Scholarly article on topic 'Unsteady rotational flows of an Oldroyd-B fluid due to tension on the boundary'

Unsteady rotational flows of an Oldroyd-B fluid due to tension on the boundary Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — A. Rauf, A.A. Zafar, I.A. Mirza

Abstract Unsteady Taylor–Couette flows of an Oldroyd-B fluid, which fills a straight circular cylinder of radius R, are studied. Flows are generated by the oscillating azimuthal tension which is given on the cylinder surface. As a novelty, authors used in this paper the governing equation related to the tension field. The closed forms of the shear stress and velocity fields corresponding to the flow problems are obtained by means of the integral transforms method. Expressions for the azimuthal tension and fluid velocity were written as sums between the “permanent component” (the steady-state component) and the transient component. By customizing values of parameters from the mathematical model were obtained the corresponding solutions of other types of fluids, namely, Maxwell fluids. By using numerical simulations and diagrams of the azimuthal stress, the fluid behavior has been analyzed. The necessary time to achieve the “steady-state” was, also, determined.

Academic research paper on topic "Unsteady rotational flows of an Oldroyd-B fluid due to tension on the boundary"

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Alexandria Engineering Journal (2015) 54, 973-979

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Alexandria University Alexandria Engineering Journal

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ORIGINAL ARTICLE

Unsteady rotational flows of an Oldroyd-B fluid due c^Ma* to tension on the boundary

A. Rauf *, A.A. Zafar, I.A. Mirza

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

Received 3 November 2014; revised 19 August 2015; accepted 1 September 2015 Available online 1 October 2015

KEYWORDS

Oldroyd-B fluid; Exact solutions; Integral transform; Velocity field; Shear stress

Abstract Unsteady Taylor-Couette flows of an Oldroyd-B fluid, which fills a straight circular cylinder of radius R, are studied. Flows are generated by the oscillating azimuthal tension which is given on the cylinder surface. As a novelty, authors used in this paper the governing equation related to the tension field. The closed forms of the shear stress and velocity fields corresponding to the flow problems are obtained by means of the integral transforms method. Expressions for the azimuthal tension and fluid velocity were written as sums between the ''permanent component" (the steady-state component) and the transient component. By customizing values of parameters from the mathematical model were obtained the corresponding solutions of other types of fluids, namely, Maxwell fluids. By using numerical simulations and diagrams of the azimuthal stress, the fluid behavior has been analyzed. The necessary time to achieve the "steady-state" was, also, determined.

© 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The Oldroyd-B fluid model is very important among the fluids of rate type due to its special behavior. Also, this model contains the Newtonian fluid model and Maxwell fluid model as special cases. The Oldroyd-B fluid model [1,2] considers the memory effects and elastic effects exhibited by a large class of fluids, such as the biological and polymeric liquids. Guillope and Saut [3] and Fontelos and Friedman [4] established the stability, existence and uniqueness results for some shearing flows of such fluids. Exact solutions for some simple flows of Oldroyd-B fluids were presented by many authors, See, for

* Corresponding author.

E-mail address: attari_ab092@yahoo.com (A. Raul).

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

example, Rajagopal and Bhatnagar [5], Hayat et al. [6,7]. Recently, various problems regarding flows of Oldroyd-B fluids through cylindrical domains have been studied. Singh and Varshney [8] have considered the unsteady laminar flow of an electrically conducting Oldroyd fluid through a circular cylinder boundary by permeable bed under the influence of an exponentially decreasing pressure gradient in porous medium. Burdujan [9] studied Taylor-Couette flows of the Oldroyd-B fluid with fractional derivatives within the annular region between two infinitely coaxial circular cylinders due to a time-dependent axial tension given on the surface of the inner cylinder. The unsteady unidirectional transient flow of Oldroyd-B fluid with fractional time derivatives, in an annular domain, produced by a constant pressure gradient and a translation with constant velocity of the inner cylinder was studied by Mathur and Khandelwal [10]. Liu et al. [11] studied some helical flows of an Oldroyd-B fluid with time-fractional

