THEORETICAL & APPLIED MECHANICS LETTERS 2, 022002 (2011)

Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders

M. Imran,a) M. Kamran, and M. Athar

Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan

(Received 03 November 2010; accepted 26 January 2011; published online 10 March 2011)

Abstract The velocity field and the adequate shear stress corresponding to the longitudinal flow of a fractional second grade fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. Initially the fluid is at rest, and at time t = 0+, the inner cylinder suddenly begins to translate along the common axis with constant acceleration. The solutions that have been obtained are presented in terms of generalized G functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary second grade and Newtonian fluids are obtained as limiting cases of the general solutions. Finally, some characteristics of the motion, as well as the influences of the material and fractional parameters on the fluid motion and a comparison between models, are underlined by graphical illustrations. © 2011 The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1102202]

Keywords second grade fluid, fractional derivative, longitudinal flow, velocity field, shear stress, Laplace and finite Hankel transforms

Navier-Stokes equations are nonlinear partial differential equations. For this reason, there exists only a limited number of exact solutions in which the nonlinear inertial terms do not disappear automatically. Exact solutions are very important not only because they are solutions of some fundamental flows but also because they serve as accuracy checks for experimental, numerical, and asymptotic methods. The inadequacy of the classical Navier-Stokes theory to describe rheologically complex fluids such as polymer solutions, blood, paints, certain oils and greases, has led to the development of several theories of non-Newtonian fluids. Amongst the many models which have been used to describe the non-Newtonian behavior exhibited by certain fluids, the fluids of different types1 have received special attention. The fluids of second grade, which form a subclass of the fluids of different types, have been studied successfully in various types of flow situations. Here, we mention some of the studies.2-11 Sometimes since the equations governing the flow of second-grade fluids are one order higher than the Navier-Stokes equations, one would require boundary conditions in addition to the "nonslip" condition to have a well-posed problem.

In recent years, fractional calculus has achieved much success in the description of complex dynamics. Fractional derivative models are used quite often to describe viscoelastic behavior of polymers in the glass transition and the glassy state. The starting point is usually classical differential equation which is modified by replacing the classical, time derivatives of an integer order by the so-called Riemann-Liouville operator. This generalization allows one to define precisely non integer order integrals or derivatives.12'13 Fractional calculus has also gained much fame in the description

of viscoelasticity.14'15 Tan and Xu16 discussed the flow of a generalized second grade fluid due to the impulsive motion of a flat plate. Relevant studies involving generalized second grade fluid in bounded domains are presented.17-24

In this work, we consider the viscoelastic fluid to be modeled as fractional second grade fluid (FSGF) and study the flow starting from rest due to the sliding of the inner cylinder along its axis with a constant acceleration. The velocity and adequate shear stress, obtained by means of the finite Hankel and Laplace transforms, are presented under series form in terms of the generalized G functions. The similar solutions for the ordinary second grade or Newtonian fluids, performing the same motion, are respectively obtained as special cases when P ^ 1, or P ^ 1 and a1 ^ 0.

The flows to be considered here have the form of9'25

v = v(r, t) = v(r, t)ez ,

where ez is the unit vector in the z-direction of the cylindrical coordinates system r, 0 and z. For such flows, the constraint of incompressibility is automatically satisfied.

Furthermore, if initially the fluid is at rest, then

v(r, 0) = 0 .

The governing equations, corresponding to such motions for second grade fluid are9'25

T (r,t) = (M + «1 fa)

d \ dv(r, t)

a) Corresponding author. Email: drmimranchaudhry@gmail.com.

dv(r,t) ( d \{ d2 1 d\ . .

-ßT = V + aW\drl + -rd~r)v(r,t)-'

where j is the dynamic viscosity, v = j/p is the kinematic viscosity, p being the constant density of the fluid, a\ is a material constant and a = ai /p.

The governing equations corresponding to an incompressible fractional second grade fluid, performing the same motion, are

^ = 1* + aDß )( £ + ^l-M)

r dr J

t(r, t) = + aiDß)

ß ) dv(r,t) 1 ) dr

where the fractional differential operator Df is defined by Ref. 26 as

V» f(t) = 1 d f (t) r(1 - ß) dt Jo (t - t)ß

Dß f (t) =

dT ; 0 < ß < 1,

and r(-) is the Gamma function. Of course, the new material constant ai , although for simplicity we keep the same notation, tends to the original a1 as 3 ^ 1. In the following the system of fractional partial differential Eqs. (5) and (6), with appropriate initial and boundary conditions, will be solved by means of Laplace and finite Hankel transforms. In order to avoid lengthy calculations of residues and contour integrals, the discrete inverse Laplace transform method will be used.

