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Numerical Solution for Variable Accelerated Flow Subject to Slip Effect

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1. Introduction

Considerable interest has been shown in the literature to investigate the steady/unsteady flows caused by eccentric rotations of disk and viscous/rheological fluid at infinity incorporating various parameters of physical interest. Erdogan [1] analyzed the transient flow resulting from eccentric rotations of disc and fluid at infinity. Some MHD boundary layer flows are mentioned in references [2] - [7] that helped to improve our understanding of astrophysical, geophysical and engineering problems. Hayat et al. [8] extended the work of Ref. [1] for porous disk and applied magnetic field. The transient flow caused by eccentric rotations of a porous disk executing oscillations and a fluid at infinity was examined by Erdogan [9] . Non-Newtonian fluid flows [10] - [16] are also very important in real world applications. Siddiqui et al. [17] and Hayat et al. [18] generalized the flow analysis of ref. [8] and [9] for non-Newtonian second and third grade MHD fluids respectively. Asghar et al. [19] [20] discussed the flow generated by eccentric rotations of constantly accelerated disk and non-Newtonian fluid at infinity using no slip boundary condition. The constant acceleration corresponds to the linear velocity; however the nonlinear velocity; that is, variable acceleration will be both interesting and realistic.

Although no slip boundary condition is well established in fluid mechanics; there are situations where this condition is not adequate in that the flow may exhibits slip effect [21] [22] [23] [24] [25] . It has been experimentally verified that the slip effects become important in nanochannels and micro channels (see [26] ). Concerning non-Newtonian fluids, no slip condition is not adequate while considering polymer melts. What we have chosen here is to study the unsteady flow generated by two independent mechanisms of non-coaxial rotation and the non-linear velocity (variable acceleration) of the disc with the slip condition. Crank-Nicolson scheme is used for the numerical results of the nonlinear governing differential equation and non-linear boundary conditions. The influence of parameters of importance is analyzed and interpreted physically putting emphasis on acceleration parameter and slip parameter.

2. Problem Statement

Consider the unsteady flow of an incompressible third grade fluid filling the semi-infinite space z > 0 above a disk at z = 0. The disk is taken to be porous and non-conducting. Both cases of suction and injection/blowing are to be analyzed. The flow is generated by eccentric rotations of variable accelerated disk and third grade fluid at infinity. The disk and the fluid at infinity initially rotate about z'-axis with angular velocity Ω (see Figure 1). At t = 0, the disk suddenly starts rotating about z-axis with same angular velocity Ω and moving linearly with variable acceleration along x-axis. However, the fluid at t = 0 continues to rotate about the z'-axis. The axes of rotation of the disk and fluid at infinity are taken in the plane x = 0. The distance between the axes of rotations is denoted by l. The fluid is electrically conducting and a constant magnetic field B_{0} is applied in the transverse direction. We consider a slip between the velocity of the fluid at the surface of the disk and the velocity of the disk. The relative velocity between the fluid and the wall is assumed to be proportional to the shear rate at the wall.

The Cauchy stress tensor T (Fosdick and Rajagopal [27] ) for a thermodynamically compatible third grade fluid is given by

$T=-pI+\mu {A}_{1}+{\alpha}_{1}{A}_{2}+{\alpha}_{2}{A}_{1}^{2}+{\beta}_{3}\left(tr{A}_{1}^{2}\right){A}_{1}$ (1)

in which p is the hydrostatic pressure, µ is the dynamic viscosity of the fluid. The equations governing the present flow are

$\nabla \cdot V=0,$ (2)

Figure 1. Geometry of problem.

$\rho \frac{DV}{Dt}=\nabla \cdot T-\sigma {B}_{0}^{2}V,$ (3)

where ρ is the density and σ is the electrical conductivity of the fluid.

The velocity field $V=\langle u,v,w\rangle $ for our problem is given by

$u=-\Omega y+f\left(z,t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}v=\Omega x+g\left(z,t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}w=-{W}_{0}$ (4)

satisfying the equation of continuity (2) identically. Here W_{0} > 0 denotes the suction and W_{0} < 0 the injection velocity.