http://dx.doi.org/10.1016/j.aej.2015.09.001

1110-0168 © 2015 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

derivatives, between two infinite concentric oscillating cylinders and within an infinite circular oscillating cylinder. Most existing solutions in the literature correspond to problems with boundary conditions on the velocity. There are several practical problems with the specified force on the boundary [12-14]. For example in [12], Renardy has studied the motion of a Maxwell fluid across a strip bounded by parallel plates and proved that, in order to formulate a well posed problem it is necessary to impose the boundary conditions on the stresses at the inflow boundary. In [13], Renardy explained how well posed boundary value problems can be formulated using boundary conditions on stresses. Waters and King [15] were among the first specialists who used the shear stress on the boundary to find exact solutions for motions of rate type fluids. Other interesting problems regarding flows of non-Newtonian fluids, in various geometry or boundary conditions, can be finding in the references [17-23]. Our goal is to investigate unsteady flows of Oldroyd-B fluids in an infinite circular cylinder. In the present paper the governing equation of the flow is related to the azimuthal tension and we considered the boundary conditions on the shear stress. The flow of the fluid is due to rotation of the cylinder around its axis, under the action of oscillating shear stress/H(t) sin(wt) or/H(t) cos(wt) given on the boundary. Finally, solutions of the Maxwell fluid flows are obtained as particular cases of our general results. Also the comparison between models is underlined by graphical illustrations.

2. Problem formulation

The constitutive equations for an Oldroyd-B fluid [1] are

T = - pI + S, S + k

LS SLT

= ^ A + k, ( ^ - LA - ALr

where T is Cauchy stress tensor, —pI is indeterminate spherical stress, S is extra stress tensor, L is velocity gradient, i is the dynamic viscosity, A = L + LT is first the Rivilin-Erickson tensor, k and kr (0 6 kr < k) are relaxation and retardation time. Assume an infinite circular cylinder of radius R with axis of rotation along z-axis. Cylinder is filled with an Oldroyd-B fluid which is at rest at time t — 0. After time t — 0+ the cylinder applies an oscillating rotational shear stress /H(t) sin(wt) or /H(t) cos(wt) to the fluid, / > 0 is constant and x is the angular frequency of oscillations. We assume that, the fluid is incompressible and homogeneous. Furthermore we assume that velocity field and extra-stress tensor are of the form

V — V(r, t) —w(r, t)e9, S — S(r, t), (2)

where eh is the unit vector in the h direction of the cylindrical coordinate system. Since the fluid and the cylinder are at rest at time t — 0, therefore,

—(r, 0) = 0, S(r, 0) = 0.

Introducing (2) in (1)2 and by using (3) we get — Srz — Szh — Szz — 0, along with the following meaningful partial differential equation

0+#(r't)—K1+k"@t) (i—1)w(r'* (4)

where s(r, t) — Srh(r, t) is one of the nonzero component of extra stress tensor. The balance of linear momentum in the absence of body forces reduces to [8]

d—(r, t) id 2\ .

qd-d^ = d,+-s(r,t),

sst(r, t) q being the constant density of the fluid. By eliminating w(r, t) between Eqs. (4) and (5) we get the following governing equation for the shear stress [24]

, , @N @r(r, t) i B \f @2 1 @ 4N , N

1 + kFt) — ^ 1 + k +1 - — s(r' 0>

du \dr2 r dr

where v — q is the kinematic viscosity of the fluid. The appropriate initial and boundary conditions are

ds(r, t)

s(r OL0 — ■

s(R, t) — fH(t) sin xt or s(R, t) — fH(t) cos rat,

H(t) being the Heaviside unit step function. Converting our problem (6)-(8) into the complex field (s — sc + hs with sc and ss being solutions for cosine, respectively sine boundary conditions), we have

^ dN ds(r, t)_ ' d @2

1 + k Ft ^T — v

(■-(£-11 - 4) sc.» (9.

s(r; OU —

ds(r, t)

t—0 dt

t—0 f > 0.

(10) (11)

s(R, t) — fH(t)e" By introducing the following dimensionless quantities

t r - s kf kr

t — t , r — —, — — —, s — -, U0 — —, p — —, x — kx,

Eqs. (5), (9)—(11) becomes (dropping the star notation)

d—(r; t)

(dr+,) s(r,t),

' -1) ^ — >■■

ds(r, t)

/d2 1 d 4\

^ U-+1 d,-r- )s(r, t),(14)

s(r, t)jt—0 — dt s(1, t)—H(t)e'xt,

— 0, —(r, 0) —0

where Re — R- is the Reynolds number.