Let us consider an incompressible FSGF at rest in the annular region between two straight circular cylinders of radii R1 and R2(> R1). At time t = 0+, the inner cylinder suddenly begins to slide along the common axis with constant accelerations A . Owing to the shear, the fluid is gradually moved, its velocity being of the form of Eq. (1). The governing equations are Eqs. (5) and (6), while the appropriate initial and boundary conditions are

v(r, 0)=0, r G [Ri,R2

j(Ri,t) = At, v(R2,t)=0 for t > 0,

where A is a real constant.

Applying the Laplace transform to Eqs. (5) and (9), we get

( d2 1 d \

qv(r,q)= (v + aqf) + rdr)^^' (10)

v(Ri,q) = ^r, v(R2,q) = 0 ,

where r G (R\,R2), v(r,q) is the Laplace transform of function v(r,t). In the following, let us denote27

[■ R2

Vh(rn, q)= rv(r, q)B(r, rn)dr , n = 1, 2, 3,.... Jr i

by the finite Hankel transform of the function v(r, q)

B(r,rn) = Jo(rrn)Yo(R2rn) - Jo(R2rn)Yo(rrn),

where rn are the positive roots of the transcendental equation B(Ri,r) = 0, and Jp(•) and Yp(•) are Bessel functions of the first and second kind of order p .

The inverse Hankel transform of wH (rn, q) is given by27

~J(r, q) = -2

n2^ r2J2(Rir„)ß(r,r„) _

2 ^ J2(Rir„) - J22(R2rn)

vh (rn, q).

Multiplying now both sides of Eq. (10) by rB(r, rn), then integrating it with respect to r from R1 to R2, and taking into account Eq. (11) along with the following relations

d-B(r: rn ) =

- rn [Ji(rrn)Y0 (R2r'n ) - Jo (R2rn)Yi(rr-n)] ,

Jo(z)Yi(z) - Ji(z)Yo(z) =--,

and the result which we can easily prove fR2 ( d2 1 d

J r^jrï + rdr )V(r,q)B(r,rn)dr =

2 Jo (R2rn)V(Ri,q) 2_ , , - rn vh (rn,q),

nJo (Ri rn) we find that

Vh (rn, q) =

—2(v + aqß )AJo (R2rn) 1

nJo (Rirn) q2(q + vrn + ar°qß )'

It can also be written in the equivalent form of VH (r q)= —2A Jo (R2rn) 1 + n nrn Jo (Rirn) q2

2AJo(R2rn) _1

nrnJo(Rirn) q(q + vrn + ar2qß) '

Applying the inverse Hankel transform to Eq. (16), and using identity

ln(R2/r) = ^ Jo(Rirn)Jo(R2rn)B(r,rn) ln(R2/Ri)= ^

J2 (Rirn) - J02(R2rn) '

we get

, N ln(R2/r) A

v(r q)=-v 2/ '---h

V{T,q) ln(R2/Ri) q2

nA \ - I Jo (Rirn) Jo (R2rn)B(r, rn) x

J2(Rirn) - J2^^)

q(q + vrn + arn qß )

Using identity

_ = {-vrl)kq-ßk-ß-1

q(q + vr"n + ar"nqß) k= (q1-ß + ar2)k+1

Equation (17) can be further simplified to

, , ln(R2/r) A v(r, q)= A2 +

lnR/Ri) q2

Jo(Rir-n)J0(R2rn)B(r, rn)

J02(Rirn) - J02(R2rn)

~ (-vr2n)kq-ßk-ß—1 k= (q1-ß + arn)k+1 •

In order to determine velocity field v(r,t), applying the discrete inverse Laplace transform to Eq. (18) and using (A1) from Appendix,28 we find that

, , ln(R2 /r) A v(r,t) = r-TTT^TT At+

ln(R2/R1)

nA ^ Jo(R1rn)Jo(R2rn)B(r, rn) x

Jo2(R1rn) - Jo2(R2rn)

(-vrDk G1-ß,-ßk-ß-1,k+1 (-arn> t)

• (19)

Applying the Laplace transform to Eq. (6), we get

T (r, q) = (m + a^) Using Eq. (18), it becomes t (r, q) =

(M + a^) x

ß) öv(r> q)

rq2 ln(R2/R1)

