Computational modeling of magnetohydrodynamic convection from a rotating cone in orthotropic Darcian porous media

Abstract

Free convective magnetohydrodynamic flow from a spinning vertical cone to an orthotropic Darcian porous medium under a transverse magnetic field is studied. The non-dimensionalized two-point boundary value problem is solved numerically using the Keller Box implicit finite difference method. The effects of spin parameter, orthotropic permeability functions, Prandtl number and hydromagnetic number on flow characteristics are presented graphically. Tangential velocity and swirl velocity are accentuated with increasing permeability owing to a corresponding decrease in porous media resistance. Temperatures are depressed with increasing permeability. Validation of the solutions is achieved with earlier studies. Applications of the study arise in electromagnetic spin coating materials processing.

Appendix 1: Keller Box scheme

A two-dimensional computational grid is imposed on the transformed boundary layer domain as shown in Fig 17. Note that since only one independent variable (Y) is employed, the streamwise variable (X) is not discredited in the numerical computations.

Fig. 17

“Keller Box” computational cell for finite difference approximation

The numerical stepping process is defined only for the Y co-ordinate by:

$$ y_{0} = \, 0; \, y_{j} = \, y_{j - 1} + \, h_{j} ,\quad j \, = \, 1,2 \ldots J, $$

(23)

where h _j denotes the step distance in the y-direction. Denoting Σ as the value of any variable at station y _j , the following central difference approximations are substituted for each reduced variable and their first order derivatives, viz:

$$ (\varSigma)_{{j \, {-} \, \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }}^{n - 1/2} = \, [\varSigma^{n}_{j} + \varSigma^{n}_{j - 1} + \varSigma^{n - 1}_{j} + \varSigma^{n - 1}_{j - 1} ]/4, $$

(24)

$$ (\partial \varSigma /\partial y)_{{j \, {-} \, \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }}^{n - 1/2} = \, [\varSigma ^{n}_{j} + \varSigma ^{n}_{j - 1} - \varSigma ^{n - 1}_{j} - \varSigma ^{n - 1}_{j - 1} ]/4h_{j} , $$

(25)

where h _j = spanwise stepping distance (y-mesh spacing) defined as follows:

$$ y_{j - 1/2} = \, \left[ {y_{j} + \, y_{j - 1} } \right]/2. $$

(26)

Phase a) reduction of the Nth order partial differential equation system to N X first-order equations

Equations (30)–(32) subject to the boundary conditions (35) constitute a seventh-order well-posed two-point boundary value problem. Equations (30)–(32) are first written as a system of seven first-order equations. For this purpose, we introduce new dependent variables u(y), v(y), t(y) and p(y). Therefore, we obtain the following seven first-order equations:

$$ F^{\prime} = u, $$

(26)

$$ u^{\prime} = v, $$

(28)

$$ G^{\prime} = t, $$

(29)

$$ H^{\prime} = p, $$

(30)

$$ v^{\prime} + 2Fv - u^{2} - \left( {\frac{1}{{\kappa_{x} }} + Nm} \right)u + \varepsilon G^{2} + H = 0, $$

(31)

$$ t^{\prime} + 2Ft - 2uG - \left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)G = 0, $$

(32)

$$ \frac{1}{\Pr }p^{\prime} + 2Fp - uH = 0, $$

(33)

where primes denote differentiation with respect to y. In terms of the dependent variables, the boundary conditions become:

$$ \begin{aligned} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F = 0,\,\,\,\,\,\,\,\,u = 0,\,\,\,\,\,\,\,\,G = 1,\,\,\,\,\,\,H = 1\,\,\,at\,\,\,\,\,y = 0, \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,u \to 0,\,\,\,\,\,\,G \to 0,\,\,\,\,\,\,H \to 0\,\,\,\,\,as\,\,\,\,\,y \to \infty \,. \hfill \\ \end{aligned} $$

(34)

Phase b) finite difference discretization

The net points are denoted by:

$$ y_{{^{0} }} = 0,\,\,\,\,\,\,\,\,\,\,\,\,y_{{^{j} }} = y_{{^{j - 1} }} + h_{j} ,\,\,\,\,\,\,\,j = 1,2, \ldots J,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y_{J} \equiv y_{\infty } , $$