3. Solution of the problem

In order to determine the exact analytical solution, we shall use the Laplace and finite Hankel transforms [25]. By applying the temporal Laplace transform to Eqs. (14) and (16) and using the initial conditions (15) we get the following transformed forms,

i @2 1 @ 4 N Re(1 + 4)4^ q) — (1 + Pq){ ^ +1 — q); (17)

t(1, q) = -

—, (18) q — im

where s(r, q) — L{s(r, t)g and q is the Laplace transform parameter. In order to find the solution of the problem (17) and (18), we use the finite Hankel transform of the order two (see (A.1) from Appendix A) along with relation (B.3) from Appendix B. We obtain the following expression for the Hankel transforms of function s(r, q)

?n(rn, q) — —r„Ji(r„)-—2 b + 1 ——2• (19)

(q — im) Req2 + (Re + pr2n)q + r2n '

Now applying inverse Hankel transform (Eq. A.2) to Eq. (24) and using identity (Eq. (B.6)) with condition J2(rn)—0 and identity (Eq. (B.7))), [27], separating real and imaginary parts we get the exact expression for shear stresses corresponding to considered problems. For cosine oscillation,

sc(r, t) = Real part {s(r, t)} = scs(r, t) + sct(r, t), t > 0,

In order to apply the inverse Laplace transform [26] we rewrite Eq. (19) into the suitable equivalent form

SH(rn, q) = -rnJl(rn)

Re q - i

(q+2R)

Req — im (q + 2Re)2 -(àe)

2Re — ¡ian 1

an \2 _ f bn \2 Г '

2Re) (2Re) J

Reb„

q — im (q + ft)

where an — Re + ftr2n, b2n — a2n — 4Rer2n. Now, applying the inverse Laplace transform [26], using the formulae (B.4) and (B.5) and the convolution theorem [25] we get

SH(rn, t) = —rnJi(rn)^(Ain + iBin) sinh^Re)e —aiRe

+(Â2n + iB2n) cosh e—2Re — (An + iB2n)eim

A in =

(r2n — Rem2)(fir2n — Re) + m2"n("nb — 2Re) bn{(r2n — Rem)2)2 + (man )2}

Bin = -

—man(brn — Re) + m(anb — nRe)(r2n — Rem2)

bn{(r2n — Rem2)2 + (man)2}

r2n + Rem2 — m2anb

(r2n — Rem2)2 + (man)2 '

mb(—r2 + Rem2) + man (r2n — Rem2)2 + (man)2

Using the identity

I I - a I

A2n + 2 = A3n + 2 '

r2 r2 r2

Аз2 = -

Rem rn + pr^m2 — m2 (a2n + Re2m2)

fn — Rem2)2 + (man)2}

i J ( rr )

scs(r, t) = r2 cos mt — 2—\ rn(A3n cos mt + B2n sin mt),

n=l Ji(rn)

sct(r,t) = —2j2 Jl}r,[n}r„(Ainsinhf^) + Anncoshf^) )e We-

n=i Ji(rn) '

For sine oscillation,

ss(r, t) = Imiginary part {s(r, t)} = sss(r, t) + sst(r, t), t > 0,

1 J rr

sss(r, t) = r2 sin mt — nV 2 2 rn(A3n sin mt — B2n cos mt), 2=0 Ji(rn)

sst (r, t) = ~2£ Jg r^Bin sinlt^) + Bnn cosh(nR))e—2«.

It is important to point out that solutions (25) and (28) are written as sums between the steady-state and transient solutions. By replacing Eqs. (25) and (28) in Eq. (13), integrating with respect to time t and using the initial condition (15)2 we get velocity corresponding to cosine and sine oscillation of Oldroyd-B fluid respectively,

t ^ Ji(rrn)r2 [e 2R' (

wc(r' °=—2 5 Jnr M

в 2Re

—(anAin + bnAnn) sinh I 2Reу

-(bnAin + anAnn) cos^nRRe) I + Xo (A3n sin(mt)—B2n cos(mt))

Jl(rrn Уп( ±

2=0 Jl(rn) \2r2

Bn J 4r

-I TT (bnAin + anA2n)+---) +--sin(mi),

Ji(rn) 2r2 m \ m

we get

sn(rn, t) = —rnJl(rn)

Ain smh(2£)+Anncosh {èt))e—-

+(A3n cos mt + B2n sin mt) cos mt

+4(Bln + B2n e^22

+(A3n sin mt — B2n cos mt) ^^sin mt

's(r, t) = ~n£ JJ

Ji(rrn)r2

2=0 Jl(rn)

' —£n£ (

an'Bin + bnBnn) sinh

-(bnBin + anBnn) cos^nrRe) i + X (—A3n cos(mt) —Bnn sin(mt))

EJi(rrn )r2 I l ,, D , „ X , A3 J 4r

, , \"< TT(bnBin + anBnn) +--> +--(l — cos(mt)).