The initial and boundary conditions of the problem are given by

$u-{\lambda}_{1}{\tau}_{xz}=-\Omega y+{c}_{0}{t}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}v-{\lambda}_{1}{\tau}_{yz}=\Omega x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{at}\text{\hspace{0.17em}}z=0;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}t>0,$ (5)

$u=-\Omega \left(y-l\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}v=\Omega x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{as}\text{\hspace{0.17em}}z\to \infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{forall}\text{\hspace{0.17em}}t,$ (6)

$u=-\Omega \left(y-l\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}v=\Omega x,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}z>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{andat}\text{\hspace{0.17em}}t=0,$ (7)

where λ_{1} is a slip parameter, c_{0} is a constant having dimension L/T^{3} and the shear stresses are

$\begin{array}{c}{\tau}_{xz}=\mu \frac{\partial u}{\partial z}+{\alpha}_{1}\left[\frac{{\partial}^{2}u}{\partial t\partial z}+w\frac{{\partial}^{2}u}{\partial {z}^{2}}+\frac{\partial v}{\partial z}\frac{\partial u}{\partial y}+2\frac{\partial v}{\partial x}\frac{\partial v}{\partial z}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{\beta}_{3}\left[{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^{2}\frac{\partial u}{\partial z}+{\left(\frac{\partial u}{\partial z}\right)}^{3}+\frac{\partial u}{\partial z}{\left(\frac{\partial v}{\partial z}\right)}^{2}\right]\end{array}$ (8)

$\begin{array}{c}{\tau}_{yz}=\mu \frac{\partial v}{\partial z}+{\alpha}_{1}\left[\frac{{\partial}^{2}v}{\partial t\partial z}+w\frac{{\partial}^{2}v}{\partial {z}^{2}}+2\frac{\partial u}{\partial y}\frac{\partial u}{\partial z}+\frac{\partial u}{\partial z}\frac{\partial v}{\partial x}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{\beta}_{3}\left[{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^{2}\frac{\partial v}{\partial z}+{\left(\frac{\partial v}{\partial z}\right)}^{3}+\frac{\partial v}{\partial z}{\left(\frac{\partial u}{\partial z}\right)}^{2}\right]\end{array}$ (9)

Substituting (4) into (3), gives

$\begin{array}{l}\frac{1}{\rho}\frac{\partial P}{\partial x}={\Omega}^{2}x+\Omega g+{W}_{0}\frac{\partial f}{\partial z}-\frac{\partial f}{\partial t}+\nu \frac{{\partial}^{2}f}{\partial {z}^{2}}-\frac{\sigma}{\rho}{B}_{0}^{2}\left[f\left(z,t\right)-\Omega y\right]\\ +\frac{{\alpha}_{1}}{\rho}\left[\frac{{\partial}^{3}f}{\partial t\partial {z}^{2}}-{W}_{0}\frac{{\partial}^{3}f}{\partial {z}^{3}}+\Omega \frac{{\partial}^{2}g}{\partial {z}^{2}}\right]+\frac{2{\beta}_{3}}{\rho}\frac{\partial}{\partial z}\left[\frac{\partial f}{\partial z}\left\{{\left(\frac{\partial f}{\partial z}\right)}^{2}+{\left(\frac{\partial g}{\partial z}\right)}^{2}\right\}\right],\end{array}$ (10)

$\begin{array}{l}\frac{1}{\rho}\frac{\partial P}{\partial y}={\Omega}^{2}y-\Omega f+{W}_{0}\frac{\partial g}{\partial z}-\frac{\partial g}{\partial t}+\nu \frac{{\partial}^{2}g}{\partial {z}^{2}}-\frac{\sigma}{\rho}{B}_{0}^{2}\left[g\left(z,t\right)+\Omega x\right]\\ +\frac{{\alpha}_{1}}{\rho}\left[\frac{{\partial}^{3}g}{\partial t\partial {z}^{2}}-{W}_{0}\frac{{\partial}^{3}g}{\partial {z}^{3}}-\Omega \frac{{\partial}^{2}f}{\partial {z}^{2}}\right]+\frac{2{\beta}_{3}}{\rho}\frac{\partial}{\partial z}\left[\frac{\partial g}{\partial z}\left\{{\left(\frac{\partial f}{\partial z}\right)}^{2}+{\left(\frac{\partial g}{\partial z}\right)}^{2}\right\}\right],\end{array}$ (11)

