Travelling waves solution for MHD aligned flow of a second grade fluid with heat transfer: A symmetry independent approach

Najeeb Alam Khan; Asmat Ara; Muhammad Jamil; Ahmet Yildirim

doi:10.1016/j.jksus.2010.08.014

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ORIGINAL ARTICLE

24 (

); 63-67

doi:

10.1016/j.jksus.2010.08.014

Travelling waves solution for MHD aligned flow of a second grade fluid with heat transfer: A symmetry independent approach

Najeeb Alam Khan^a,*, Asmat Ara^a, Muhammad Jamil^b,c, Ahmet Yildirim^d

a Department of Mathematics, University of Karachi, Karachi 75270, Pakistan

b Abdul Salam School of Mathematical Science, GC University, Lahore, Pakistan

c Department of Mathematics, NED University of Engineering & Technology, Karachi 75270, Pakistan

d Department of Mathematics, Faculty of Science, Ege University, 35100 Bornova-Izmir, Turkey

*Corresponding author. Tel.: +92 2135040516 njbalam@yahoo.com (Najeeb Alam Khan),

Received: 2010-6-22, Accepted: 2010-8-26, Published: 2010-01-01

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

Available online 6 September 2010

Abstract

In this work, an approach is implemented for finding exact solutions of an incompressible MHD aligned flow with heat transfer in a second grade fluid. This approach based on travelling wave phenomenon. The partial differential equations (PDEs) are reduced to ordinary differential equations (ODEs) by using wave parameter. The methodology used in this work is independent of perturbation, symmetry consideration and other restrictive assumption. Comparison is made with the results obtained previously.

Keywords

Travelling waves

Heat transfer

Second grade fluid

Aligned flows

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1 Introduction

It is an established fact that the flow characteristics of non-Newtonian fluids are quite different when compared with the viscous fluids. Therefore, the well-known Navier–Stokes equations (Berker, 1963; Chandna et al., 1982; Chandna and Oku-Ukpong, 1994a; Hui, 1987; Naeem and Jamil, 2006 ) are inappropriate for the non-Newtonion fluids. There are numerous models of fluids are mainly classified in the literature. The flows of non-Newtonion fluids (Khan et al., 2009; Fetecau et al., 2008; Fetecau et al., 2006 ) are mainly classified into differential, integral and rate type fluids. Amongst these, the flows of differential type fluids have attracted the mathematician, computer programmers and numerical solvers. The constitutive equations of differential type fluids are very complex. Several authors in fluid dynamics are engaged with equations of motion of second grade fluid (Hayat et al., 2005; Siddiqui et al., 1985; Labropulu, 2003; Siddiqui, 1990; Rajagopal, 1980; Rajagopal and Gupta, 1984; Kaloni and Huschilt, 1984; Siddiqui and Kaloni, 1986; Mohyuddin et al., 2005; Ali and Hasan, 2007; Ali et al., 2007 ).

Ting (1963) has studied unsteady flows of a second grade fluid in a bounded region. He studied that the solution exists only if the coefficient of the higher order derivative in the governing equation positive. In 1994, Chandna and Oku-Ukpong (Chandna and Oku-Ukpong, 1994b) studied MHD aligned flow of a second grade fluid by assuming the prescribed form of vorticity. By taking the vorticity to be proportional to the stream function perturbed by a uniform stream, Lin and Tobak (Lin and Tobak, 1986), Benharbit and Siddiqui (Benharbit and Siddiqui, 1992), Labropulu (Labropulu, 2000), they investigated the exact solutions for second grade fluid. Under the slow motion assumption Dolapçi and Pakdemirli (Dolapçi and Pakdemirli, 2004) found the exact solution by Lie group method. Yürüsoy (Yürüsoy, 2004) found exact solutions of steady creeping flow of a second grade fluid with heat transfer via Lie group. Recently, for creeping flow Afify (Afify, 2009) studied the Lie group analysis approach and found the exact solutions for steady MHD aligned second grade fluid with heat transfer.

