Stability of natural convection in a vertical non-Newtonian fluid layer with an imposed magnetic field

Abstract

A numerical study has been conducted to analyze the influence of a uniform horizontal magnetic field on the stability of buoyancy driven parallel shear flow in a differentially heated vertical layer of an electrically conducting couple stress fluid; a type of non-Newtonian fluid. Within the framework of linear stability theory, the resulting complex generalized eigenvalue problem is solved numerically using the Chebyshev collocation method with QZ algorithm. The critical Grashof number $G_{c}$ and the corresponding wave number $\alpha_{c}$ and wave speed $c_{c}$ are computed for a wide range of couple stress parameter $\varLambda_{c}$, Prandtl number $Pr$ and Hartmann number $M$. It is found that the value of $Pr$ at which the instability switches over from stationary to travelling-wave mode increases with increasing $M$ and decreasing $\varLambda_{c}$. The effect of magnetic field is to delay the onset of instability while an opposite kind of behavior is observed with increasing $\varLambda_{c}$. The streamlines presented herein demonstrate the development of complex dynamics at the transition mode.

Appendix

For the given physical domain right and left wall corresponds to $j = 0$ and $j = N$, respectively. Substituting Eq. (28) into Eqs. (25) and (26), the resulting discretized form of governing equations are

$$\begin{aligned} i\alpha G\left[ {(w_{b} - c)\left( {\left( {1 - x_{j}^{2} } \right)^{2} \sum\limits_{k = 0}^{N} {B_{jk} \phi_{k} } - 8x_{j} \left( {1 - x_{j}^{2} } \right)\sum\limits_{k = 0}^{N} {A_{jk} \phi_{k} } + \left( {8x_{j}^{2} - 4\left( {1 - x_{j}^{2} } \right)} \right)\phi_{j} } \right) - \alpha^{2} (w_{b} - c)\phi_{j} - \frac{{d^{2} w_{b} }}{{dx^{2} }}\phi_{j} } \right] \\ \quad & = \,\left[ {\left( {x_{j}^{2} - 1} \right)^{2} \sum\limits_{k = 0}^{N} {E_{jk} \phi_{k} } + 16x_{j} \left( {x_{j}^{2} - 1} \right)\sum\limits_{k = 0}^{N} {C_{jk} \phi_{k} } + 24\left( {3x_{j}^{2} - 1} \right)\sum\limits_{k = 0}^{N} {B_{jk} \phi_{k} } + 96x_{j} \sum\limits_{k = 0}^{N} {A_{jk} \phi_{k} } + 24\phi_{j} } \right] \\ & \quad \quad + \,\alpha^{4} \phi_{j} - 2\alpha^{2} \left[ {\left( {1 - x_{j}^{2} } \right)^{2} \sum\limits_{k = 0}^{N} {B_{jk} \phi_{k} } - 8x_{j} \left( {1 - x_{j}^{2} } \right)\sum\limits_{k = 0}^{N} {A_{jk} \phi_{k} } + \left( {8x_{j}^{2} - 4\left( {1 - x_{j}^{2} } \right)} \right)\phi_{j} } \right] \\ & \quad \quad - \,\frac{1}{{\varLambda_{c}^{2} }}\left\{ {\begin{array}{*{20}l} {\left( {x_{j}^{2} - 1} \right)^{2} \sum\limits_{k = 0}^{N} {H_{jk} \phi_{k} } + 24x_{j} \left( {x_{j}^{2} - 1} \right)\sum\limits_{k = 0}^{N} {F_{jk} \phi_{k} } } \hfill \\ { + 60\left( {3x_{j}^{2} - 1} \right)\sum\limits_{k = 0}^{N} {E_{jk} \phi_{k} } + 480x_{j} \sum\limits_{k = 0}^{N} {C_{jk} \phi_{k} } + 360\sum\limits_{k = 0}^{N} {B_{jk} \phi_{k} } } \hfill \\ { - 3\alpha^{2} \left( \begin{aligned} \left( {x_{j}^{2} - 1} \right)^{2} \sum\limits_{k = 0}^{N} {E_{jk} \phi_{k} } + 16x_{j} \left( {x_{j}^{2} - 1} \right)\sum\limits_{k = 0}^{N} {C_{jk} \phi_{k} } + 24\left( {3x_{j}^{2} - 1} \right)\sum\limits_{k = 0}^{N} {B_{jk} \phi_{k} } + 96x_{j} \sum\limits_{k = 0}^{N} {A_{jk} \phi_{k} } + 24\phi_{j} \hfill \\ \end{aligned} \right)} \hfill \\ { + 3\alpha^{4} \left( {\left( {1 - x_{j}^{2} } \right)^{2} \sum\limits_{k = 0}^{N} {B_{jk} \phi_{k} } - 8x_{j} \left( {1 - x_{j}^{2} } \right)\sum\limits_{k = 0}^{N} {A_{jk} \phi_{k} } + \left( {8x_{j}^{2} - 4\left( {1 - x_{j}^{2} } \right)} \right)\phi_{j} } \right) - \alpha^{6} \phi_{j} } \hfill \\ \end{array} } \right. \\ & \quad \quad - \,\sum\limits_{K = 0}^{N} {A_{jk} \theta_{k} - M^{2} \left( {\left( {1 - x_{j}^{2} } \right)^{2} \sum\limits_{k = 0}^{N} {B_{jk} \phi_{k} } - 8x_{j} \left( {1 - x_{j}^{2} } \right)\sum\limits_{k = 0}^{N} {A_{jk} \phi_{k} } + \left( {8x_{j}^{2} - 4\left( {1 - x_{j}^{2} } \right)} \right)\phi_{j} } \right)} , \quad j = 1\left( 1 \right)N - 1 \\ \end{aligned}$$