$\frac{1}{\rho}\frac{\partial P}{\partial z}=\frac{\sigma}{\rho}{B}_{0}^{2}{W}_{0},$ (12)

where the modified pressure P is expressed as

$P=p-\left(2{\alpha}_{1}+{\alpha}_{2}\right)\left\{{\left(\frac{\partial f}{\partial z}\right)}^{2}+{\left(\frac{\partial g}{\partial z}\right)}^{2}\right\}.$ (13)

Equation (12) shows that P is not a function of z. We can eliminate P from Equations (10)-(11) by differentiating and then integrating with respect to z, and combining the resulting equations to obtain the following equation

$\begin{array}{l}\frac{{\alpha}_{1}}{\rho}\frac{{\partial}^{3}{F}^{*}}{\partial t\partial {z}^{2}}-\frac{{\alpha}_{1}{W}_{0}}{\rho}\frac{{\partial}^{3}{F}^{*}}{\partial {z}^{3}}+\left(\nu -i\frac{{\alpha}_{1}\Omega}{\rho}\right)\frac{{\partial}^{2}{F}^{*}}{\partial {z}^{2}}+{W}_{0}\frac{\partial {F}^{*}}{\partial z}\\ -\frac{\partial {F}^{*}}{\partial t}-\Omega \left(i+\frac{\sigma {B}_{0}^{2}}{\rho \Omega}\right){F}^{*}+2{\beta}_{3}\frac{\partial}{\partial z}\left[{\left(\frac{\partial {F}^{*}}{\partial z}\right)}^{2}\frac{\partial \stackrel{\xaf}{{F}^{*}}}{\partial z}\right]=0,\end{array}$ (14)

$\begin{array}{c}{F}^{*}\left(0,t\right)=\Omega {c}_{0}{t}^{2}-\Omega l+{\lambda}_{1}\left[\frac{\partial {F}^{*}}{\partial z}+\frac{{\alpha}_{1}}{\rho \nu}\left(\frac{{\partial}^{2}{F}^{*}}{\partial t\partial z}-\frac{{W}_{0}}{\Omega}\frac{{\partial}^{2}{F}^{*}}{\partial {z}^{2}}-i\frac{\partial {F}^{*}}{\partial z}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2{\beta}_{3}}{\rho \sqrt{\Omega \nu}}{\left(\frac{\partial {F}^{*}}{\partial z}\right)}^{2}\left(\frac{\partial \stackrel{\xaf}{{F}^{*}}}{\partial z}\right),\end{array}$ (15)

${F}^{*}\left(\infty ,t\right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{F}^{*}\left(z,0\right)=0,$ (16)

where

${F}^{*}\left(z,t\right)=f\left(z,t\right)+ig\left(z,t\right)-\Omega l.$ (17)

Equation (15) together with the Conditions (16) and (17), in dimensionless variables, can be expressed as

$\begin{array}{l}\alpha \frac{{\partial}^{3}F}{\partial \tau \partial {\eta}^{2}}-\alpha S\frac{{\partial}^{3}F}{\partial {\eta}^{3}}+\left(1-i\alpha \right)\frac{{\partial}^{2}F}{\partial {\eta}^{2}}+2S\frac{\partial F}{\partial \eta}-2\frac{\partial F}{\partial \tau}\\ -\text{\hspace{0.17em}}2\left(i+N\right)F+\beta \frac{\partial}{\partial \eta}\left[{\left(\frac{\partial F}{\partial \eta}\right)}^{2}\frac{\partial \stackrel{\xaf}{F}}{\partial \eta}\right]=0,\end{array}$ (18)

$F\left(0,\tau \right)=c{\tau}^{2}-1+\lambda \left[\frac{\partial F}{\partial \eta}+\alpha \left(\frac{{\partial}^{2}F}{\partial \tau \partial \eta}-S\frac{{\partial}^{2}F}{\partial {\eta}^{2}}-i\frac{\partial F}{\partial \eta}\right)\right]+{\beta}_{3}{\left(\frac{\partial F}{\partial \eta}\right)}^{2}\left(\frac{\partial \stackrel{\xaf}{F}}{\partial \eta}\right),$ (19)

