**2. Mathematical model of the problem**

We introduce a Cartesian coordinate system with z-axis normal to the plane of the parallel plates. The plates are located at 0 *z* and *z L* and the plates and the fluid bounded between them are in a rigid body rotation with constant angular velocity about the zaxis. The fluid is electrically conducting and assumed to be permeated by an imposed magnetic field *B*0 perpendicular to the parallel plates. The disturbance in the fluid is produced by small amplitude non-torsional oscillations of the lower plate. For the present model we take the velocity field of the form.

$$\mathbf{V} = \begin{bmatrix} \mu(z, t), \ v(z, t), \ 0 \end{bmatrix} \tag{1}$$

where *u* and *v* are the x and y components of the velocity field. The Cauchy stress tensor for the fourth grade fluid can be obtained by the model introduced by Coleman and Noll [58]

$$\mathbf{T} = -p\mathbf{I} + \sum\_{j=1}^{n} \mathbf{S}\_{j}.\tag{2}$$

For the fourth grade fluid we have 4 *n* and the first four tensors *Sj* are given by

$$\mathbf{S}\_1 = \mu \mathbf{A}\_{1'} \tag{3}$$

$$\mathbf{S}\_2 = a\_1 \mathbf{A}\_2 + a\_2 \mathbf{A}\_1^2. \tag{4}$$

$$\mathbf{S}\_3 = \beta\_1 \mathbf{A}\_3 + \beta\_2 (\mathbf{A}\_2 \mathbf{A}\_1 + \mathbf{A}\_1 \mathbf{A}\_2) + \beta\_3 (\text{tr} \mathbf{A}\_1^2) \mathbf{A}\_1. \tag{5}$$

$$\begin{aligned} \mathbf{S}\_4 &= \gamma\_1 \mathbf{A}\_4 + \gamma\_2 (\mathbf{A}\_3 \mathbf{A}\_1 + \mathbf{A}\_1 \mathbf{A}\_3) + \gamma\_3 \mathbf{A}\_2^2 \\ &+ \gamma\_4 (\mathbf{A}\_2 \mathbf{A}\_1^2 + \mathbf{A}\_1^2 \mathbf{A}\_2) + \gamma\_5 (\text{tr} \mathbf{A}\_2) \mathbf{A}\_2 + \gamma\_6 (\text{tr} \mathbf{A}\_2) \mathbf{A}\_1^2 \\ &+ \left\{ \gamma\_7 (\text{tr} \mathbf{A}\_3) + \gamma\_8 (\text{tr} \mathbf{A}\_2 \mathbf{A}\_1) \right\} \mathbf{A}\_1. \end{aligned} \tag{6}$$

where is the co-efficient of shear viscosity; and

$$\alpha\_i \text{ ( $i = 1, 2$ ),  $\beta\_j$  ( $j = 1, 2, 3$ ),  $\gamma\_k$  ( $k = 1, 2, ..., 8$ )}$$

are material constants. The Rivlin- Ericken tensors **A***n* are defined by the recursion relation

$$\mathbf{A}\_n = \frac{d\mathbf{A}\_{n-1}}{dt} + \mathbf{A}\_{n-1}(grad\mathbf{V}) + (grad\mathbf{V})^T \mathbf{A}\_{n-1}, n > 1,\tag{7}$$

$$\mathbf{A}\_1 = \begin{pmatrix} \operatorname{grad} \mathbf{V} \end{pmatrix} + \begin{pmatrix} \operatorname{grad} \mathbf{V} \end{pmatrix}^T,\tag{8}$$

where

192 Topics in Magnetohydrodynamics

uniform heat source on viscoelastic boundary layer flow over a linear stretching sheet. Abel and Nandeppanavar [54] studied the effect of non-uniform heat source/sink on MHD viscoelastic boundary layer flow, further Nandeppanavar et al. [55] studied the effects of elastic deformation and non-uniform heat source on viscoelastic boundary layer flow. Motivated by these studies, Mahantesh et al. [56] extended the results of researchers [53,54,55] for MHD viscoelastic boundary layer flow with combined effects of viscous dissipation, thermal radiation and non-uniform heat source which was ignored by [53,54,55]. Furthermore, they analyzed the effects of radiation, viscous dissipation, viscoelasticity, magnetic field on the heat transfer characteristics in the presence of nonuniform heat source with variable PST and PHF temperature boundary conditions. Kayvan Sadeghy et al. [57] have investigated theoretically the applicability of magnetic fields for controlling hydrodynamic separation in Jeffrey-Hamel flows of viscoelastic fluids. It is shown that for viscoelastic fluids, it is possible to delay flow separation in a diverging channel provided that the magnetic field is sufficiently strong. It is also shown that the effect of magnetic field on flow separation becomes more pronounced the higher