(35)

where $h_{j}$ is the $\Delta y -$spacing. Here, j is just the sequence number that indicates the co-ordinate location. We approximate the quantities (F, u, v, G, t, H, p) at points $(y_{j} )$ of the net by $(F_{j}^{n} ,\,u_{j}^{n} ,\,v_{j}^{n} ,\,G_{j}^{n} ,\,t_{j}^{n} ,H_{j}^{n} ,p_{j}^{n} )$, which we denote as net functions. We also employ the notion $(\,\,)_{j}^{n}$ for points and quantities midway between net points and for any net function:

$$ \,\,y_{{^{j - 1/2} }} \equiv \frac{1}{2}\left( {y_{j} + y_{j - 1} } \right),\left( {} \right)_{j}^{n - 1/2} = \frac{1}{2}\left[ {\left( {} \right)_{j}^{n} + \left( {} \right)_{j}^{n - 1} } \right]\,\,\,and\,\,\left( {} \right)_{j - 1/2}^{n} = \frac{1}{2}\left[ {\left( {} \right)_{j}^{n} + \left( {} \right)_{j - 1}^{n} } \right]\,. $$

(36)

We start by writing the finite difference approximations of the ordinary differential equations (27–30) using centered-difference derivatives. This process is called “centering about $(y_{j - 1/2} )$”. This gives:

$$ \frac{{\left( {F_{j}^{n} - F_{j - 1}^{n} } \right)}}{{h_{j} }} = \frac{1}{2}\left( {u_{j}^{n} + u_{j - 1}^{n} } \right) = u_{j - 1/2}^{n} , $$

(37)

$$ \frac{{\left( {u_{j}^{n} - u_{j - 1}^{n} } \right)}}{{h_{j} }} = \frac{1}{2}\left( {v_{j}^{n} + v_{j - 1}^{n} } \right) = v_{j - 1/2}^{n} , $$

(38)

$$ \frac{{\left( {G_{j}^{n} - G_{j - 1}^{n} } \right)}}{{h_{j} }} = \frac{1}{2}\left( {t_{j}^{n} + t_{j - 1}^{n} } \right) = t_{j - 1/2}^{n} , $$

(39)

$$ \frac{{\left( {H_{j}^{n} - H_{j - 1}^{n} } \right)}}{{h_{j} }} = \frac{1}{2}\left( {p_{j}^{n} + p_{j - 1}^{n} } \right) = p_{j - 1/2}^{n} , $$

(40)

The differential equations (31)–(33) take the form:

$$ \begin{aligned} \left( {v^{\prime}} \right)^{n} + 2\left( {Fv} \right)^{n} - \left( {u^{2} } \right)^{n} - \left( {\frac{1}{{\kappa_{x} }} + Nm} \right)u^{n} + \varepsilon \left( {G^{2} } \right)^{n} + H^{n} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \left[ {} \right.\left( {v^{\prime}} \right) + 2\left( {Fv} \right) - \left( {u^{2} } \right) - \left( {\frac{1}{{\kappa_{x} }} + Nm} \right)u + \varepsilon \left( {G^{2} } \right) + H\left. {} \right]^{n - 1} , \hfill \\ \end{aligned} $$

(41)

$$ \left( {t^{\prime}} \right)^{n} + 2\left( {Ft} \right)^{n} - 2\left( {uG} \right)^{n} - \left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)G^{n} \,\, = - \left[ {\left( {t^{\prime}} \right) + 2\left( {Ft} \right) - 2\left( {uG} \right) - \left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)G} \right]^{n - 1} , $$

(42)

$$ \frac{1}{\Pr }\left( {p^{\prime}} \right)^{n} + 2\left( {Fp} \right)^{n} - \left( {uH} \right)^{n} \,\, = \left[ {\frac{1}{\Pr }\left( {p^{\prime}} \right) + 2\left( {Fp} \right) - \left( {uH} \right)\,} \right]^{n - 1} , $$

