (—ar?, t) —a1 w2 ^ sin(w2 (t — s))G1— ?,—?k— 1,k+1 (—ar?,t)-t|

— W1W1J1 (R2r„)jij^ cos(w1(t — s))

G1—?,—?k—?,k+1 (—arj;, s)ds + a1G1—?,—?k—1,k+1(—arn, t)

—a1«W sin(w1 (t — s))G1— ?,—?k— 1,k+1 (—ar;;,s)ds

3. Limiting cases

3.1. Solutions for ordinary second grade fluid (p fi 1)

Making ? fi 1 into Eqs. (23) and (27) and using (A4) and (A7), we obtain the velocity field

vs(r,t) =

W1R1 (R? — r2)sin(w11) + W2R2(r2 — R?) sin(w2t)

(R —R1)r

J1(R1r„)B1(rr„)

1 J? (R1r„) — j\(R2rn)

W2w2J1(R1r„

[v2^ + w2(1 + arn) +«2(1 + arn)sin((w2t))

x < vr^ ( cos(w1t)—exp

vr^ ^cos(«2t)—exp ^—1 W1X1J1(R2r„)

v2r4 + «1(1 +arn)2

1 +ar2n

+«1 (1 + ar;;)sin(w1t)

and associated shear stress 2R R

ss(r, t) = R r1 J 2 [i(R1 W2 sin(w2t)— R2 W1 sin(«1t)) (R2 R1)r

+a1(R1 W2w2 cos(w2t)— R2 W1w1 cos(w1t))] J1(R1rB)[(1/r)B1(rr„)—r„i?1(rr„)]

n=1 J?(R1r„)— J?(R2r„)

x (J1(R1r„)g? — J1(R2r„)g1),

where in above Eq. (29)

gm = Wm<Xm

v2rn + wm(1 + ar?)'

■{vrKcos(c°mt)—exp(—iran))

■ / ■.] a1 / vrh \

+wm(1 + arJ Sm^O j + 1+022 ex^— T+Orl)

j«m ^1 + ar^ (^exp(— 1 +'2'gtr2)— cos(Wmt)^)

v2r4 + <0 + arn)?

+vrn sin(«mt)

corresponding to the ordinary or classical second grade fluid, performing the same motion. The velocity field (28) and shear stress field (29) are same as obtained in the Ref. Mahmood et al. (in press).

Eqs. (28) and (29) correspond to the transient solutions of the velocity field and shear stress, respectively, of the second grade fluid. As time goes to infinity, these terms vanish and we are left with the corresponding steady-state solutions.

3.2. Solutions for Newtonian fluid

Making a fi 0 (equivalently a1 fi 0) into Eqs. (28) and (29), velocity field and associated shear stress for Newtonian fluid, performing the same motion, can be obtained. For instance, the velocity field is

W1R1(Rl — r2) sin(w1t) + WiRi(r2 — R?) sin(wit) VN(r'0 = (R? — R])r

^ J1(R1 rn)B1 (rrn) [Wi«?^^) f 2, , A

— "n=1 J?^ —J^n) lurivr"(cos(x2t)

— exp (—vr?t)) + «2 sin((«2t))|

W1X1 J1(R2rnH 2( / ^ / \

--v2 r4 + «2 |Vri(cos(«1t) — exp(—Vrnt))

+«1 sin(«1t)

4. Concluding remarks

Our purpose in this paper was to establish exact analytic solutions for the velocity field and shear stress corresponding to the flow of a generalized second grade fluid between two infinite coaxial circular cylinders, by using Laplace and Hankel transforms. The motion of fluid was due to the simple harmonic sine oscillations of both cylinders around their common axis, with different angular frequencies mi and m2 of their velocities. It is important to point out that the velocity field and the shear stress for the oscillatory motion between the cylinders, when one of them is at rest, can be obtained from our general

solutions by making Wi = 0, W2 = W and m2 = m (when inner cylinder is at rest) or Wi = W, W2 = 0 and mi = m (when outer cylinder is at rest). For instance, the velocity field for the flow of generalized second grade fluid, when inner cylinder is at rest and the outer cylinder is oscillating, is given by (from Eq. (23))

. . WR2(r2 — R?) sin(mt) v^ v^

V(f't) =-fR _RV PWm££l

\R2 Ri) r n=i k=0

■72(Rir„)£i(rr„

Ji(Rir„)-/i(R2r„) jo x Gi-bt-bk-b,k+i(-ar2n, s)ds.

/ cos(m(t — s)) Jo

Figure 4 Velocity profiles for different values of time.

The solutions that have been obtained, presented under integral and series forms in terms of the generalized G and R functions, satisfy the governing equation and all imposed initial and boundary conditions and for b fi 1 reduce to the similar solutions for the second grade fluid. Finally, the solutions for the flow of Newtonian fluid for the similar flow between cylinders have been also recovered as a special case of our general solutions, when b fi 1 and a fi 0.

Finally the graphical illustrations, Figs. 2-4, are given to show the comparison between the flow of generalized second grade (curves vG1(r) and vG2(r) for b = 0.9 and 0.6,

respectively), second grade (curve vS(r) for b = 1) and that of Newtonian fluid (curve vN(r) for b =1 and a fi 0). These graphs also show the influence of the fractional coefficient b on the velocity v(r, t). In all figures we consider Ri = 1, r2 = 4, V1 = 1, V2 = 4, X1 = 5, X2 = 7, a = 9 x 10-3 and v = 1.1746 x 10-3 while SI units of parameters are used.

Appendix A. Some results used in the text: The finite Hankel transform of the function AR^R? - r2)+BR2(r2 - R?)

a(r) = '- , i ,,

w (R2 - R>

satisfying a(R1) = A and a(R2) = B is

iR2 w 2B 2AJ1R rn

an = ra(r)B1 (rrn)dr = —? -

pr2 %r2nJ1(R1 rn

{{q^-d)^ = t); Re(ac - b) > 0,

Re(q) > 0, — < 1.

y ' qa

c)(t - d)i)+1)a-b-1

\q"- n , j—0 C[() +1)a - b]

d P 0, Re((j + 1)a - b) > 0, Re(q) > 0.

R2;1(-a2,0, t) — cos(at).

If u1(t) — L 1 {u1(q)} and u2(t) — L 1 {u2(q)} then (A5)

L-1{u1(q)ii2(q)} — (u1 * u2)(t) — u1(t — s)u2(s)ds

— / u1(s)u2(t — s)ds. J0

J2(-1)kak

œ / \ k

J2 ( - Vr2n) G0,-k-1,k+1(—arn, t) k—0 \ /

— X ( 2^k +1),(-<)) tk

— XX

(1 + arl)k+l k!

1 V U- Vr2nt V 1 + ar2 k—0 k!\ 1 + O 1

1 + ar2

1 + ar2,)

[B1 (rrn)] — rn[J0(rrn) Y1 Rrn)- J1 (R2rn) Y0K)] 1

-- B1(rrn). r

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