In the present paper we have modeled the unsteady flow equations of a fourth grade fluid bounded between two non-conducting rigid plates in a rotating frame of reference with imposed uniform transverse magnetic field. It is interesting to note that we are able to couple the equations arising for the velocity field. The steady rotating flow of the non-Newtonian fluid subject to a uniform transverse magnetic field is studied. The non-linear differential equations resulting from the balance of momentum and mass are solved numerically. The effects of exerted magnetic field, Ekman number and material parameter on the velocity distribution are presented graphically. The results for Newtonian and non-

We introduce a Cartesian coordinate system with z-axis normal to the plane of the parallel plates. The plates are located at 0 *z* and *z L* and the plates and the fluid bounded between them are in a rigid body rotation with constant angular velocity about the zaxis. The fluid is electrically conducting and assumed to be permeated by an imposed magnetic field *B*0 perpendicular to the parallel plates. The disturbance in the fluid is produced by small amplitude non-torsional oscillations of the lower plate. For the present

where *u* and *v* are the x and y components of the velocity field. The Cauchy stress tensor for the fourth grade fluid can be obtained by the model introduced by Coleman and Noll [58]

*p*

For the fourth grade fluid we have 4 *n* and the first four tensors *Sj* are given by

1 . *n j j*

*uzt vzt* ( , ), ( , ), 0 , **V** (1)

**TIS** (2)

the fluid's elasticity.

Newtonian fluids are compared.

**2. Mathematical model of the problem** 

model we take the velocity field of the form.

$$\frac{d}{dt}\mathbf{(.)} = \left(\frac{\partial}{\partial t} + \mathbf{V}.\nabla\right) \mathbf{(.)}.\tag{9}$$

When 0 ( 1,2,...,8), *<sup>k</sup> k* the fourth grade model reduces to third grade model, when 0 ( 1,2,3) *<sup>j</sup> j* and 0 ( 1,2,...,8) *<sup>k</sup> k* then above model reduces to second grade model and if 0 ( 1,2), 0 ( 1,2,3), 0 ( 1,2,...,8) *ij k ij k* the flow model reduces to classical Navier-Stokes viscous fluid model.

The hydromagnetic flow is generated in the uniformly rotating fluid by small amplitude non-torsional oscillations of the plate located at 0. *z* With the Cartesian coordinate system *O*xyz the unsteady motion of the incompressible fourth grade conducting fluid in the presence of magnetic field **B** is governed by the law of balance of linear momentum and balance of mass i.e.

$$\frac{d\mathbf{V}}{dt} + \mathbf{2}(\boldsymbol{\Omega} \times \mathbf{V}) + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r}) = \frac{1}{\rho} \text{div}\, \mathbf{T} + \frac{1}{\rho}(\mathbf{J} \times \mathbf{B}),\tag{10}$$

$$\text{div}\,\mathbf{V} = 0 \,,\tag{11}$$

2

*u v*

2 2

*uu vv z zt z zt*

2 2 2

2 2

2 2

2

2

2

456

2

*z zt z zt*

*v*

5 1 2 2 3 0

 

> 

2 2

2 2

*uu vv z zt z zt*

2 2

<sup>2</sup> 2 2

(17)

*u v zz z*

*uu vv*

3 3 2 2

*uu vv z z zt zt*

 

(4 4 2 .

*u v*

*z z <sup>v</sup> B v*

2

2 2

*u v t z z*

2 2 2

*z t z zt z zt*

*z t uu vv*

,

(16)

*vu v z zt z z*

2

 

*v tz z*

2 2

 

*u*

2 2

*p u v tz z*

1

1 2 1

*z zz z z*

2 2

*uu vv z z zt z zt*

<sup>2</sup> 2 2

 

4

*u v z z*

*u v zt zt*

2

<sup>2</sup> 2 2

7 8

23 4

2 2 22

 

> 

*vu v zzz z*

2 2

2

2 2

 

*z z*

*z z*

 

 

2

3

Defining the modified pressure

*z*

22

1 2 0

3 5

2

2

 

<sup>1</sup> <sup>2</sup>

2 3

2 1 1

7

 

2

*t v*

*v p vv v u y t y z zt zt*

 

where is the density, **J** is the current density and 0 **B B bb** ( , being the induced magnetic field) is the total magnetic field.