Eq. (21), we get the shear stress of the form of

'(r,t) = A

a1t1-ß

r ln (R2/R1) V r(2 - ß)

A ^ r-nJo(R1rn)Jo(R2rn)B*(r, rn) 71 n=1 J2(R1rn) - Jo2 (R2rn) X

E (-vr~n) [vG1-ß,-ßk-ß-1, k+1 ( arn, 0 +

a1G1-ß,-ßk-1,k+1 (-arn•

By introducing Eq. (19) into Eq. (6) and using the results of

DßGa, b, c(d, t) = Ga,b+ß,c(d, t),

Dß tY

r(7 + 1)tY-ß

r(7 - ß + 1) '

we can also find shear stress as T (r,t) =

r ln R2/R1

a1t1-ß

lft + fc-) )

r-nJo(R1rn)Jo(R2rn)B*(r, rn). Jo2(R1rn) - Jo2(R2rn) '

E <y-vr'2n) [fG1-ß,-ßk-ß-1, k+1 ( arn>

a1G1-ß,-ßk-1,k+1 {-arln,t)] •

(1) Making P ^ 1 in Eqs. (19) and (23) we find the velocity field and the shear stress

vso(r,t) =

ln(R2/r)

ln(R2/R1)

Jo(R1rn)Jo(R2rn)B(r, rn) Jo2(R1 rn) - Jo2(R2rn)

r-nJo(R1rn)Jo(R2rn)B*(r, rn)

Jo2 (R1rn) - Jo2(R2rn)

(-vr2n )kq-ßk-ß-1

(q1-ß + arn )k+1

B*(r,rn) = J1(rrn)Yo(R2rn) - Jo(R2rn)Y1(rrn) •

Now, applying the inverse Laplace transform to

(-vrl) Go,-k-2,k+1 (-arn> t)

Tso(r,t) =

_-M_(t + 01) _

r ln (R2/R1) V f)

A ^ rnJo(R1rn)Jo(R2T-n)B*(r, rn)

n n=i Jo(R1rn) - Jo2 (R2rn) X

E^) k f Go,-k-2, k+1 (-arn+

a1Go,-k-1,k+1 (-arn>

corresponding to an ordinary second grade fluid, performing the same motion. These solutions can also be simplified to (see also Eqs. (A2) and (A3) from Appendix)

VSG (r,t) =

ln(R2/r) ln(R2/Ri )

nA ^ Jo (Rirn) Jo (R2rn)B(r, rn) v ( [Jo2 (Ri rn ) -Jo2 (R2 rn )] '

( vrn t

1 - exp - n

( vrnt \

V 1 + arl)

TSG (r,t) =

r ln R2/R

0 (t+ai) -

Jo (Rirn)Jo (R2rn)B* (r,rn) ,

=1 rn [Jo2 (Ri rn ) - Jo2 (R2 rn )] •

1 + ar2

( ^ \ V 1 + arnJ

(2) Now making a ^ 0 in Eqs. (29) and (30), we find the velocity field and the shear stress

VN (r,t) =

ln(R2 /r) ln(R2/Ri ) nA ^ Jo (Rirn ) Jo (R2rn)B (r, rn) v rn [Jo2(Rirn) - Jo2(R2rn)] X

(1 - e-^n*) ,

Tn (r, t) =

r ln R/Ri)

Jo (Ri rn)Jo (R2rn )B* (r,rn ) = rn [Jo2(Rirn) - Jo2(R2rn)]

(1 - e-**)

corresponding to the Newtonian fluid, performing the same motion.

(3) For large values of t, these solutions as well as those corresponding to second grade fluids approach to large time solutions

VL (r, t) =

ln(R2/Ri )

i) At +

ln(R2/Ri )

nA ^ Jo (Rirn) Jo (R2rn)B(r, rn) v ¿^ rn [Jo2 (Rirn) - Jo2 (R2rn)] , ( )

Tl (r,t) =

m (' + 7)-

r ln (R2/Ri

" Jo (Rirn)Jo (R2rn)B* (r,rn)

= rn [Jo2 (Rirn) - Jo2 (R2rn )] '

which are the same for both types of fluids.

Fig. 1. The time required to reach large time solutions for Newtonian, second grade and fractional second grade fluids, for Ri = 0.3, R2 = 0.5, A = 2, a = 0.003, v = 0.001 and

f3 = 0.5.