$F\left(\infty ,\tau \right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}F\left(\eta ,0\right)=0,$ (20)

where

$\begin{array}{l}F=\frac{{F}^{\ast}}{\Omega l},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}S=\frac{{w}_{0}}{\sqrt{2\nu \Omega}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\eta =\sqrt{\frac{\Omega}{2\nu}}z,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\tau =\Omega t,\\ N=\frac{\sigma {B}_{0}^{2}}{\rho \Omega},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =\frac{{\alpha}_{1}\Omega}{\rho \nu},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta =\frac{{\Omega}^{3}{l}^{2}{\beta}_{3}}{\rho {\nu}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}c=\frac{{c}_{0}}{{\Omega}^{2}l},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\lambda ={\lambda}_{1}\sqrt{\frac{\Omega}{\nu}}.\end{array}$ (21)

3. Numerical Solution

We obtain a numerical solution of the problem given by the governing Equation (18) together with Conditions (19)-(21). The discretized form of Equation (18) is presented using a finite difference scheme in ref. [19] . However, the boundary Condition (19) is discretized as

${F}_{0,j}={r}_{1}{F}_{1,j}+{r}_{2}{F}_{2,j}+{r}_{3}\left[{F}_{0,j-1}-{F}_{1,\text{\hspace{0.17em}}j-1}\right]+{r}_{4}\left[B{T}_{j}+c{j}^{2}{k}^{2}-1\right]$ (22)

where

$\begin{array}{l}{r}_{0}={h}^{2}k+\lambda \left(1-i\alpha \right)hk+\alpha \lambda h+\alpha S\lambda k,\\ {r}_{1}=\left[\lambda \left(1-i\alpha \right)hk+\alpha \lambda h+2\alpha S\lambda k\right]/{r}_{0},\\ {r}_{2}=-\left[\alpha S\lambda k\right]/{r}_{0},\\ {r}_{3}=\left[\alpha \lambda h\right]/{r}_{0},\\ {r}_{4}=\left[{h}^{2}k\right]/{r}_{0},\\ B{T}_{j}=\frac{\beta \lambda}{{h}^{3}}\left[{\left({F}_{1,j}-{F}_{0,j}\right)}^{2}\left({\stackrel{\xaf}{F}}_{1,j}-{\stackrel{\xaf}{F}}_{0,j}\right)\right].\end{array}$ (23)

To evaluate ${F}_{0,j+1}$ , we firstly take $B{T}_{j+1}=B{T}_{j}$ in the system of algebraic equations and the solution of the system is sought which results in known values of ${F}_{i,j+1};i=1,2,3,\cdots ,M-1$ . Secondly, we update ${F}_{0,j+1}$ by using iterative method as follows:

$\begin{array}{c}{F}_{0,j+1}^{k+1}={r}_{1}{F}_{1,j+1}+{r}_{2}{F}_{2,j+1}+{r}_{3}\left[{F}_{0,j}^{k}-{F}_{1,j}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{r}_{4}\frac{\beta \lambda}{{h}^{3}}\left[{\left({F}_{1,j+1}-{F}_{0,j+1}^{k}\right)}^{2}\left({\stackrel{\xaf}{F}}_{1,j+1}-{\stackrel{\xaf}{F}}_{0,j+1}^{k}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{r}_{4}\left[c{\left(j+1\right)}^{2}{k}^{2}-1\right],\end{array}$ (24)

where

${F}_{0,j+1}^{0}={F}_{0,j+1}$ (25)

and this iterative procedure is continued until ${F}_{0,j+1}^{k+1}\approx {F}_{0,j+1}^{k}$ . Furthermore, ${F}_{0,0}$ is evaluated by letting ${F}_{0,-1}={F}_{1,-1}=0$ and by using iterative method as described above with ${F}_{0,0}^{0}=0$ as initial guess.

