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

A. Mahmood; N.A. Khan; I. Siddique; A. Zada; A.U. Awan

doi:10.1016/j.jksus.2010.07.022

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ORIGINAL ARTICLE

23 (

); 341-347

doi:

10.1016/j.jksus.2010.07.022

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

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

Received: 2010-7-2, Accepted: 2010-7-24, Published: 2011-10-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Available online 23 March 2011

Peer review under responsibility of King Saud University.

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 ω₁ and ω₂. 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.

Keywords

Second grade fluid

Shear stress

Torsional oscillatory flow

Fractional calculus

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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, rheological 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 ω₁ and ω₂. 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

(1)

T = - p I + μ A_{1} + α_{1} A_{2} + α_{2} A_{1}^{2},

where −p is the pressure, I is the unit tensor, μ is the coefficient of viscosity, α₁ and α₂ are the normal stress moduli and A₁ and A₂ are the kinematic tensors defined through

(2)

A_{1} = grad v + (grad v)^{T}, A_{2} = \frac{d A_{1}}{dt} + A_{1} (grad v) + (grad v)^{T} A_{1} .

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

(3)

div v = tr A_{1} = 0 .

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

(4)

μ ⩾ 0, α_{1} ⩾ 0 and α_{1} + α_{2} = 0 .

The sign of the material moduli α₁ and α₂ 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

(5)

A_{2} = D_{t}^{β} A_{1} + A_{1} (grad v) + (grad v)^{T} A_{1},

where

D_{t}^{β}

is the Riemann–Liouville fractional calculus operator of order β with respect to t defined by

(6)

D_{t}^{β} f (t) = \frac{1}{Γ (1 - β)} \frac{d}{dt} \int_{0}^{t} (t - τ)^{- β} f (τ) d τ, 0 < β ⩽ 1,

where Γ(·) is the Gamma function. When β = 1, Eq. (5) may be simplified as (2)₂ while for α₁ = 0 the constitutive relationship (1) describes the Rainer–Rivlin viscous fluid.

2.1

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 shownin 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(ω₁t) and W₂sin(ω₂t), where ω₁ is the frequency of velocity of inner cylinder and ω₂ is that of outer cylinder. Owing to the shear, the fluid between the cylinders is gradually moved, its velocity being of the form

(7)

v = v (r, t) = v (r, t) e_{θ},

where e_θ is the unit vector along θ-direction. For such flows the constraint of incompressibility is automatically satisfied.

Introducing (7) into the constitutive equation, we find that

(8)

τ (r, t) = (μ + α_{1} D_{t}^{β}) (\frac{\partial}{\partial r} - \frac{1}{r}) v (r, t),

where τ(r, t) = S_rθ(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

(9)

ρ \frac{\partial v (r, t)}{\partial t} = (\frac{\partial}{\partial r} + \frac{2}{r}) τ (r, t) .

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

(10)

\frac{\partial v (r, t)}{\partial t} = (ν + α D_{t}^{β}) (\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} - \frac{1}{r^{2}}) v (r, t); r \in (R_{1}, R_{2}), t > 0,

where α = α₁/ρ and ν = μ/ρ is the kinematic viscosity of the fluid (ρ being its constant density).

The appropriate initial and boundary conditions are

(11)

v (r, 0) = 0; r \in (R_{1}, R_{2}),

(12)

v (R_{1}, t) = W_{1} \sin (ω_{1} t), v (R_{2}, t) = W_{2} \sin (ω_{2} t) for t > 0 .

2.2

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

(13)

\frac{\partial^{2} \bar{v} (r, q)}{\partial r^{2}} + \frac{1}{r} \frac{\partial \bar{v} (r, q)}{\partial r} - \frac{1}{r^{2}} \bar{v} (r, q) - \frac{q}{α q^{β} + ν} \bar{v} (r, q) = 0; r \in (R_{1}, R_{2}),

where the image function

\bar{v} (r, q)

of v(r, t) has to satisfy the conditions

(14)

\bar{v} (R_{1}, q) = \frac{W_{1} ω_{1}}{q^{2} + ω_{1}^{2}}, \bar{v} (R_{2}, q) = \frac{W_{2} ω_{2}}{q^{2} + ω_{2}^{2}} .