(34)

$$(w_{b} - c)\theta_{j} + \phi_{j} = \frac{1}{i\alpha \,Pr\,G}\left[ {\sum\limits_{k = 0}^{N} {B_{jk} \theta_{k} - \alpha^{2} \theta_{j} } } \right],\,\,j = 1\left( 1 \right)N - 1$$

(35)

with the corresponding boundary conditions

$$(1 - x_{j}^{2} )^{2} \sum\limits_{k = 0}^{N} {C_{jk} \phi_{k} } - 12x_{j} (1 - x_{j}^{2} )\sum\limits_{k = 0}^{N} {B_{jk} \phi_{k} } - 12(1 - 3x_{j}^{2} )\sum\limits_{k = 0}^{N} {A_{jk} \phi_{k} } + 24x_{j} \phi_{j} = 0, {\text{j}} = 0 \& {\text{N}}$$

(36)

$$\theta_{j} = 0,j = 0 \& {\text{N}}$$

(37)

where

$$A_{jk} = \left\{ {\begin{array}{*{20}l} {\frac{{\varGamma_{j} \left( { - 1} \right)^{k + j} }}{{\varGamma_{k} \left( {x_{j} - x_{k} } \right)}}} \hfill & {j \ne k} \hfill \\ {\frac{{x_{j} }}{{2\left( {1 - x_{j}^{2} } \right)}}} \hfill & {1 \le j = k \le N - 1} \hfill \\ {\frac{{2N^{2} + 1}}{6}} \hfill & {j = k = 0} \hfill \\ { - \frac{{2N^{2} + 1}}{6}} \hfill & {j = k = N} \hfill \\ \end{array} } \right.$$

(38)

$$B_{jk} = A_{jm} \cdot A_{mk} ,\quad C_{jk} = B_{jm} \cdot A_{mk} ,\quad E_{jk} = C_{jm} \cdot A_{mk} ,\quad F_{jk} = E_{jm} \cdot A_{mk} ,\quad H_{jk} = F_{jm} \cdot A_{mk}$$

(39)

with

$$\varGamma_{j} = \left\{ {\begin{array}{*{20}l} 2 \hfill & {j = 0,N} \hfill \\ 1 \hfill & {1 \le j \le N - 1} \hfill \\ \end{array} .} \right.$$

The above equations form the following system of linear algebraic equations

$$\Delta_{1} X = c\,\Delta_{2} X$$

(40)