In this work, the MHD aligned unsteady flow of a second grade fluid with heat transfer is studied. The study of flow for an electrically conducting fluid has applications in many engineering problems such as MHD generators, plasma studied, geothermal energy extractions and electromagnetic propulsion. The electromagnetic propulsion system is closely associated with magneto chemistry, requires a complete understanding of the equation of state and transport properties such as diffusion, stress–shear rate relationship, thermal conductivity and radiation. The investigations of the travelling wave solution of nonlinear equations play an important role in the study of nonlinear physical phenomena. Travelling wave phenomenon, that appears in many areas such as physics (El-Wakil and Abdou, 2008; Yildirim and Gülkanat, 2010), mathematical biology (Mohyud-Din et al., 2010), chemical kinetics, fiber optics, fluid mechanics, etc.

To the best of our knowledge only few studies, which deal with the flows of MHD aligned second grade fluid available in the literature. For the present paper, the method adopted is as follows. In Section 2, we put forward the statement of the problem. A theoretical development of Section 3 is illustrated by solution of the equations of motion of MHD aligned second grade fluid with heat transfer in Section 4. Section 5 synthesis the concluding remarks.

2 Flow development

The equations of motion of an unsteady MHD aligned flow of a second grade fluid with heat transfer is governed by

(1)

ρ_{t} + div (ρ V) = 0 (Continuity)

(2)

ρ \frac{dV}{dt} = div τ + Λ (Curl H) \times H + ρ r_{1} (Linear momentum)

(3)

divH = 0 (Solenoidal)

(4)

H_{t} = Curl (V \times H) - \frac{1}{Λ δ} Curl (Curl H) (Diffusion)

(5)

ρ \frac{dE}{dt} = τ . L - div q + ρ r_{2} (Energy)

The constitutive equations for the stress of a second grade fluid given by Coleman and Noll (Coleman and Noll, 1960)

(6)

τ = - pI + μ L_{1} + α_{1} L_{2} + α_{2} L_{1}^{2}

(7)

L_{1} = (grad V) + (grad V)^{*}

(8)

L_{2} = \frac{{dL}_{1}}{dt} + L_{1} (grad V) + (grad V)^{*} L_{1}

Here V is the velocity, p is the fluid pressure function,

ρ

the density,

μ

the constant viscosity, grad the gradient operator, ‘∗’ the transpose,

\frac{d}{dt}

the material time derivative,

α_{1}, α_{2}

are the constant normal stress moduli,

L_{1}

and

L_{2}

are the first two Rivlin–Ericksen tensors,

r_{1}

the body force per unit mass,

r_{2}

is the radiant heating (assumed to be zero),

E = C_{p} T

is the specific internal energy,

C_{p}

is the specific heat and T is the temperature,

q = - k \nabla T

is the heat flux vector, k is the constant thermal conductivity.

If an incompressible fluid of second grade is to have motions that are compatible with thermodynamics in the sense of the Clausius–Duhem inequality and the condition that the Helmholtz free energy be a minimum when the fluid is at rest, then the following must be satisfied (Dunn and Fosdick, 1974)

(9)

μ ⩾ 0, α_{1} ⩾ 0, α_{1} + α_{2} = 0

In this paper, we shall consider a second grade fluid which under goes isochoric motion in a plane and body force is negligible, we take

V = (u (x, y, t), v (x, y, t), 0)

p = p (x, y, t) H = (H_{1} (x, y, t), H_{2} (x, y, t), 0)

and

T = T (x, y, t)

so that our flow Eqs. (1)–(5) takes the form

(10)

\frac{\partial u}{\partial x} = - \frac{\partial v}{\partial y}

(11)

\begin{matrix} \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \\ = - \frac{1}{ρ} \frac{\partial p}{\partial x} + ν \nabla^{2} u + β [\frac{\partial}{\partial t} (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}) \\ + 5 \frac{\partial u}{\partial x} \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial u}{\partial x} \frac{\partial^{2} u}{\partial y^{2}} + u \frac{\partial^{3} u}{\partial x^{3}} + v \frac{\partial^{3} u}{\partial y^{3}} + u \frac{\partial^{3} u}{\partial x \partial y^{2}} \\ + 2 \frac{\partial v}{\partial x} \frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial u}{\partial y} \frac{\partial^{2} u}{\partial x \partial y} + \frac{\partial u}{\partial y} \frac{\partial^{2} v}{\partial x^{2}} + v \frac{\partial^{3} u}{\partial y \partial x^{2}}] \end{matrix}