In the following, let us denote by

(15)

{\bar{v}}_{n} (q) = \int_{R_{1}}^{R_{2}} r \bar{v} (r, q) B_{1} ({rr}_{n}) dr; n = 1, 2, 3, \dots,

the finite Hankel transforms of

\bar{v} (r, q)

, where r_n are the positive roots of the transcendental equation B₁(R₁r) = 0 and

(16)

B_{1} ({rr}_{n}) = J_{1} ({rr}_{n}) Y_{1} (R_{2} r_{n}) - J_{1} (R_{2} r_{n}) Y_{1} ({rr}_{n}) .

In the above relation, J₁(·) 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

(17)

\frac{2 W_{2} ω_{2}}{π (q^{2} + ω_{2}^{2})} - \frac{2 W_{1} ω_{1}}{π (q^{2} + ω_{1}^{2})} \frac{J_{1} (R_{2} r_{n})}{J_{1} (R_{1} r_{n})} - r_{n}^{2} {\bar{v}}_{n} (q) - \frac{q}{α q^{β} + ν} {\bar{v}}_{n} (q) = 0,

or equivalently,

(18)

{\bar{v}}_{n} (q) = \frac{2 W_{2} ω_{2} (α q^{β} + ν)}{π (q^{2} + ω_{2}^{2}) (α r_{n}^{2} q^{β} + q + ν r_{n}^{2})} - \frac{2 W_{1} ω_{1} (α q^{β} + ν)}{π (q^{2} + ω_{1}^{2}) (α r_{n}^{2} q^{β} + q + ν r_{n}^{2})} \frac{J_{1} (R_{2} r_{n})}{J_{1} (R_{1} r_{n})} .

In order to determine

\bar{v} (r, q)

, we firstly write

{\bar{v}}_{n} (q)

under the suitable form

(19)

{\bar{v}}_{n} (q) = \frac{2 W_{2} ω_{2}}{π r_{n}^{2} (q^{2} + ω_{2}^{2})} - \frac{2 W_{1} ω_{1}}{π r_{n}^{2} (q^{2} + ω_{1}^{2})} \frac{J_{1} (R_{2} r_{n})}{J_{1} (R_{1} r_{n})} - \frac{2 W_{2} ω_{2} q}{π r_{n}^{2} (q^{2} + ω_{2}^{2}) (α r_{n}^{2} q^{β} + q + ν r_{n}^{2})} + \frac{2 W_{1} ω_{1} q}{π r_{n}^{2} (q^{2} + ω_{1}^{2}) (α r_{n}^{2} q^{β} + q + ν r_{n}^{2})} \frac{J_{1} (R_{2} r_{n})}{J_{1} (R_{1} r_{n})}

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

(20)

\bar{v} (r, q) = \frac{π^{2}}{2} \sum_{n = 1}^{\infty} \frac{r_{n}^{2} J_{1}^{2} (R_{1} r_{n}) B_{1} ({rr}_{n})}{J_{1}^{2} (R_{1} r_{n}) - J_{1}^{2} (R_{2} r_{n})} {\bar{v}}_{n} (q)

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

(21)

\frac{1}{α r_{n}^{2} q^{β} + q + ν r_{n}^{2}} = \frac{1}{q^{β} [ν r_{n}^{2} q^{- β} + (q^{1 - β} + α r_{n}^{2})]} = \sum_{k = 0}^{\infty} \frac{(- ν r_{n}^{2})^{k} q^{- β k - β}}{(q^{1 - β} + α r_{n}^{2})^{k + 1}}

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

(22)

G_{a, b, c} (d, t) = \sum_{j = 0}^{\infty} \frac{(c)_{j} d^{j} t^{(j + c) a - b - 1}}{Γ (j + 1) Γ [(j + c) a - b]}