In the absence of displacement currents, the Maxwell equations and the generalized Ohm's law can be written as

$$
\nabla \cdot \mathbf{B} = 0, \ \nabla \times \mathbf{B} = \mu\_m \mathbf{J}, \ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \tag{12}
$$

$$\mathbf{J} = \sigma(\mathbf{E} + \nabla \times \mathbf{B}),\tag{13}$$

where *m* is the magnetic permeability, **E** is the electric field and is the electrical conductivity of the fluid.

The magnetic Reynolds number is assumed to be very small so that the induced magnetic field is negligible [14]. This assumption is reasonable for the flow of liquid metals, e.g. mercury or liquid sodium (which are electrically conducting under laboratory conditions). The electron–atom collision frequency is assumed to be relatively high so that the Hall effect can be included [14]. The Lorentz force per unit volume is given by

$$\mathbf{J} \times \mathbf{B} = -\sigma B\_0^2 \mathbf{V}.\tag{14}$$

For the velocity field defined in Eq. (1), the equation of continuity (11) is identically satisfied and Eq. (10) in component form can be written as

 23 4 2 1 1 2 2 22 2 2 2 3 2 2 2 2 2 <sup>1</sup> <sup>2</sup> 2 2 2 2 2 *u p uu u v x t x z zt zt uu v zzz z u v u tz z z z uu vv z zt z zt* 2 2 2 3 5 2 2 7 2 2 7 8 2 2 *uu v z zt z z u v u tz z z z uu vv z zt z zt* 5 1 2 2 3 <sup>0</sup> , *<sup>u</sup> B u z t* (15)

 23 4 2 1 1 2 2 22 2 2 2 3 2 2 <sup>1</sup> <sup>2</sup> 2 2 2 *v p vv v u y t y z zt zt vu v zzz z u v v tz z z z* 2 2 2 2 2 2 3 5 7 2 2 2 2 *uu vv z zt z zt vu v z zt z z u t v z z* 2 2 5 1 2 2 3 0 2 2 7 8 , *v z z <sup>v</sup> B v z t uu vv z zt z zt* (16)

$$\begin{split} 0 = -\frac{1}{\rho} \frac{\partial p}{\partial z} + \frac{\left(\frac{2a\_1 + a\_2}{\rho}\right) \frac{\partial}{\partial z} \left[\left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2\right] + \frac{\rho\_1}{\rho} \frac{\partial}{\partial z} \begin{bmatrix} 2\left\frac{\partial}{\partial t} \left(\left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2\right) \\ + \frac{\rho\_1}{\rho} \frac{\partial}{\partial z} \left(2\frac{\partial u}{\partial z} \frac{\partial^2 u}{\partial z \partial t} + 2\frac{\partial v}{\partial z} \frac{\partial^2 v}{\partial z \partial t} \end{bmatrix} \\ + \frac{\rho\_2}{\rho} \frac{\partial}{\partial z} \left[2\frac{\partial u}{\partial z} \frac{\partial^2 u}{\partial z \partial t} + 2\frac{\partial v}{\partial z} \frac{\partial^2 v}{\partial z \partial t}\right] + \frac{\gamma\_1}{\rho} \frac{\partial}{\partial z} \\ + 2\frac{\partial}{\partial z} \left(\frac{\partial u}{\partial z} \frac{\partial^2 u}{\partial z \partial t} + \frac{\partial v}{\partial z} \frac{\partial^2 v}{\partial z \partial t} \right) + \frac{\gamma\_2}{\rho} \frac{\partial}{\partial z} \\ + 2\left(\frac{\partial u}{\partial z} \frac{\partial^2 u}{\partial z \partial t} + \frac{\partial v}{\partial z} \frac{\partial^2 v}{\partial z \partial t}\right) \\ + \frac{\gamma\_3}{\rho} \frac{\partial}{\partial z} \left[\left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2\right] \end{bmatrix} \tag{17}$$

Defining the modified pressure

194 Topics in Magnetohydrodynamics

In the absence of displacement currents, the Maxwell equations and the generalized Ohm's