(43)

where the notation $\left[ {} \right]^{n - 1}$ corresponds to quantities in the square bracket evaluated at $y = y^{n - 1}$. Discretization gives:

$$ \begin{aligned} \left( {\frac{{v_{j}^{n} - v_{j - 1}^{n} }}{{h_{j} }}} \right) + 2\left( {F_{j - 1/2}^{n} v_{j - 1/2}^{n} } \right) - \left( {u_{j - 1/2}^{n} } \right)^{2} - \left( {\frac{1}{{\kappa_{x} }} + Nm} \right)u_{j - 1/2}^{n} + \varepsilon \left( {G_{j - 1/2}^{n} } \right)^{2} + H_{j - 1/2}^{n} \hfill \\ \,\,\, = \,\, - \left[ {\left( {\frac{{v_{j}^{n - 1} - v_{j - 1}^{n - 1} }}{{h_{j} }}} \right) + 2\left( {F_{j - 1/2}^{n - 1} v_{j - 1/2}^{n - 1} } \right) - \left( {u_{j - 1/2}^{n - 1} } \right)^{2} - \left( {\frac{1}{{\kappa_{x} }} + Nm} \right)u_{j - 1/2}^{n - 1} + \varepsilon \left( {G_{j - 1/2}^{n - 1} } \right)^{2} + H_{j - 1/2}^{n - 1} } \right], \hfill \\ \end{aligned} $$

(44)

$$ \begin{aligned} \left( {\frac{{t_{j}^{n} - t_{j - 1}^{n} }}{{h_{j} }}} \right) + 2\left( {F_{j - 1/2}^{n} t_{j - 1/2}^{n} } \right) - 2\left( {u_{j - 1/2}^{n} G_{j - 1/2}^{n} } \right) - \left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)G_{j - 1/2}^{n} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \left[ {\left( {\frac{{t_{j}^{n - 1} - t_{j - 1}^{n - 1} }}{{h_{j} }}} \right) + 2\left( {F_{j - 1/2}^{n - 1} t_{j - 1/2}^{n - 1} } \right) - 2\left( {u_{j - 1/2}^{n - 1} G_{j - 1/2}^{n - 1} } \right) - \left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)G_{j - 1/2}^{n - 1} } \right], \hfill \\ \end{aligned} $$

(45)

$$ \begin{aligned} \frac{1}{\Pr }\left( {\frac{{p_{j}^{n} - p_{j - 1}^{n} }}{{h_{j} }}} \right) + 2\left( {F_{j - 1/2}^{n} p_{j - 1/2}^{n} } \right) - \left( {u_{j - 1/2}^{n} H_{j - 1/2}^{n} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \left[ {\frac{1}{\Pr }\left( {\frac{{p_{j}^{n - 1} - p_{j - 1}^{n - 1} }}{{h_{j} }}} \right) + 2\left( {F_{j - 1/2}^{n - 1} p_{j - 1/2}^{n - 1} } \right) - \left( {u_{j - 1/2}^{n - 1} H_{j - 1/2}^{n - 1} } \right)} \right]. \hfill \\ \end{aligned} $$

(46)

Equations (A21) and (A23) are imposed for j= 1, 2… J at given n, and the transformed boundary layer thickness,$y_{J}$ is to be sufficiently large so that it is beyond the edge of the boundary layer.

$$ F_{0}^{n} = u_{0}^{n} = 0,\,\,\,\,\,\,\,\,\,G_{0}^{n} = 1,\,\,\,\,\,\,\,\,\,\,H_{0}^{n} = 1,\,\,\,\,\,\,\,\,\,\,u_{J}^{n} = 0,\,\,\,\,\,\,\,\,\,\,G_{J}^{n} = 0,\,\,\,\,\,\,\,\,\,\,H_{J}^{n} = 0. $$

(47)

Phase c) quasi-linearization of nonlinear Keller algebraic equations

Newton’s method is then employed to quasi-linearize the equations (A21) to (A23). If we assume $(F_{j}^{n - 1} ,\,u_{j}^{n - 1} ,\,v_{j}^{n - 1} ,\,G_{j}^{n - 1} ,\,t_{j}^{n - 1} ,H_{j}^{n - 1} ,p_{j}^{n - 1} )$ to be known for $0 \le j \le J$, then Eqs. (A14)–(A17) and (A21)–(A23) with (A24) are a system of equations for the solution of the unknowns $(F_{j}^{n} ,\,u_{j}^{n} ,\,v_{j}^{n} ,\,G_{j}^{n} ,\,t_{j}^{n} ,H_{j}^{n} ,p_{j}^{n} )$, j = 0, 1, 2,…J. For simplicity of notation, we shall write the unknowns at $y = y^{n}$ as:

$$ (F_{j}^{n} ,\,u_{j}^{n} ,\,v_{j}^{n} ,\,G_{j}^{n} ,\,t_{j}^{n} ,H_{j}^{n} ,p_{j}^{n} ) \equiv (F_{j}^{{}} ,\,u_{j}^{{}} ,\,v_{j}^{{}} ,\,G_{j}^{{}} ,\,t_{j}^{{}} ,H_{j}^{{}} ,p_{j}^{{}} ). $$