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

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

(23)

v (r, t) = \frac{W_{1} R_{1} (R_{2}^{2} - r^{2}) \sin (ω_{1} t) + W_{2} R_{2} (r^{2} - R_{1}^{2}) \sin (ω_{2} t)}{(R_{2}^{2} - R_{1}^{2}) r} - π \sum_{n = 1}^{\infty} \sum_{k = 0}^{\infty} (- ν r_{n}^{2})^{k} \frac{J_{1} (R_{1} r_{n}) B_{1} ({rr}_{n})}{J_{1}^{2} (R_{1} r_{n}) - J_{1}^{2} (R_{2} r_{n})} \times [W_{2} ω_{2} J_{1} (R_{1} r_{n}) \int_{0}^{t} \cos (ω_{2} (t - τ)) G_{1 - β, - β k - β, k + 1} (- α r_{n}^{2}, τ) d τ - W_{1} ω_{1} J_{1} (R_{2} r_{n}) \int_{0}^{t} \cos (ω_{1} (t - τ)) G_{1 - β, - β k - β, k + 1} (- α r_{n}^{2}, τ) d τ] .

2.3

2.3 Calculation of the shear stress

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

(24)

\bar{τ} (r, q) = (μ + α_{1} q^{β}) (\frac{\partial}{\partial r} - \frac{1}{r}) \bar{v} (r, q),

where

(25)

(\frac{\partial}{\partial r} - \frac{1}{r}) \bar{v} (r, q) = \frac{2 R_{1} R_{2}}{(R_{2}^{2} - R_{1}^{2}) r^{2}} (\frac{R_{1} W_{2} ω_{2}}{q^{2} + ω_{2}^{2}} - \frac{R_{2} W_{1} ω_{1}}{q^{2} + ω_{1}^{2}}) + π \sum_{n = 1}^{\infty} \sum_{k = 0}^{\infty} (- ν r_{n}^{2})^{k} \times \frac{J_{1} (R_{1} r_{n}) [(2 / r) B_{1} ({rr}_{n}) - r_{n} {\tilde{B}}_{1} ({rr}_{n})]}{J_{1}^{2} (R_{1} r_{n}) - J_{1}^{2} (R_{2} r_{n})} [W_{2} ω_{2} J_{1} (R_{1} r_{n}) \frac{q}{q^{2} + ω_{2}^{2}} \frac{(k + 1)_{j} (- α r_{n}^{2})^{j}}{j! q^{k + j (1 - β) + 1}} - W_{1} ω_{1} J_{1} (R_{2} r_{n}) \frac{q}{q^{2} + ω_{1}^{2}} \frac{(k + 1)_{j} (- α r_{n}^{2})^{j}}{j! q^{k + j (1 - β) + 1}}]

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

{\tilde{B}}_{1} ({rr}_{n}) = J_{0} ({rr}_{n}) Y_{1} (R_{2} r_{n}) - J_{1} (R_{2} r_{n}) Y_{0} ({rr}_{n}) .

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

(26)

R_{a, b} (c, d, t) = \sum_{j = 0}^{\infty} \frac{c^{j} (t - d)^{(j + 1) a - b - 1}}{Γ [(j + 1) a - b]}

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

(27)

τ (r, t) = \frac{2 R_{1} R_{2}}{(R_{2}^{2} - R_{1}^{2}) r^{2}} [μ (R_{1} W_{2} \sin (ω_{2} t) - R_{2} W_{1} \sin (ω_{1} t)) + α_{1} \{R_{1} W_{2} ω_{2} R_{2, β} (- ω_{2}^{2}, 0, t) - R_{2} W_{1} ω_{1} R_{2, β} (- ω_{1}^{2}, 0, t)\}] + π \sum_{n = 1}^{\infty} \sum_{k = 0}^{\infty} (- ν r_{n}^{2})^{k} \frac{J_{1} (R_{1} r_{n}) [(1 / r) B_{1} ({rr}_{n}) - r_{n} {\tilde{B}}_{1} ({rr}_{n})]}{J_{1}^{2} (R_{1} r_{n}) - J_{1}^{2} (R_{2} r_{n})} \times [W_{2} ω_{2} J_{1} (R_{1} r_{n}) \{μ \int_{0}^{t} \cos (ω_{2} (t - τ)) G_{1 - β, - β k - β, k + 1} (- α r_{n}^{2}, τ) d τ + α_{1} G_{1 - β, - β k - 1, k + 1} (- α r_{n}^{2}, t) - α_{1} ω_{2} \int_{0}^{t} \sin (ω_{2} (t - τ)) G_{1 - β, - β k - 1, k + 1} (- α r_{n}^{2}, τ) d τ\} - W_{1} ω_{1} J_{1} (R_{2} r_{n}) \{μ \int_{0}^{t} \cos (ω_{1} (t - τ)) G_{1 - β, - β k - β, k + 1} (- α r_{n}^{2}, τ) d τ + α_{1} G_{1 - β, - β k - 1, k + 1} (- α r_{n}^{2}, t) - α_{1} ω_{1} \int_{0}^{t} \sin (ω_{1} (t - τ)) G_{1 - β, - β k - 1, k + 1} (- α r_{n}^{2}, τ) d τ\}] .