For $i=\text{1}$ , $i=2$ , $3\le i\le M-3$ , $i=M-\text{2}$ and $i=M-\text{1}$ , we respectively have

${{C}^{\prime}}_{1}{F}_{1,j+1}+{{D}^{\prime}}_{1}{F}_{2,j+1}+{{E}^{\prime}}_{1}{F}_{3,j+1}={{G}^{\prime}}_{1},$ (26)

${{B}^{\prime}}_{2}{F}_{1,j+1}+{{C}^{\prime}}_{2}{F}_{2,j+1}+{D}_{2}{F}_{3,j+1}+{E}_{2}{F}_{4,j+1}={{G}^{\prime}}_{2},$ (27)

${A}_{i}{F}_{i-2,j+1}+{B}_{i}{F}_{i-1,j+1}+{C}_{i}{F}_{i,j+1}+{D}_{i}{F}_{i+1,j+1}+{E}_{i}{F}_{i+2,j+1}={G}_{i}$ (28)

${A}_{M-2}{F}_{M-4,j+1}+{B}_{M-2}{F}_{M-3,j+1}+{C}_{M-2}{F}_{M-2,j+1}+{D}_{M-2}{F}_{M-1,j+1}={{G}^{\prime}}_{M-2},$ (29)

${A}_{M-1}{F}_{M-3,j+1}+{B}_{M-1}{F}_{M-2,j+1}+{C}_{M-1}{F}_{M-1,j+1}={{G}^{\prime}}_{M-1},$ (30)

where ${A}_{i},{B}_{i},{C}_{i},{D}_{i},{E}_{i},{G}_{i},{{C}^{\prime}}_{1},{{D}^{\prime}}_{1},{{E}^{\prime}}_{1},{{B}^{\prime}}_{2},{{C}^{\prime}}_{2},{{G}^{\prime}}_{M-2}$ and ${{G}^{\prime}}_{M-1}$ are given in ref. [2] and

$\begin{array}{l}{{G}^{\prime}}_{1}={G}_{1}-\left({A}_{1}{L}_{0}+{B}_{1}\right)\text{\hspace{0.17em}}\left[{r}_{3}\left({F}_{0,j}-{F}_{1,j}\right)+{r}_{4}\left(B{T}_{j+1}+c{\left(j+1\right)}^{2}{k}^{2}-1\right)\right],\\ {{G}^{\prime}}_{2}={G}_{2}-{A}_{2}\left[{r}_{3}\left({F}_{0,j}-{F}_{1,j}\right)+{r}_{4}\left(B{T}_{j+1}+c{\left(j+1\right)}^{2}{k}^{2}-1\right)\right].\end{array}$

In matrix form, the above set of M-1equationscan be finally written as

4. Discussion

Recalling that we have introduced the new features of the slip boundary condition and the variable acceleration of the disc in non-coaxial rotation; we will lay emphasis to investigate the variation of slip parameter λ and the acceleration parameter c on the velocity profiles. However, for completeness we will show the variations of other parameters as well to look for the joint effects of already discussed and the new phenomena. Firstly, in order to ensure reliability of our numerical scheme; the numerical and the analytical results [20] are compared in Table 1, showing a good agreement.

Table 1. τ = 1, α = 0.1, β = 0, c = 1, S = 0.5, N = 0.5.

4.1. Variation of Slip Parameter

Figure 2 and Figure 3 show the effects of slip parameter λ on the flow pattern when $\alpha =0.\text{2}$ , $\beta =1.0$ , $c=0.25$ , $S=N=0$ for $\tau =0.5$ and $\tau =1.5$ . We note that increasing in λ enhances the velocity profile f/Ωl; whereas the velocity profile f/Ωl increases close to the disk and decreases away from the disk. It is further observed that the slip reduces the boundary layer thickness for the two velocity profiles. Thus the slip condition helps to control the boundary layer for rotating disc.

4.2. Variation of Acceleration Parameter

Figure 4(a) reveals that the velocity profile f/Ωl increases with small increase in acceleration parameter c keeping the values of the parameters $\alpha =0.\text{2}$ , $\beta =1.0$ , $\lambda =0.\text{5}$ , $S=N=0$ and $\tau =1.5$ fixed. However, for larger values of acceleration parameter; it is decreasing close to the disk and increasing away from the disk. The boundary layer thickness for the velocity profile f/Ωl decreases

Figure 2. Velocity profiles against distance from the disk for different values of the slip parameter λ when α = 0.2, β = 1.0, c = 0.25, S = N = 0 and τ = 0.5.