**JE B** 

The magnetic Reynolds number is assumed to be very small so that the induced magnetic field is negligible [14]. This assumption is reasonable for the flow of liquid metals, e.g. mercury or liquid sodium (which are electrically conducting under laboratory conditions). The electron–atom collision frequency is assumed to be relatively high so that the Hall effect can be included [14]. The Lorentz force per unit volume is

> 2 <sup>0</sup> **JB V**

For the velocity field defined in Eq. (1), the equation of continuity (11) is identically satisfied

23 4

2

2

 

*u tz z*

 

2 2

2

7 8

2

*u tz z z z uu vv*

7

 

2 2 22

*uu v zzz z*

2 2

2 2

2 2 2

*uu v z zt z z*

2 2

*u v*

2 2

*z zt z zt*

*uu vv z zt z zt*

*u v*

2 2

 

> 

*m* is the magnetic permeability, **E** is the electric field and

0, , , *<sup>m</sup> <sup>t</sup>* 

 **<sup>B</sup> B B JE** (12)

( ), (13)

*B* . (14)

5 1 2 2 3 <sup>0</sup> , *<sup>u</sup> B u*

 

  (15)

 

*z t*

is the electrical

is the density, **J** is the current density and 0 **B B bb** ( , being the induced

where

where

given by

conductivity of the fluid.

law can be written as

magnetic field) is the total magnetic field.

and Eq. (10) in component form can be written as

2

 

<sup>1</sup> <sup>2</sup>

*v x*

2

 

2 3

*z z*

3 5

2 1 1

*u p uu u*

 

*t x z zt zt*

23 4

2

*v uv zz t z z*

2 2 22

*vu v zzz z*

2 2 2

*vu v z zt z z*

<sup>ˆ</sup> 0 . *<sup>p</sup> z*

In view these substitutions we

23 4

2 2

 

2 2 22

 

> 

7 8

*x y*

2 2 1 1

2 2

2 2 2

*uu v z zt z z*

2 2

*uu v zzz z*

 

> 

> > 2 2

2 2 2

*uu vv z zt z zt*

(20)

(22)

2 2

 2 2 2 2 7 8

 

*v uu vv z z zt z zt*

(21)

2 2

2 2

*uu vv z zt z zt*

*uu vv z zt z zt*

2 2

 

2 2

 

2

 

,

can write Eqs. (19)-(21) in the following manner:

 

2

3 5

2

2

5 1 2

 

*z t*

2 3 0 ,

2

 

<sup>1</sup> <sup>2</sup> <sup>ˆ</sup> <sup>2</sup>

2 3

2 2

2

3 5

2

5 1 2 2 3 0

 

*z t*

Since <sup>222</sup> 1 1 <sup>2</sup> <sup>2</sup> , therefore . 2 2 *r x y x r and y r*

<sup>ˆ</sup> <sup>2</sup>

2

2 3

*v p vv v u y t y z zt zt*

2 1 1

7

*<sup>v</sup> B v*

 

*u uu u v pr t x z zt zt*

*u uv zz t z z*

*u uv zz t z z*

7

*<sup>u</sup> B u*

 

 

 

*v u zz t z*

$$
\begin{split}
\hat{p} = \frac{p}{\rho} - \frac{\left(2\alpha\_1 + \alpha\_2\right)}{\rho} \left[\left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2\right] - \frac{\beta\_1}{\rho} \left[\frac{2\left\{\left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2\right\}}{\rho^2}\right] \\
\quad \times \frac{\partial \left(u\right)^2}{\partial z} \frac{\partial \left(u\right)^2}{\partial z \partial t} + 2 \frac{\partial v}{\partial z} \frac{\partial^2 v}{\partial z \partial t} \\
\quad \left. - \frac{\beta\_2}{\rho} \left(2\frac{\partial u}{\partial z}\frac{\partial^2 u}{\partial z \partial t} + 2\frac{\partial v}{\partial z}\frac{\partial^2 v}{\partial z \partial t}\right) - \frac{\gamma\_1}{\rho} \right] + 2\frac{\partial}{\partial z} \left[\left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2\right] \\
\quad \left. + 2\frac{\partial}{\partial z} \left(\frac{\partial u}{\partial z}\frac{\partial^2 u}{\partial z \partial t} + \frac{\partial v}{\partial z}\frac{\partial^2 v}{\partial z \partial t}\right) \right] \\
\quad \left. + 2\left(\frac{\partial u}{\partial z}\frac{\partial^3 u}{\partial z \partial t} + \frac{\partial v}{\partial z}\frac{\partial^3 v}{\partial z \partial t}\right) \right] \\
\quad - \frac{\gamma\_3}{\rho} \left[\left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2\right] \\
\end{split}
$$