3 Limiting cases

3.1

3.1 Solutions for ordinary second grade fluid (β → 1)

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

(28)

v_{S} (r, t) = \frac{W_{1} R_{1} (R_{2}^{2} - r^{2}) \sin (ω_{1} t) + W_{2} R_{2} (r^{2} - R_{1}^{2}) \sin (ω_{2} t)}{(R_{2}^{2} - R_{1}^{2}) r} - π \sum_{n = 1}^{\infty} \frac{J_{1} (R_{1} r_{n}) B_{1} ({rr}_{n})}{J_{1}^{2} (R_{1} r_{n}) - J_{1}^{2} (R_{2} r_{n})} \times [\frac{W_{2} ω_{2} J_{1} (R_{1} r_{n})}{ν^{2} r_{n}^{4} + ω_{2}^{2} (1 + α r_{n}^{2})^{2}} \{ν r_{n}^{2} (\cos (ω_{2} t) - \exp (- \frac{ν r_{n}^{2} t}{1 + α r_{n}^{2}})) + ω_{2} (1 + α r_{n}^{2}) \sin ((ω_{2} t))\} - \frac{W_{1} ω_{1} J_{1} (R_{2} r_{n})}{ν^{2} r_{n}^{4} + ω_{1}^{2} (1 + α r_{n}^{2})^{2}} \{ν r_{n}^{2} (\cos (ω_{1} t) - \exp (- \frac{ν r_{n}^{2} t}{1 + α r_{n}^{2}})) + ω_{1} (1 + α r_{n}^{2}) \sin (ω_{1} t)\}],

and associated shear stress

(29)

τ_{S} (r, t) = \frac{2 R_{1} R_{2}}{(R_{2}^{2} - R_{1}^{2}) r^{2}} [μ (R_{1} W_{2} \sin (ω_{2} t) - R_{2} W_{1} \sin (ω_{1} t)) + α_{1} (R_{1} W_{2} ω_{2} \cos (ω_{2} t) - R_{2} W_{1} ω_{1} \cos (ω_{1} t))] + π \sum_{n = 1}^{\infty} \frac{J_{1} (R_{1} r_{n}) [(1 / r) B_{1} ({rr}_{n}) - r_{n} {\tilde{B}}_{1} ({rr}_{n})]}{J_{1}^{2} (R_{1} r_{n}) - J_{1}^{2} (R_{2} r_{n})} \times (J_{1} (R_{1} r_{n}) g_{2} - J_{1} (R_{2} r_{n}) g_{1}),

where in above Eq. (29)

g_{m} = W_{m} ω_{m} [\frac{μ}{ν^{2} r_{n}^{4} + ω_{m}^{2} (1 + α r_{n}^{2})^{2}} \{ν r_{n}^{2} (\cos (ω_{m} t) - \exp (- \frac{ν r_{n}^{2} t}{1 + α r_{n}^{2}})) + ω_{m} (1 + α r_{n}^{2}) \sin (ω_{m} t)\} + \frac{α_{1}}{1 + α r_{n}^{2}} \exp (- \frac{ν r_{n}^{2} t}{1 + α r_{n}^{2}}) - \frac{α_{1} ω_{m}}{ν^{2} r_{n}^{4} + ω_{m}^{2} (1 + α r_{n}^{2})^{2}} \times \{ω_{m} (1 + α r_{n}^{2}) (\exp (- \frac{ν r_{n}^{2} t}{1 + α r_{n}^{2}}) - \cos (ω_{m} t)) + ν r_{n}^{2} \sin (ω_{m} t)\}]; m = 1, 2,

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).