Ji (rn) ^2r2 ^ m

Figure 1 Decay time of transient component for the shear stress sct(r, t) and sst(r, t) of Oldroyd-B fluids given by Eqs. (27) and (30) for Re = 25, b = 0.5 and different values of angular frequency x.

r = 0.5

/ M-» p = 0.4 -

ess p - Q 6

p = 0.8 _

(a) for cosine oscillations

r = 0.5

= 0 .4 "

p = 0.6

^^ (3 = 0.8 -

(b)for sine oscillations

Figure 2 Decay time of transient component for the shear stress sct(r, t) and sst(r, t) of Oldroyd-B fluids given by Eqs. (27) and (30) for Re = 50, x = 0.393 and different values of the ratio ft.

By now letting x ! 0 into Eqs. (25) and (31), as it was to be expected, we recover the known results [16, Eqs. (15) and (17)] corresponding to the motion induced by the flat plate that applies a constant shear fH(t) to the fluid.

4. Limiting cases

4.1. The case b ! 0 (Maxwell fluids)

By taking b ! 0 into Eqs. (25)-(32) we get exact expressions of the shear stress and velocity for both, cosine and sine cases corresponding to the motion of a Maxwell fluid in an infinite circular cylinder which is rotating under the boundary conditions (16)

< A u 4 .JMR2 — 4Rern^

ScM(r, t) = —2 , - rA lAllnsmh «=1 '(rn) \

+A22n cosh

e2 - 4Rer2)t 1

+(A33ncosxt + B22nsinxt)}+ r2cosxt, t > 0, (33)

sSM(r, t) = —'J(rr") rJ Bu« sinh

n=i Jl(rn) \

e2 4Rer2 t

+B22n cosh

e2 4Rer2 t

+(A33n sin xt — B22n cos xt)} + r2sin xt, t > 0, (34)

/ A Jl(rrn)r2n

WcM(r, t) = —2 § JW

— ( ReAiin + \j (Re2 — 4Rer2) A22nj J ( Re2 — 4Rer2)A / ,--\

*-—- H V (Re2 — 4Rer2)Aiin + ReA^

J(Re2 — 4Rer«)^ 1

--1 M—(A33n sin(xt) —B22n cos(xt))

'1(rn) 2r2n

— J (Re2 — 4Rer2)Aiin + ReA22n

)+ B2f}

H— sin(xt), x

WsM(r, t) = —2^

Ji (rrn)r2

2=1 Ji(rn.

(Re2 — 4Rer 2)t

2—^ I — ^ ReBiin + (Re2 — 4Rer2)B22nj

— (a/ (Re2 — 4Rer2)Biin + ReB22n)j

\J (Re2 — 4Rern)^ 1

Ji(rrn )r2n\ 1

H— — A33n cos(xi) — B22n sin(xt) x 1

'(rn) l^1 V (Re2 — 4Rer2)B11n + ReB22n I

H—(1 — cos(xt)), x

Re? + Re2x2

y/1 — 4Rer2 {(rn — Rex2)2 + x2}'

Figure 3 Decay time of transient component for the shear stress sct(r, t) and sst(r, t) of Oldroyd-B fluids given by Eqs. (27) and (30) for x = 0.785, b = 0.6 and different values of the Reynolds number Re.

Figure 4 Profiles of both cosine and sine shear stress for Oldroyd-B fluid and Maxwell fluid for b = 0.6, x = 0.785, r = 0.5 and different values of Reynolds number.

xRe{Re - 2Re(r2n - Rex2)}

Bun — —, —-,

yjl - 4Rernf(rn - Rex2)2 + x2}

-r2n + Rex2

A22n —

(r2n - Rex2)2 + x2'

A33n —

(rn - Rex2)2 + x2'

Rex2r2n - x2Re2(1 + x2) r2n{(rl - Rex2)2 + x2} "