#### then Eqs. (15)-(17) become

 23 4 2 1 1 2 2 22 2 2 2 3 2 2 <sup>ˆ</sup> <sup>2</sup> 2 2 2 *u p uu u v x t x z zt zt uu v zzz z u uv zz t z z* 2 2 2 2 2 2 3 5 2 7 2 2 2 2 *uu vv z zt z zt uu v z zt z z u u zz t z* <sup>2</sup> 2 2 7 8 5 1 2 2 3 0 , *v uu vv z z zt z zt <sup>u</sup> B u z t* (19)

196 Topics in Magnetohydrodynamics

2 2

2

2

2

 

 

> 

2

23 4

2 2 22

*uu v zzz z*

2 2 2

*uu v z zt z z*

 

> 

2

2

*u uv zz t z z*

2 2 1 2 1

*p u v tz z*

2 2 <sup>ˆ</sup>

*z z*

2 2 2 1

*uu vv z zt z zt*

2 2 2 2

 

*u v zt zt*

4

2 2

2 1 1

7

*<sup>u</sup> B u*

 

 

*u u zz t z*

*u v z z*

 

2 2

 

2

 

2 3 0 ,

2

2

2

5 1 2

 

*z t*

3 5

2 3

*u p uu u*

*t x z zt zt*

 

 

2 2

2 2

2 2

*v z*

<sup>2</sup> 2 2

(18)

(19)

*z z*

2 2

*uu vv t z zt z zt uu vv z z zt zt*

3 3 2 2

(4 4 2 ) , *u v*

2 2 2

*uu vv z zt z zt*

<sup>2</sup> 2 2

*v uu vv z z zt z zt*

7 8

 

2 2

 

456

2 2

*uu vv z zt z zt*

*u v*

2 2

*u t z*

 

 

 

*p*

22

3

<sup>ˆ</sup> <sup>2</sup>

*v x*

then Eqs. (15)-(17) become

 23 4 2 1 1 2 2 22 2 2 2 3 2 2 <sup>ˆ</sup> <sup>2</sup> 2 2 2 *v p vv v u y t y z zt zt vu v zzz z v uv zz t z z* 2 2 2 2 2 2 3 5 7 2 2 2 2 *uu vv z zt z zt vu v z zt z z v u zz t z* 2 2 2 2 7 8 5 1 2 2 3 0 , *v uu vv z z zt z zt <sup>v</sup> B v z t* (20)

$$0 = -\frac{\partial \hat{p}}{\partial z}.\tag{21}$$

Since <sup>222</sup> 1 1 <sup>2</sup> <sup>2</sup> , therefore . 2 2 *r x y x r and y r x y* In view these substitutions we can write Eqs. (19)-(21) in the following manner:

 23 4 2 2 1 1 2 2 22 2 2 2 3 2 2 2 <sup>1</sup> <sup>2</sup> <sup>ˆ</sup> <sup>2</sup> 2 2 2 *u uu u v pr t x z zt zt uu v zzz z u uv zz t z z* 2 2 2 2 2 3 5 2 2 7 2 2 2 2 *uu vv z zt z zt uu v z zt z z u uv zz t z z* 2 2 7 8 5 1 2 2 3 0 , *uu vv z zt z zt <sup>u</sup> B u z t* (22)

*vu v zzz z*

2 2 2

*vu v z zt z z*

2

2

0 . *<sup>p</sup> z* 

Differentiating Eqs. (26) and (27) with respect to z and making use of Eq. (28), and then

2 2

2 2

2 2 2

*uu v z zt z z*

2 2

*u uv zz t z z*

*u uv zz t z z*

*<sup>u</sup> Bu t*

7

 

> 

*uu v zzz z*

*v uv*

 

> 

*v uv zz t z z*

7

*zz t z*

*<sup>v</sup> B v*

 

 

2 2

 

2 2 2

*uu vv z zt z zt*

<sup>2</sup> 2 2

*z z zt z zt*

(28)

2 2

2 2

*uu vv z zt z zt*

*uu vv z zt z zt*

2 2

7 8

 

*uu vv*

(27)

(29)

2 2

7 8

 

2 2

 

 

*v p vv v*

*t y z zt zt*

2

 