Figure 3. Velocity profiles against distance from the disk for different values of the slip parameter λ when α = 0.2, β = 1.0, c = 0.25, S = N = 0 and τ = 1.5.

Figure 4. Velocity profiles against distance from the disk for different values of the acceleration c when α = 0.2, β = 1.0, λ = 0.5, S = N = 0 and τ = 1.5.

with c. Furthermore, it is evident from Figure 4(b) that the velocity profile g/Ωl increases with increasing c close to the disk while it decreases away from the disk. The momentum boundary layer thickness for the velocity profile g/Ωl, decreases with increasing c.

4.3. Time Evolution of the Velocity Field

The evolution of the velocity profiles f/Ωl and g/Ωl for different values of time parameter τ in the case of rigid plate and non-conducting third grade fluid $\beta $ , are shown in Figure 5. Here we take $\alpha =0.\text{2}$ , $\beta =1.0$ , $c=0.25$ and $\lambda =0.\text{5}$ . This leads us to believe that; as the time evolves, the velocity profile f/Ωl increases close to the disk and decreases nominally farther from the disk. We note from Figure 5(a) that thickness of the boundary layer increases as the time $\tau $ progresses. It is evident from Figure 5(b) that the velocity profile g/Ωl and thickness of the boundary layer are increasing with time $\tau $ .

4.4. Variation of Other Physical Parameters

Figure 6 is drawn to examine the behavior of third grade parameter $\beta $ when $\alpha =0.\text{2}$ , $c=0.25$ , $\lambda =S=N=0$ and $\tau =0.5$ . It is observed from Figure 6(a) that an increase in $\beta $ slightly reduces the velocity profile f/Ωl and a slight increase in the thickness of boundary layer. The behavior is reversed for the velocity profile g/Ωl but a slight increase in the thickness of boundary layer is seen from Figure 6(b). However, in the presence of slip parameter λ = 0.25, we refer to Figure 7. This Fig. shows that there is slight increase in the velocity profiles f/Ωl and g/Ωl at the surface of the disk due to the presence of the slip parameter; otherwise the behavior is almost the same as has been seen in Figure 6.

The effects of injection/suction can be seen from Figure 8. The fluid velocities f/Ωl and g/Ωl are found to increase and decrease respectively at the surface of the disk; for slip parameter $\lambda =0.\text{5}$ , magnetic parameter $N=0$ for $\alpha =0.\text{2}$ , $\beta =1.0$ , $c=0.25$ and $\tau =0.5$ .This figure also shows that increasing porosity

Figure 5. Velocity profiles against distance from the disk for different values of time τ when α = 0.2, β = 1.0, c = 0.25, λ = 0.5 and S = N = 0.

Figure 6. Velocity profiles against distance from the disk for different values of the third grade parameter β when α = 0.2, c = 0.25, λ = S = N = 0 and τ = 0.5.

Figure 7. Velocity profiles against distance from the disk for different values of the third grade parameter β when α = 0.2, c = 0.25, λ = 0.25, S = N = 0 and τ = 0.5.

parameter S the velocity profile f/Ωl increases whereas the velocity profile g/Ωl decreases. However the boundary layer thickness reduces for f/Ωl and g/Ωl. The effects of magnetic field parameter N are identical to that of suction (see Figure 9).

To see the effects of rotation Ω; we draw the Figure 10 and Figure 11 in the absence and presence of the slip parameter λ when $\alpha =0.\text{2}$ , $\beta =1.0$ , $c=\text{1}$ , $S=N=0$ , $l=0.\text{25}$ and $\tau =1.5$ . It is found that increasing rotation Ω the velocity profile f/Ωl increases whereas g/Ωl decreases. Moreover the velocity profile f/Ωl becomes stable at some values of Ω. It is further noticed that in the presence of the slip parameter $\lambda =0.2\text{5}$ both velocity profiles f/Ωl and g/Ωl decrease at the surface of the disk as compared to the absence of slip parameter $\lambda =0$ . Otherwise, the behaviours of both f/Ωl and g/Ωl are the same as shown in Figure 10 and Figure 11. Remembering that this problem is essentially a three dimensional; three dimensional velocity profiles are plotted in Figures 12-16. These streamlines are drawn by taking into consideration various physical parameters mentioned in Figure captions.