(12)

\begin{matrix} \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \\ = - \frac{1}{ρ} \frac{\partial p}{\partial y} + ν \nabla^{2} v + β [\frac{\partial}{\partial t} (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}) \\ + 5 \frac{\partial u}{\partial x} \frac{\partial^{2} u}{\partial x \partial y} - \frac{\partial u}{\partial x} \frac{\partial^{2} v}{\partial x^{2}} - v \frac{\partial^{3} u}{\partial x \partial y^{2}} + u \frac{\partial^{3} v}{\partial x^{3}} - v \frac{\partial^{3} u}{\partial x^{3}} \\ + 2 \frac{\partial u}{\partial y} \frac{\partial^{2} u}{\partial y^{2}} - \frac{\partial v}{\partial x} \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial v}{\partial x} \frac{\partial^{2} u}{\partial y^{2}} - u \frac{\partial^{3} u}{\partial x^{2} \partial y}] \end{matrix}

(13)

\frac{\partial H_{1}}{\partial x} + \frac{\partial H_{2}}{\partial y} = 0

(14)

\frac{\partial}{\partial t} (\frac{\partial H_{2}}{\partial x} - \frac{\partial H_{1}}{\partial y}) = \nabla^{2} [\frac{1}{Λ δ} (\frac{\partial H_{2}}{\partial x} - \frac{\partial H_{1}}{\partial y}) + {vH}_{1} - {uH}_{2}]

(15)

\begin{matrix} ρ C_{p} (\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y}) \\ = k \nabla^{2} T + μ [{(\frac{\partial u}{\partial y})}^{2} + 2 {(\frac{\partial v}{\partial y})}^{2} + {(\frac{\partial v}{\partial x})}^{2} + 2 {(\frac{\partial u}{\partial x})}^{2} + 2 \frac{\partial u}{\partial y} \frac{\partial v}{\partial x}] \\ + α_{1} [\frac{\partial u}{\partial y} \frac{\partial^{2} u}{\partial t \partial y} + 2 \frac{\partial v}{\partial y} \frac{\partial^{2} v}{\partial t \partial y} + 2 v \frac{\partial v}{\partial y} \frac{\partial^{2} v}{\partial y^{2}} + \frac{\partial v}{\partial x} \frac{\partial^{2} v}{\partial t \partial y} + v \frac{\partial v}{\partial x} \frac{\partial^{2} v}{\partial y^{2}} \\ + 2 \frac{\partial u}{\partial x} \frac{\partial^{2} u}{\partial t \partial x} + \frac{\partial u}{\partial y} \frac{\partial^{2} v}{\partial t \partial x} + \frac{\partial v}{\partial x} \frac{\partial^{2} v}{\partial t \partial x} + u \frac{\partial u}{\partial y} \frac{\partial^{2} u}{\partial x \partial y} + v \frac{\partial u}{\partial x} \frac{\partial^{2} u}{\partial y^{2}}] \end{matrix}

3 Travelling waves

Let Lx be a linear form of the independent variables.

(16)

Lx = L_{\bar{α}} x_{\bar{α}}

The representation of a solution is assumed in the form

(17)

ϑ (x) = υ (Lx)

where the function

υ (ζ)

depends on one independent variable

ζ

. The value of

ζ

is called a phase of the wave. Fixing the phase

ζ = Lx

, one obtains the front of the wave, where the values of the dependent variables are constant. Hence, the front of the wave is a plane propagating in the space

R^{n}

A solution $u (x_{1}, x_{2}, \dots, x_{n})$ is called an r-multiple travelling wave, if it has the representation

(18)

ϑ (x) = υ (Lx)

where Lx be a vector, which coordinates are linear forms of independent variables

(19)

(Lx)_{i} = L_{i \bar{α}} x_{\bar{α}}, i = 1, 2, \dots r, r < n

The variables

\bar{ζ} = Lx \in R^{r}

are called parameters of the wave. It is note that for an r-multiple travelling wave the rank of the Jacobi matrix of the dependent variables with respect to the independent variables is less or equal then r.