2

2

2 3 0 ,

integrating with respect to z to obtain

3 5

2

5 1 2

 

*z t*

2 2

2

 

*u uu u*

 

*t z zt zt*

 

2

2

2 3 0 ( ),

3 5

2

5 1 2

 

*z t*

2 3

23 4 1 1 2 2 22

 

> 

2

2

*v*

*u*

2 3

23 4 1 1 2 2 22

 23 4 2 2 1 1 2 2 22 2 2 2 3 2 2 2 <sup>1</sup> <sup>2</sup> <sup>ˆ</sup> <sup>2</sup> 2 2 2 *v vv v u pr t y z zt zt vu v zzz z v uv zz t z z* 2 2 2 2 2 3 5 2 2 7 7 2 2 2 2 *uu vv z zt z zt vu v z zt z z v uv zz t z z* 2 2 8 5 1 2 2 3 0 , *uu vv z zt z zt <sup>v</sup> B v z t* (23)

$$0 = -\frac{\partial}{\partial z} \left( \hat{p} - \frac{1}{2} \Omega^2 r^2 \right). \tag{24}$$

Redefining the modified pressure

$$
\tilde{p} = \hat{p} - \frac{1}{2}\Omega^2 r^2,\tag{25}
$$

Eqs. (22)-(24) become

 23 4 1 1 2 2 22 2 2 2 3 2 2 2 2 2 2 2 *u uu u <sup>p</sup> <sup>v</sup> t x z zt zt uu v zzz z u uv zz t z z* 2 2 2 2 2 3 5 2 7 2 2 2 2 *uu vv z zt z zt uu v z zt z z u uv zz t z z* <sup>2</sup> 2 2 7 8 5 1 2 2 3 0 , *uu vv z zt z zt <sup>u</sup> B u z t* (26)

198 Topics in Magnetohydrodynamics

2 2 1 1

*v vv v u pr t y z zt zt*

*v uv zz t z z*

*v uv zz t z z*

> 23 4 1 1 2 2 22

*u uv zz t z z*

7

*<sup>u</sup> B u*

 

 

 

> 

2 2

2 2 2

*vu v z zt z z*

2 2 7 7

> <sup>1</sup> 2 2 0 . <sup>ˆ</sup> <sup>2</sup> *<sup>p</sup> <sup>r</sup> z*

*vu v zzz z*

23 4

2 2

 

2 2 22

 

> 

2 2

 

2 2

2 2 2

*uu v z zt z z*

2

*u uv zz t z z*

*uu v zzz z* 2 2

2 2

<sup>1</sup> 2 2 <sup>ˆ</sup> , <sup>2</sup> *<sup>p</sup> p r* (25)

2 2

*uu vv z zt z zt*

<sup>2</sup> 2 2

*uu vv z zt z zt*

2 2

7 8

 

*uu vv z zt z zt*

(24)

(23)

(26)

*uu vv z zt z zt*

2 2

8

 

2

3 5

2

5 1 2

 

*z t*

*u uu u <sup>p</sup> <sup>v</sup> t x z zt zt*

 

2 2

2

 

2 3 0 ,

2

3 5

2

5 1 2

 

*z t*

2

 

2 3

2 3 0 ,

Redefining the modified pressure

Eqs. (22)-(24) become

2

2

*<sup>v</sup> B v*

 

 

2

 

<sup>1</sup> <sup>2</sup> <sup>ˆ</sup> <sup>2</sup>

2 3

2 2

 23 4 1 1 2 2 22 2 2 2 3 2 2 2 2 2 2 *v p vv v u t y z zt zt vu v zzz z v uv zz t z z* 2 2 2 2 2 2 3 5 2 7 2 2 2 2 *uu vv z zt z zt vu v z zt z z v uv zz t z* <sup>2</sup> 2 2 7 8 5 1 2 2 3 0 , *uu vv z z zt z zt <sup>v</sup> B v z t* (27)

$$0 = -\frac{\partial \tilde{p}}{\partial z}.\tag{28}$$

Differentiating Eqs. (26) and (27) with respect to z and making use of Eq. (28), and then integrating with respect to z to obtain

 23 4 1 1 2 2 22 2 2 2 3 2 2 2 2 2 2 2 *u uu u v t z zt zt uu v zzz z u uv zz t z z* 2 2 2 2 2 3 5 2 2 7 2 2 2 2 *uu vv z zt z zt uu v z zt z z u uv zz t z z* 2 2 7 8 5 1 2 2 3 0 ( ), *uu vv z zt z zt <sup>u</sup> Bu t z t* (29)