Figure 8. Velocity profiles against distance from the disk for different values of porosity S when α = 0.2, β = 1.0, c = 0.25, λ = 0.5, N = 0 and τ = 0.5.

Figure 9. Velocity profiles against distance from the disk for different values of the magnetic parameter N when α = 0.2, β = 1.0, c = 0.25, λ = 0.5, S = 0 and τ = 0.5.

Figure 10. Velocity profiles against distance from the disk for different values of the rotation parameter Ω when α = 0.2, β = 1.0, c = 1, λ = S = N = 0, l_{2} = 0.25 and τ = 1.5.

Figure 11. Velocity profiles against distance from the disk for different values of the rotation parameter Ω when α = 0.2, β = 1.0, c = 1, λ = 0.25, S = N = 0, l_{2} = 0.25 and τ = 1.5.

Figure 12. Streams lines for (a) τ = 0.5 and (b) τ = 1.5 when α = 0.2, β = 1.0, c = 1.0, λ = 0.25, S = 0.5 and N = 0.

Figure 13. Streams lines for (a) λ = 0 and (b) λ = 1.0 when α = 0.2, β = 1.0, c = 1.0, τ = 0.75, S = 0.5 and N = 0.

Figure 14. Streams lines for (a) c = 0 and (b) c = 1.0 when α = 0.2, β = 1.0, λ = 1.0, τ = 1.25, S = 0.5 and N = 0.

5. Conclusion

In the present work, the unsteady three-dimensional MHD flow caused by a porous disk rotating non-coaxially with fluid at infinity is investigated numerically. Both the disk and the fluid at infinity are rotating with the same angular velocity. Particular emphasis has been given to non-Newtonian third grade fluid, slip boundary condition and variable acceleration of the disk. We observe that

Figure 15. Streams lines for (a) S = 0 and (b) S = 1.0 when α = 0.2, β = 1.0, λ = 1.0, τ = 1.25, S = 0.5, N = 0 and c = 1.

Figure 16. Streams lines for (a) N = 0 and (b) N = 2.0 when α = 0.2, β = 1.0, λ = 1.0, τ = 0.75, S = 1.0 and c = 1.

the rotation gives rise to a three dimensional problem. The velocity, expressed in terms of real (f/Ωl) and imaginary (g/Ωl), will remain the focus of this study. Main findings of the performed analysis are presented as follows: The behavior of third grade fluid and no-slip boundary condition is to increase the boundary layer thicknesses for both velocity fields f/Ωl and g/Ωl. On the contrary, the slip condition generates thin boundary layer thickness for the velocity f/Ωl and thick boundary layer thickness for the velocity g/Ωl. The slip and rotation parameters have increasing effects on f/Ωl but the third grade fluid has an opposite effect. The effects of these quantities on g/Ωl are exactly the opposite. The effects of suction and injuction/blowing on the boundary layer thickness remain the same irrespective of third grade fluid and the slip condition. That is, the suction causes thinning and blowing causes thickening of the boundary layer thickness. We remember that the magnetic field is applied parallel to the axis of rotation. This causes a decrease in the thickness of boundary layer for f/Ωl and an increase for the velocity profile g/Ωl. The comparison of the velocity profile for variable acceleration with constant acceleration of the disc discussed earlier [2] reveals important observation. For time τ < 1, the velocity profiles for constant accelerated flow are greater than that of variable accelerated flow. However, for the time τ > 1, the velocity profiles for variable accelerated flow are much larger than constant accelerated flow.

Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (41-130-35-HiCi). The author, therefore, acknowledge technical and financial support of KAU.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Open Journal of Fluid Dynamics*,

**7**, 485-500. doi: 10.4236/ojfd.2017.74033.

[1] |
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