4 Solution

Without loss of generality one can assume that the travelling wave type solution has the representation

(20)

u = u (ξ), v = v (ξ), p = p (ξ), T = T (ξ), H_{1} = H_{1} (ξ), H_{2} = H_{2} (ξ)

where

ξ = bx + ay + Ct

is the wave parameter

b, a

are constants and C is the constant phase velocity.

On substituting the representation of the solution (20) into (10)–(14), we have

(21)

{bu}^{'} + {av}^{'} = 0

(22)

\begin{matrix} (C + bu + av) u^{'} \\ = - \frac{b}{ρ} p^{'} + ν (a^{2} + b^{2}) u^{″} \\ + β [(C + bu + av) (a^{2} + b^{2}) u^{‴} + \frac{2 b}{a^{2}} (a^{2} + b^{2})^{2} u^{'} u^{″}] \\ - η H_{2} ({bH}_{2}^{'} - {aH}_{1}^{'}) \end{matrix}

(23)

\begin{matrix} (C + bu + av) v^{'} \\ = - \frac{a}{ρ} p^{'} + ν (a^{2} + b^{2}) v^{″} \\ + β [- (C + bu + av) (\frac{a^{2} + b^{2}}{a}) {bu}^{‴} + \frac{2}{a} (a^{2} + b^{2})^{2} u^{'} u^{″}] \\ + η H_{2} ({bH}_{2}^{'} - {aH}_{1}^{'}) \end{matrix}

(24)

{bH}_{1}^{'} + {aH}_{2}^{'} = 0

(25)

\begin{matrix} C ({bH}_{2}^{″} - {aH}_{1}^{″}) \\ = (a^{2} + b^{2}) [\frac{1}{Λ δ} ({bH}_{2}^{‴} - {aH}_{1}^{‴}) + {vH}_{1}^{″} - {uH}_{2}^{″} + H_{1} v^{″} - H_{2} u^{″}] \end{matrix}

(26)

\begin{matrix} (C + bu + av) T^{'} = \frac{k}{ρ C_{p}} (a^{2} + b^{2}) T^{″} + \frac{μ}{ρ C_{p}} (a^{2} + b^{2}) u^{' 2} \\ + \frac{α_{1}}{ρ C_{p}} [a^{2} (bu + av) + \frac{C}{a^{2}} (a^{2} + b^{2})^{2}] u^{'} u^{″} \end{matrix}

where

u = \frac{μ}{ρ}

and

β = \frac{α_{1}}{ρ}

are kinematical viscosity and non-Newtonian parameter respectively and

η = \frac{Λ}{ρ}

On integrating the Eqs. (21) and (24), we have

(27)

bu + av = m_{1}

(28)

{bH}_{1} + {aH}_{2} = m_{2}

where

m_{1}, m_{2}

are arbitrary constants. Here we assumed

m_{2} = 0

On utilizing Eqs. (27) and (28) in Eqs. (21), (22), (25) and (26), we have

(29)

\begin{matrix} (C + m_{1}) u^{'} = - \frac{b}{ρ} p^{'} + ν (a^{2} + b^{2}) u^{″} \\ + β [(C + m_{1}) (a^{2} + b^{2}) u^{‴} + \frac{2 b}{a^{2}} (a^{2} + b^{2})^{2} u^{'} u^{″}] \\ + η \frac{(a^{2} + b^{2})}{b} H_{1} H_{1}^{'} \end{matrix}

(30)

\begin{matrix} (C + m_{1}) v^{'} = - \frac{a}{ρ} p^{'} + ν (a^{2} + b^{2}) v^{″} \\ + β [- (C + m_{1}) (\frac{a^{2} + b^{2}}{a}) {bu}^{‴} + \frac{2}{a} (a^{2} + b^{2})^{2} u^{'} u^{″}] \\ - η \frac{(a^{2} + b^{2})}{b} H_{2} H_{2}^{'} \end{matrix}

(31)

(a^{2} + b^{2}) H_{1}^{‴} - {CH}_{1}^{″} = 0

(32)

\begin{matrix} (C + m_{1}) T^{'} = \frac{k}{ρ C_{p}} (a^{2} + b^{2}) T^{″} + \frac{μ}{ρ C_{p}} (a^{2} + b^{2})^{2} u^{' 2} \\ + \frac{α_{1}}{ρ C_{p}} [\frac{U (a^{2} + b^{2}) + a^{4} m_{1}}{a^{2}}] u^{'} u^{″} \end{matrix}