*q u iv q u iv t t i t* , , ( ) ( ) ( ) 

2 2 2

2 2 2

22 2 2 2 2 1 2 22 2 <sup>2</sup> 2 43 <sup>2</sup> *u uu uv uv v vE H u zz z z z zz z*

22 2 2 2 2 1 2 22 2 <sup>2</sup> 2 43 <sup>2</sup> *v vv vu uv u E u H v zz z z z zz z*

4

 

To solve the boundary value problem the derivatives *u* and *u* involved in the problem are approximated by finite differences of appropriate order. If we employ second order

> <sup>2</sup> ( )( ) ( ) ( ), <sup>2</sup> *uz h uz h u z h h*

*L* 

,

in above equations and simplifying the resulting equations and dropping '\*' to obtain

2( ) 22 , *<sup>v</sup> vu v u B v z zzz z*

2( ) 22 , *<sup>u</sup> uu v v B u z zzz z*

For steady state the Equations (29) and (30) reduce to

 

 

> 

\* *z z*

*<sup>L</sup>* , \* *uL u*

 

<sup>1</sup> *<sup>L</sup> <sup>E</sup>* 

Consider a simplest boundary value problem

central difference formulation, then we can write

**3. Numerical procedure** 

2

and *E* is the Ekman number while H is the Hartmann number.

, 2 3

4

where

Introducing the dimensionless variables

2 3 2 2 0

2 3 2 2 0

> , and \* *vL v*

2 <sup>2</sup> *nL <sup>H</sup>* , <sup>2</sup> *<sup>B</sup>*<sup>0</sup> *<sup>n</sup>* 

*Fu u uz* ( , , , ) 0, (39)

*ua A ub B* ( ) and ( ) . (40)

(41)

(38)

 

> 

 

(35)

(32)

(33)

(34)

(36)

(37)

 23 4 1 1 2 2 22 2 2 2 3 2 2 2 2 2 2 2 *v vv v u t z zt zt vu v zzz z v uv zz t z z* 2 2 2 2 2 3 5 2 7 2 2 2 2 *uu vv z zt z zt vu v z zt z z v u zz t z* <sup>2</sup> 2 2 7 8 5 1 2 2 3 0 ( ). *v uu vv z z zt z zt <sup>v</sup> Bv t z t* (30)

On multiplying Eq. (30) by *i* and then adding the resulting equation in Eq. (29) we get

$$\begin{split} \frac{\partial q}{\partial t} + 2i\Omega q &= \frac{\mu}{\rho} \frac{\partial^2 q}{\partial z^2} + \frac{\alpha\_1}{\rho} \frac{\partial^3 q}{\partial z^2 \partial t} + \frac{\beta\_1}{\rho} \frac{\partial^4 q}{\partial z^2 \partial t^2} \\ &\quad + \frac{4(\beta\_2 + \beta\_3)}{\rho} \frac{\partial}{\partial z} \left[ \left[ \left( \frac{\partial q}{\partial z} \right)^2 \left( \frac{\partial \overline{q}}{\partial z} \right) \right] \right] \\ &\quad + \frac{\gamma\_2}{\rho} \frac{\partial}{\partial z} \left[ \left[ 3 \left( \frac{\partial q}{\partial z} \right) \frac{\partial}{\partial t} \left( \frac{\partial q}{\partial z} \frac{\partial \overline{q}}{\partial z} \right) \right] \right] + \frac{\gamma\_1}{\rho} \frac{\partial^5 q}{\partial z^2 \partial t} \\ &\quad + \frac{(\gamma\_3 + \gamma\_5)}{\rho} \frac{\partial}{\partial z} \left[ \left[ 2 \left( \frac{\partial^2 q}{\partial z \partial t} \right) \left( \frac{\partial q}{\partial z} \frac{\partial \overline{q}}{\partial z} \right) \right] \right] \\ &\quad + \frac{\partial}{\partial z} \left[ \frac{2 \gamma\_7}{\rho} \left( \frac{\partial q}{\partial z} \right) \frac{\partial}{\partial t} \left( \frac{\partial q}{\partial z} \frac{\partial \overline{q}}{\partial z} \right) \right] - \frac{\sigma}{\rho} B\_0^2 q + \eta^\rho(t), \end{split} \tag{31}$$

where

$$
\overline{q} = \mu + i\upsilon, \quad \overline{q} = \mu - i\upsilon, \quad \wp(t) = \mathcal{X}(t) + i\delta(t) \tag{32}
$$