It is important to point out here that the terms containing exp(·) in 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

3.2 Solutions for Newtonian fluid

Making α → 0 (equivalently α₁ → 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

(30)

v_{N} (r, t) = \frac{W_{1} R_{1} (R_{2}^{2} - r^{2}) \sin (ω_{1} t) + W_{2} R_{2} (r^{2} - R_{1}^{2}) \sin (ω_{2} t)}{(R_{2}^{2} - R_{1}^{2}) r} - π \sum_{n = 1}^{\infty} \frac{J_{1} (R_{1} r_{n}) B_{1} ({rr}_{n})}{J_{1}^{2} (R_{1} r_{n}) - J_{1}^{2} (R_{2} r_{n})} [\frac{W_{2} ω_{2} J_{1} (R_{1} r_{n})}{ν^{2} r_{n}^{4} + ω_{2}^{2}} \{ν r_{n}^{2} (\cos (ω_{2} t) - \exp (- ν r_{n}^{2} t)) + ω_{2} \sin ((ω_{2} t))\} - \frac{W_{1} ω_{1} J_{1} (R_{2} r_{n})}{ν^{2} r_{n}^{4} + ω_{1}^{2}} \{ν r_{n}^{2} (\cos (ω_{1} t) - \exp (- ν r_{n}^{2} t)) + ω_{1} \sin (ω_{1} t)\}] .

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 ω₁ and ω₂ 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 W₁ = 0, W₂ = W and ω₂ = ω (when inner cylinder is at rest) or W₁ = W, W₂ = 0 and ω₁ = ω (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))

(31)

v (r, t) = \frac{{WR}_{2} (r^{2} - R_{1}^{2}) \sin (ω t)}{(R_{2}^{2} - R_{1}^{2}) r} - π W ω \sum_{n = 1}^{\infty} \sum_{k = 0}^{\infty} {(- ν r_{n}^{2})}^{k} \frac{J_{1}^{2} (R_{1} r_{n}) B_{1} ({rr}_{n})}{J_{1}^{2} (R_{1} r_{n}) - J_{1}^{2} (R_{2} r_{n})} \int_{0}^{t} \cos (ω (t - τ)) G_{1 - β, - β k - β, k + 1} (- α r_{n}^{2}, τ) d τ .

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 β → 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 β → 1 and α → 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 β = 0.9 and 0.6, respectively), second grade (curve vS(r) for β = 1) and that of Newtonian fluid (curve vN(r) for β = 1 and α → 0). These graphs also show the influence of the fractional coefficient β on the velocity v(r, t). In all figures we consider R₁ = 1, R₂ = 4, V₁ = 1, V₂ = 4, Ω₁ = 5, Ω₂ = 7, α = 9 × 10⁻³ and ν = 1.1746 × 10⁻³ while SI units of parameters are used.

Velocity profiles for different values of time.