On eliminating pressure p from Eqs. (29) and (30), we get

(33)

u^{‴} - {Mu}^{″} - {Nu}^{'} = 0

where

M = \frac{μ}{α_{1} (C + m_{1})}, N = \frac{ρ}{α_{1} (a^{2} + b^{2})}

The components of velocity are

(34)

u (ξ) = m_{3} e^{n_{1} ξ} + m_{4} e^{n_{2} ξ} - \frac{m_{5}}{N}

(35)

v (ξ) = \frac{m_{1}}{a} - \frac{b}{a} (m_{3} e^{n_{1} ξ} + m_{4} e^{n_{2} ξ} - \frac{m_{5}}{N})

where

n_{1}, n_{2}

are the roots of the equation

n^{2} - Mn - N = 0

From Eq. (31), the components of magnetic field are

(36)

H_{1} (ξ) = m_{6} e^{\frac{C}{(a^{2} + b^{2})} ξ} + m_{7} ξ + m_{8}

(37)

H_{2} (ξ) = - \frac{b}{a} (m_{6} e^{\frac{C}{(a^{2} + b^{2})} ξ} + m_{7} ξ + m_{8})

Substituting Eqs. (34) and (36) into Eq. (29), we obtain

(38)

p (ξ) = M_{1} e^{n_{1} ξ} + M_{2} e^{n_{2} ξ} + M_{3} e^{2 n_{1} ξ} + M_{4} e^{2 n_{2} ξ} + M_{5} e^{(n_{1} + n_{2}) ξ} + M_{6} {(m_{6} e^{\frac{C}{(a^{2} + b^{2})} ξ} + m_{7} ξ + m_{8})}^{2} + M_{7}

Where

n_{1}, n_{2}

are the roots of the equation

\begin{matrix} M_{1} = \frac{m_{3} [- ρ (c + m_{1}) + (a^{2} + b^{2}) \{μ n_{1} + α_{1} m_{4} n_{1}^{2} (C + m_{1})\}]}{b} \\ M_{2} = \frac{m_{4} [- ρ (c + m_{1}) + (a^{2} + b^{2}) \{μ n_{2} + α_{1} m_{4} n_{2}^{2} (C + m_{1})\}]}{b} \\ M_{3} = \frac{α_{1} n_{1}^{2} m_{3}^{2} {(a^{2} + b^{2})}^{2}}{a^{4}} \\ M_{4} = \frac{α_{1} n_{2}^{2} m_{4}^{2} {(a^{2} + b^{2})}^{2}}{a^{4}} \\ M_{5} = \frac{2 α_{1} m_{3} m_{4} n_{1} n_{2} (a^{2} + b^{2})}{a^{4}} \\ M_{6} = \frac{Λ (a^{2} + b^{2})}{2 b^{2}} \\ M_{7} = \frac{{bNm}_{9} + ρ m_{5} (C + m_{1})}{bN} \end{matrix}

Utilizing Eqs. (27) and (34) in Eq. (26) and integrating, we obtain

(39)

T (ξ) = M_{9} e^{2 n_{1} ξ} + M_{10} e^{2 n_{2} ξ} + M_{11} e^{(n_{1} + n_{2}) ξ} + m_{10} e^{M_{8} ξ} - \frac{m_{9}}{M_{8}}

\begin{matrix} M_{8} = \frac{ρ C_{p} (C + m_{1})}{k (a^{2} + b^{2})} \\ M_{9} = \frac{- n_{1} λ_{2}^{2} (a^{2} + b^{2})}{k (2 n_{1} - A_{7})} [μ (a^{2} + b^{2}) + \frac{n_{1} α_{1} {U (a^{2} + b^{2}) + a^{4} λ_{1}}}{2 (a^{2} + b^{2})}] \\ M_{10} = \frac{- n_{2} λ_{3}^{2} (a^{2} + b^{2})}{k (2 n_{2} - A_{7})} [μ (a^{2} + b^{2}) + \frac{n_{2} α_{1} {U (a^{2} + b^{2}) + a^{4} λ_{1}}}{2 (a^{2} + b^{2})}] \\ M_{11} = \frac{- 2 λ_{2} λ_{3} n_{1} n_{2} (a^{2} + b^{2})}{k (n_{1} + n_{2} - A_{7})} [\frac{μ (a^{2} + b^{2})}{(n_{1} + n_{2})} + \frac{α_{1} {U (a^{2} + b^{2}) + a^{4} λ_{1}}}{2 (a^{2} + b^{2})}] \end{matrix}