For steady state the Equations (29) and (30) reduce to

$$-2\Omega v = \frac{\mu}{\rho} \frac{\partial^2 u}{\partial z^2} + \frac{2(\beta\_2 + \beta\_3)}{\rho} \frac{\partial}{\partial z} \left[ 2 \frac{\partial u}{\partial z} \left\{ \left( \frac{\partial u}{\partial z} \right)^2 + \left( \frac{\partial v}{\partial z} \right)^2 \right\} \right] - \frac{\sigma}{\rho} B\_0^2 u\_\prime \tag{33}$$

$$2\Omega\Omega = \frac{\mu}{\rho}\frac{\partial^2 v}{\partial z^2} + \frac{2(\beta\_2 + \beta\_3)}{\rho}\frac{\partial}{\partial z}\left[2\frac{\partial v}{\partial z}\left\{\left(\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2\right\}\right] - \frac{\sigma}{\rho}B\_0^2 v\_\prime \tag{34}$$

Introducing the dimensionless variables

$$z^\* = \frac{z}{L}, \; u^\* = \frac{uL}{\nu}, \; \text{and} \; v^\* = \frac{vL}{\nu} \tag{35}$$

in above equations and simplifying the resulting equations and dropping '\*' to obtain

$$-2\upsilon E^{-1} = \frac{\hat{\sigma}^2 u}{\hat{\sigma} z^2} + 4\beta \left[ 3 \left( \frac{\hat{\sigma}^2 u}{\hat{\sigma} z^2} \right) \left( \frac{\partial u}{\partial z} \right)^2 + \left( \frac{\hat{\sigma}^2 u}{\hat{\sigma} z^2} \right) \left( \frac{\partial v}{\partial z} \right)^2 + 2 \left( \frac{\partial u}{\partial z} \right) \left( \frac{\partial v}{\partial z} \right) \left( \frac{\partial^2 v}{\partial z^2} \right) \right] - H^2 u \tag{36}$$

$$2E^{-1}u = \frac{\partial^2 v}{\partial z^2} + 4\beta \left[ 3\left(\frac{\partial^2 v}{\partial z^2}\right)\left(\frac{\partial v}{\partial z}\right)^2 + \left(\frac{\partial^2 v}{\partial z^2}\right)\left(\frac{\partial u}{\partial z}\right)^2 + 2\left(\frac{\partial u}{\partial z}\right)\left(\frac{\partial v}{\partial z}\right)\left(\frac{\partial^2 u}{\partial z^2}\right) \right] - H^2 v \tag{37}$$

where

200 Topics in Magnetohydrodynamics

2 2

 

2 2

(30)

*uu vv z zt z zt*

<sup>2</sup> 2 2

*v uu vv z z zt z zt*

> 5 1

 

 

3 2

2 <sup>0</sup> ( ),

*Bq t*

(31)

*q z t*

2 2

7 8

 

2

*q qq*

*q qq z t zz z q qq*

 

2

 

*z zt z z*

 

*z z t zz*

 

> 

*q q zz z*

*q qq*

*z t zz*

2 2

2 2 2

*vu v z zt z z*

*v uv zz t z z*

2

On multiplying Eq. (30) by *i* and then adding the resulting equation in Eq. (29) we get

23 4 1 1 2 2 22

7

 

> 

*v u zz t z*

*<sup>v</sup> Bv t*

3 5

2

7

 

7 8

2

2

 

2

( )

*q qq q i q <sup>t</sup> z zt zt*

 

 

2 3

*vu v zzz z*

2 2

 

2

 

*v vv v*

 

*t z zt zt*

2

2

2 3 0 ( ).

2

where

4

3

3 5

2

5 1 2

 

*z t*

2

*u*

2 3

23 4 1 1 2 2 22

 

> 

$$E^{-1} = \frac{\Omega L^2}{\nu}, \; \beta = \frac{4\left(\beta\_2 + \beta\_3\right)\nu}{\rho L^4}, \; H^2 = \frac{nL^2}{\nu}, \; n = \frac{\sigma B\_0^2}{\rho} \tag{38}$$

and *E* is the Ekman number while H is the Hartmann number.