5. Numerical results and discussion

Generally, in the rheological measurements, the transient parts of the starting solutions are neglected. Practically, it is of (37) interest to find the required time to reach the large-time state for solutions, consequently, to approximate the time after which the fluid is moving according to the large-time state is an important problem regarding the technical relevance of the solutions. In our problem the required time for the decay of transient part of solutions for both cosine and sine shear stresses depends only of three quantities, namely, x the angular frequency of oscillations, b the ratio of the retardation time

and the relaxation time and Re the Reynolds number. Fig. 1 depicts that the decaying of transient part for the solutions (27) and (30), corresponding to both cosine and sine shear stresses is faster for increasing angular frequency x. Fig. 2 shows that the decay time of transient part for the solutions (27) and (30), corresponding to both cosine and sine shear stresses decreases with the increase of ratio p. Fig. 3 depicts that the decay time of transient part for the solutions (27) and (30), corresponding to both cosine and sine shear stresses increases with the increase in Reynolds number. In each of the examined cases, it can be determined from the chart, the approximate values of the time t after which the transient solutions can be neglected. Fig. 4 shows the comparison between Oldroyd-B and Maxwell fluids. It shows that the amplitude of the wave cycle for shear stress corresponding to Oldroyd-B and Maxwell fluids decreases with increasing the Reynolds number Re. Moreover, amplitude of wave cycle for Oldroyd-B fluids is always lesser in comparison with that of Maxwell fluids for both sine and cosine shear stresses.

Also, it is important to note that, for a short time-interval, the flow of Maxwell fluid exhibits instability for cosine oscillations. In the case of sine oscillations, for a short time interval, the Maxwell fluid is not moving. The length of this interval increases for increasing of Reynold's number.

6. Conclusions

where J2(-) is the Bessel function of the first kind of order 2 and 0 < ri < r2 < ... are the positive roots of the equation J2(r) — 0. The inverse finite Hankel transform of order 2 of function gH(rn) is defined by

H—1{gn(rn)}— g(r) — 2^ gH(rn),

2—1 J3(rn)

where the summation is defined over all positive roots of

J2 (r) — 0.

Appendix B. By using the following formulae [27,28] and definition of finite Hankel transform of order 2 (A.1)

dJ1[m(r)] ( 1

— {W) J1[m(r)]— J2[m(r)]}m'(r), (B.1)

), (B:2)

dJ2[m(r)] 2

— Mr)^[m(r)] + J1[m(r)] }m'(r.

we get

f1 / d2 1 d 4 N. w. w

J0 \dr2 + ? — s(r,q)J2(rrn)dr

— —rlsn(rn, q) — rnS(1, q)J1 (rn)•

In this paper we have studied unsteady flows on Oldroyd-B fluids through a circular cylinder on whose boundary was given the azimuthal tension in the form f sin(xt) or f cos(xt). If, usually in literature the governing equation of the flow is related at the velocity field, in the present paper, the basic flow equation is related at the azimuthal tension. Solutions of the initial-boundary value flow problem have been obtained by means of the Laplace and Hankel transforms. Expressions of the shear stress corresponding to both types of oscillations have been written as a sum between ''the steady-state" (or permanent solution) and the transient solution which tends to zero for large values of the time t.

Analyzing some specific situations, the decreasing of transient solutions was studied. The approximate values of the time for which the transient solution can be neglected were determined. Corresponding solutions for Maxwell fluids have been obtained as particular case and a comparison between both models was presented. The roots of the equations J2(x) —0, numerical calculations and graphs were obtained using subroutines from Mathcad15.

Acknowledgments

The authors Abdul Rauf, Azhar Ali Zafar and Itrat Abbas Mirza are highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan, and also Higher Education Commission of Pakistan for supporting and facilitating this research work.

Appendix A. For the function g(r) the finite Hankel transform of order 2 [25] is defined by

H2{g(r)} — gH(rn) — i rg(r)J2 (rrn)dr; J0

eas cosh(bs)ds —-2 facosh(bs) — b

a2 — b2

x sinh(bs)}, a2 — b2 •

eas sinh(bs)ds —-2 fa sinh(bs) — b

a2 — b2

x cosh(bs)}, a2 — b2 •

J1 (rn) +J3(rn) — — J2(rn)• rn

For finite inverse Hankel transform of order 2 with J2(rn) — 0 we have the following identity

H ^ — - J1 (rn)\— r2.

Proof Since,

H2{r2} — / r3J2(rrn)dr. J0

Replace z — rrn, then 1 {rn

H2{r2}—- z3J2(z)dz.

Using J0z t3J2(t)dt — z3J3(z) [28] and J2(rn)— 0 along with J3(z)—lJ2(z) — J1(z) [28] we get,

H2{r2} — —- J1 (rn) •

Hence,

H ^ — - J1 (rn)\— r2.

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[27] N.W. McLachlan, Bessel Functions for Engineers, Oxford Univeristy Press, London, 1955.

[28] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, in: NBS, Appl. Math. Series, vol. 55. Washington, DC, 1964.