The velocity components, pressure, temperature and magnetic field components in the original variables are

(40)

u (x, y, t) = m_{3} e^{n_{1} (bx + ay + Ct)} + m_{4} e^{n_{2} (bx + ay + Ct)} - \frac{m_{5}}{N}

(41)

v (x, y, t) = \frac{m_{1}}{a} - \frac{b}{a} (m_{3} e^{n_{1} (bx + ay + Ct)} + m_{4} e^{n_{2} (bx + ay + Ct)} - \frac{m_{5}}{N})

(42)

\begin{matrix} p (x, y, t) = M_{1} e^{n_{1} (bx + ay + Ct)} + M_{2} e^{n_{2} (bx + ay + Ct)} + M_{3} e^{2 n_{1} (bx + ay + Ct)} \\ + M_{4} e^{2 n_{2} (bx + ay + Ct)} + M_{5} e^{(n_{1} + n_{2}) (bx + ay + Ct)} \\ + M_{6} {(m_{6} e^{\frac{C}{(a^{2} + b^{2})} (bx + ay + Ct)} + m_{7} (bx + ay + Ct) + m_{8})}^{2} + M_{7} \end{matrix}

(43)

\begin{matrix} T (x, y, t) = A_{8} e^{2 n_{1} (bx + ay + Ct)} + A_{9} e^{2 n_{2} (bx + ay + Ct)} + A_{9} e^{(n_{1} + n_{2}) (bx + ay + Ct)} \\ + (λ_{6} (bx + ay + Ct) + λ_{7}) e^{2 A_{7} (bx + ay + Ct)} \end{matrix}

(44)

H_{1} (x, y, t) = m_{6} e^{\frac{C}{(a^{2} + b^{2})} (bx + ay + Ct)} + m_{4} (bx + ay + Ct) + m_{8}

(45)

H_{2} (x, y, t) = - \frac{b}{a} (m_{6} e^{\frac{C}{(a^{2} + b^{2})} (bx + ay + Ct)} + m_{4} (bx + ay + Ct) + m_{8})

5 Concluding remarks

The goal to obtain exact solution of an incompressible MHD aligned flow with heat transfer of a second grade fluid. The methodology (Meleshko, 2004) used in this work is easy for linearizing the flow equations by considering travelling wave phenomenon. The method was used in a direct way without transformation, symmetry consideration and restrictive assumptions. It is observed that exponential type solution have been obtained. Finally, compared the velocity and magnetic field in our work by Ali and Mahmood (Ali and Mahmood, 2005) the results are found to be excellent. This guarantees the correctness of the mathematical calculation.

Acknowledgement

The author M. Jamil highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan; Department of Mathematics, NED University of Engineering & Technology, Karachi-75270, Pakistan and also Higher Education Commission of Pakistan for supporting and facilitating this research work.

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Introduction

Flow development

Travelling waves

Solution

Concluding remarks

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[Google Scholar]

[28] Rajagopal K.R., Gupta A.S., . On a class of exact solutions to the equations of motion of a second grade fluid. International Journal of Engineering Sciences. 1984;19:373-381.
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[29] Siddiqui A.M., . Some more inverse solutions of a non-Newtonian fluid. Mechanics Research Communication. 1990;17:157-163.
[Google Scholar]

[30] Siddiqui A.M., Kaloni P.N., . Certain-inverse solutions of a non-Newtonian fluid. International Journal of Nonlinear Mechanics. 1986;21:459-473.
[Google Scholar]

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[Google Scholar]

[32] Ting T.W., . Certain nonsteady flows of second-order fluids. Archive of Rational Mechanics Analysis. 1963;14(1963):1-26.
[Google Scholar]

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[34] Yürüsoy M., . Similarity solutions for creeping flow and heat transfer of a second grade fluids. International Journal of Nonlinear Mechanics. 2004;39:665-672.
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