References

Casarella M.J., Laura P.A., . Drag on oscillating rod with longitudinal and torsional motion. J. Hydronautics. 1969;3:180-183.
[Google Scholar]
Dunn J.E., Rajagopal K.R., . Fluids of differential type: critical review and thermodynamic analysis. Int. J. Eng. Sci.. 1995;33:689-729.
[Google Scholar]
Fetecau C., Fetecau C., . Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder. Int. J. Eng. Sci.. 2006;44:788-796.
[Google Scholar]
Fetecau C., Hayat T., Fetecau C., . Starting solutions for oscillating motions of Oldroyd-B fluids in cylindrical domains. J. Non-Newtonian Fluid Mech.. 2008;153:191-201.
[Google Scholar]
Khan M., Ali S.H., Qi H., . Exact solutions of starting flows for a fractional Burgers'fluid between coaxial cylinders. Nonlinear Anal. Real World Appl.. 2009;10:1775-1783.
[Google Scholar]
Khan, M., Asghar, S., Hayat, T., 2005. Oscillating flow of a Burgers’ fluid in a pipe. The Abdus Salam International Center for Theoretical Physics, IC/2005/071.
Lorenzo, C.F., Hartley, T.T., 1999. Generalized functions for the fractional calculus. NASA/TP-1999-209424/Rev1.
Mahmood A., Parveen S., Ara A., Khan N.A., . Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model. Commun. Nonlinear Sci. Numer. Simulat.. 2009;14:3309-3319.
[Google Scholar]
Mahmood A., Fetecau C., Khan N.A., Jamil M., . Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders. Acta Mech. Sin.. 2010;26:541-550.
[Google Scholar]
Mahmood, A., Khan, N.A., Parveen, S., in press. Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cyclinders. Acta Mech. Sin. doi:10.1007/s10409-011-0443-y.
Massoudi M., Phuoc T.X., . On the motion of a second grade fluid due to longitudinal and torsional oscillations of a cylinder: a numerical study. Appl. Math. Comput.. 2008;203:471-481.
[Google Scholar]
Podlubny I., . Fractional Differential Equations. San Diego: Academic Press; 1999.
Rajagopal K.R., . Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid. Acta Mech.. 1983;49:281-285.
[Google Scholar]
Rajagopal K.R., Bhatnagar R.K., . Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mech.. 1995;113:233-239.
[Google Scholar]
Sneddon I.N., . Functional analysis. In: Encyclopedia of Physics. Vol vol. II. Berlin: Springer; 1955.
[Google Scholar]
Stokes G.G., . On the Effect of the Rotation of Cylinders and Spheres about their Axis in Increasing the Logarithmic Decrement of the Arc of Vibration. Cambridge: Cambridge University Press; 1886.
Vieru D., Akhtar W., Fetecau C., Fetecau C., . Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains. Meccanica. 2007;42:573-583.
[Google Scholar]

Appendix A

Some results used in the text:

The finite Hankel transform of the function

(A1)

a (r) = \frac{{AR}_{1} (R_{2}^{2} - r^{2}) + {BR}_{2} (r^{2} - R_{1}^{2})}{(R^{2} - R_{1}^{2}) r}

satisfying a(R₁) = A and a(R₂) = B is

a_{n} = \int_{R_{1}}^{R_{2}} ra (r) B_{1} ({rr}_{n}) dr = \frac{2 B}{π r_{n}^{2}} - \frac{2 A}{π r_{n}^{2}} \frac{J_{1} (R_{2} r_{n})}{J_{1} (R_{1} r_{n})} .

(A2)

L^{- 1} \{\frac{q^{b}}{(q^{a} - d)^{c}}\} = G_{a, b, c} (d, t); Re (ac - b) > 0, Re (q) > 0, |\frac{d}{q^{a}}| < 1 .

(A3)

L^{- 1} \{\frac{e^{- dq} q^{b}}{q^{a} - c}\} = R_{a, b} (c, d, t) = \sum_{j = 0}^{\infty} \frac{c^{j} (t - d)^{(j + 1) a - b - 1}}{Γ [(j + 1) a - b]};

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

(A4)

R_{2, 1} (- a^{2}, 0, t) = \cos (at) .

(A5)

If u_{1} (t) = L^{- 1} {{\bar{u}}_{1} (q)} and u_{2} (t) = L^{- 1} {{\bar{u}}_{2} (q)} then

L^{- 1} {{\bar{u}}_{1} (q) {\bar{u}}_{2} (q)} = (u_{1} * u_{2}) (t) = \int_{0}^{t} u_{1} (t - s) u_{2} (s) ds = \int_{0}^{t} u_{1} (s) u_{2} (t - s) ds .

(A6)

\frac{1}{z + a} = \sum_{k = 1}^{\infty} (- 1)^{k} \frac{z^{k}}{a^{k + 1}} .