In conclusion, after some time, the behavior of the second grade fluid can be well enough approximated by that of a Newtonian fluid. Our interest here is to show that such a property also holds for fractional second grade fluid whose exact solutions, as well as those of the second grade fluid (see Eqs. (27) and (28) ), are written in terms of generalized Ga,bc(•, t) functions. More exactly, we must show that large time solutions corresponding to fractional second grade fluid is also given by Eqs. (33) and (34). Indeed from Fig. 1 we also have the time required to reach the large-time solutions for these fluids. This time (20s) is the lowest for Newtonian fluid and the highest (30s) for fractional second grade fluid.

In this paper, we find the velocity field and the adequate tangential shear stress, corresponding to the flow of a fractional second grade fluid between two infinite circular cylinders in which the inner cylinder slides along the axis, are determined. At time t = 0+, the inner cylinder begins to move along the common axis with constant acceleration A. The solutions, determined by employing the Laplace and finite Hankel transforms, and presented in terms of Bessel functions J0(•), Y0(•), Ji(•) and Yl(•) and generalized G functions, satisfy the corresponding governing equations as well as all imposed initial and boundary conditions. In the special cases, when ft ^ 1 or ft ^ 1 and aL ^ 0, the corresponding solutions for a second grade fluid or for the Newtonian fluid, performing the same motion, are respectively obtained from the general solutions.

1 1 1 1

¿t^ML ^=0.001

+++ ^=0.005

w u=0.009

1 1 i i

0.34 0.38 0.42 0.46 0.50

Fig. 3. Profiles of the velocity field v(r,t) given by Eq. (19) for Ri = 0.3, R2 = 0.5, t = 15s, A = 1, a = 0.002, f3 = 0.8 and different values of v

•¿Jo •f o

0 0.30

i i i i

AAA Oi = 0.10

a = 0.20

QQO Of = 0.25 -

i i I I X.

0.34 0.38 0.42 0.46 0.50

Fig. 4. Profiles of the velocity field v(r,t) given by Eq. (19) for Ri = 0.3, R2 = 0.5, t = 15s, A = 1, v = 0.005, f3 = 0.2 and different values of a

motion. Its effect on the fluid motion is qualitatively the same as that of a. More exactly, the velocity v(r, t) is a decreasing function with regards to ft.

Finally, for comparison, the profiles of v(r,t) corresponding to the motion of the Newtonian, ordinary second grade and fractional second grade are depicted together in Fig. 6 for the same t and the same common

Fig. 2. Profiles of the velocity field v(r,t) given by Eq. (19) for Ri = 0.3, R2 = 0.5, A = 1,a = 0.0003, v = 0.003,^ = 0. 2 and different values of t

Now, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity v (r,t) are depicted against r for different values of time t, and for different material and fractional parameters. Figure 2 clearly shows that the velocity is an increasing function of t. The influence of the kinematic viscosity v on the fluid motion is shown in Fig. 3. The velocity v (r,t) is a decreasing function of v.

The influences of material parameter a on the velocity is shown in Fig. 4. As expected, the velocity is a decreasing function with respect to a. Figure 5 shows the influence of the fractional parameter ft on the fluid

Fig. 5. Profiles of the velocity field v(r,t) given by Eq. (19) for Ri = 0.3, R2 = 0.5, t = 6s, A = 1, v = 0.003, a = 0.02 and different values of f3

Fig. 6. Profiles of the velocity field v(r, t) corresponding to the Newtonian, Second grade and fractional second grade fluids, for Ri = 0.3, R2 = 0.5, t = 3 s, A = 2, a = 0.2, v = 0.003 and 3 = 0.2

material and fractional parameters. The fractional second grade fluid, as resulting from these figures, is the slowest and the Newtonian fluid is the swiftest on the whole flow domain. In practice, it is necessary to know the approximate time after which the fluid is moving according to the large time solutions. This time, as resulting from Fig. 1, is the smallest for the Newtonian fluid and the highest for fractional second grade fluid. The units of the material constants are SI units in all figures, and the roots rn have been approximated by nn/R - Ri).

Appendix

-1! g i \(aa - d)c J

.(aa _d)cj - Gabc(d,t) dj r(c + j) t(c+j)a-b-1

= r(c)r(j + 1) r[(c + j)a _ b] where Re(ac _ b) > 0, \d/qa\ < 1,

J2(_vr2n)k Go, -k-1, k+1 (_ar2n, t)

1 + arn

( "ft \ V 1+ arn

Y^(_vr2n)k Go, -k-2, k+1 (_arl, t)

1 _ ex^ _ iVS )

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