(A7)

\sum_{k = 0}^{\infty} {(- ν r_{n}^{2})}^{k} G_{0, - k - 1, k + 1} (- α r_{n}^{2}, t) = \sum_{k = 0}^{\infty} (- ν r_{n}^{2})^{k} \sum_{j = 0}^{\infty} \frac{(k + 1)_{j} (- α r_{n}^{2})^{j}}{j!} \frac{t^{k}}{k!} = \sum_{k = 0}^{\infty} (- ν r_{n}^{2})^{k} \frac{1}{(1 + α r_{n}^{2})^{k + 1}} \frac{t^{k}}{k!} = \frac{1}{1 + α r_{n}^{2}} \sum_{k = 0}^{\infty} \frac{1}{k!} {(- \frac{ν r_{n}^{2} t}{1 + α r_{n}^{2}})}^{k} = \frac{1}{1 + α r_{n}^{2}} \exp (- \frac{ν r_{n}^{2} t}{1 + α r_{n}^{2}}) .

(A8)

\frac{d}{dr} [B_{1} ({rr}_{n})] = r_{n} [J_{0} ({rr}_{n}) Y_{1} (R_{2} r_{n}) - J_{1} (R_{2} r_{n}) Y_{0} ({rr}_{n})] - \frac{1}{r} B_{1} ({rr}_{n}) .

Introduction

Torsional oscillations between two cylinders

Mathematical formulation of the problem and governing equation
Calculation of the velocity field
Calculation of the shear stress

Limiting cases

Solutions for ordinary second grade fluid (β → 1)
Solutions for Newtonian fluid

Concluding remarks

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[1] Casarella M.J., Laura P.A., . Drag on oscillating rod with longitudinal and torsional motion. J. Hydronautics. 1969;3:180-183.
[Google Scholar]

[2] Dunn J.E., Rajagopal K.R., . Fluids of differential type: critical review and thermodynamic analysis. Int. J. Eng. Sci.. 1995;33:689-729.
[Google Scholar]

[3] Fetecau C., Fetecau C., . Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder. Int. J. Eng. Sci.. 2006;44:788-796.
[Google Scholar]

[4] Fetecau C., Hayat T., Fetecau C., . Starting solutions for oscillating motions of Oldroyd-B fluids in cylindrical domains. J. Non-Newtonian Fluid Mech.. 2008;153:191-201.
[Google Scholar]

[5] Khan M., Ali S.H., Qi H., . Exact solutions of starting flows for a fractional Burgers'fluid between coaxial cylinders. Nonlinear Anal. Real World Appl.. 2009;10:1775-1783.
[Google Scholar]

[6] Khan, M., Asghar, S., Hayat, T., 2005. Oscillating flow of a Burgers’ fluid in a pipe. The Abdus Salam International Center for Theoretical Physics, IC/2005/071.

[7] Lorenzo, C.F., Hartley, T.T., 1999. Generalized functions for the fractional calculus. NASA/TP-1999-209424/Rev1.

[8] Mahmood A., Parveen S., Ara A., Khan N.A., . Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model. Commun. Nonlinear Sci. Numer. Simulat.. 2009;14:3309-3319.
[Google Scholar]

[9] Mahmood A., Fetecau C., Khan N.A., Jamil M., . Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders. Acta Mech. Sin.. 2010;26:541-550.
[Google Scholar]

[10] Mahmood, A., Khan, N.A., Parveen, S., in press. Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cyclinders. Acta Mech. Sin. doi:10.1007/s10409-011-0443-y.

[11] Massoudi M., Phuoc T.X., . On the motion of a second grade fluid due to longitudinal and torsional oscillations of a cylinder: a numerical study. Appl. Math. Comput.. 2008;203:471-481.
[Google Scholar]

[12] Podlubny I., . Fractional Differential Equations. San Diego: Academic Press; 1999.

[13] Rajagopal K.R., . Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid. Acta Mech.. 1983;49:281-285.
[Google Scholar]

[14] Rajagopal K.R., Bhatnagar R.K., . Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mech.. 1995;113:233-239.
[Google Scholar]

[15] Sneddon I.N., . Functional analysis. In: Encyclopedia of Physics. Vol vol. II. Berlin: Springer; 1955.
[Google Scholar]

[16] Stokes G.G., . On the Effect of the Rotation of Cylinders and Spheres about their Axis in Increasing the Logarithmic Decrement of the Arc of Vibration. Cambridge: Cambridge University Press; 1886.

[17] Vieru D., Akhtar W., Fetecau C., Fetecau C., . Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains. Meccanica. 2007;42:573-583.
[Google Scholar]