(48)

Then the system of equations considered then reduces to (after multiplying with$h_{j}$):

$$ F_{j}^{{}} - F_{j - 1}^{{}} - \frac{{h_{j} }}{2}\left( {u_{j}^{{}} + u_{j - 1}^{{}} } \right) = 0, $$

(49)

$$ u_{j}^{{}} - u_{j - 1}^{{}} - \frac{{h_{j} }}{2}\left( {v_{j}^{{}} + v_{j - 1}^{{}} } \right) = 0, $$

(50)

$$ G_{j}^{{}} - G_{j - 1}^{{}} - \frac{{h_{j} }}{2}\left( {t_{j}^{{}} + t_{j - 1}^{{}} } \right) = 0, $$

(51)

$$ H_{j}^{{}} - H_{j - 1}^{{}} - \frac{{h_{j} }}{2}\left( {p_{j}^{{}} + p_{j - 1}^{{}} } \right) = 0, $$

(52)

$$ \begin{aligned} \left( {v_{j} - v_{j - 1} } \right) + \frac{{h_{j} }}{2}\left[ {\left( {F_{j} + F_{j - 1} } \right)\left( {v_{j} + v_{j - 1} } \right)} \right] - \frac{{h_{j} }}{4}\left( {u_{j} + u_{j - 1} } \right)^{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{{h_{j} }}{2}\left( {\frac{1}{{\kappa_{x} }} + Nm} \right)\left( {u_{j} + u_{j - 1} } \right) + \frac{{\varepsilon h_{j} }}{4}\left( {G_{j} + G_{j - 1} } \right)^{2} + \frac{{h_{j} }}{2}\left( {H_{j} + H_{j - 1} } \right) = \left[ {R_{1} } \right]_{j - 1/2}^{n - 1} , \hfill \\ \end{aligned} $$

(53)

$$ \begin{aligned} \left( {t_{j} - t_{j - 1} } \right) + \frac{{h_{j} }}{2}\left[ {\left( {F_{j} + F_{j - 1} } \right)\left( {t_{j} + t_{j - 1} } \right)} \right] - \frac{{h_{j} }}{2}\left[ {\left( {u_{j} + u_{j - 1} } \right)\left( {G_{j} + G_{j - 1} } \right)} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)\frac{{h_{j} }}{2}\left( {u_{j} + u_{j - 1} } \right) = \left[ {R_{2} } \right]_{j - 1/2}^{n - 1} , \hfill \\ \end{aligned} $$

(54)

$$ \frac{1}{\Pr }\left( {p_{j} - p_{j - 1} } \right) + \frac{{h_{j} }}{2}\left[ {\left( {F_{j} + F_{j - 1} } \right)\left( {p_{j} + p_{j - 1} } \right)} \right] - \frac{{h_{j} }}{4}\left[ {\left( {u_{j} + u_{j - 1} } \right)\left( {H_{j} + H_{j - 1} } \right)} \right] = \left[ {R_{3} } \right]_{j - 1/2}^{n - 1} , $$

(55)

where:

$$ \left[ {R_{1} } \right]_{j - 1/2}^{n - 1} = - h_{j} \left[ {\left( {\frac{{v_{j}^{{}} - v_{j - 1}^{{}} }}{{h_{j} }}} \right) + 2\left( {F_{j - 1/2}^{{}} v_{j - 1/2}^{{}} } \right) - \left( {u_{j - 1/2}^{{}} } \right)^{2} - \left( {\frac{1}{{\kappa_{x} }} + Nm} \right)u_{j - 1/2}^{{}} + \varepsilon \left( {G_{j - 1/2}^{{}} } \right)^{2} + H_{j - 1/2}^{{}} } \right], $$

(56)

$$ \left[ {R_{2} } \right]_{j - 1/2}^{n - 1} = - h_{j} \left[ {\left( {\frac{{t_{j}^{{}} - t_{j - 1}^{{}} }}{{h_{j} }}} \right) + 2\left( {F_{j - 1/2}^{{}} t_{j - 1/2}^{{}} } \right) - 2\left( {u_{j - 1/2}^{{}} G_{j - 1/2}^{{}} } \right) - \left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)G_{j - 1/2}^{{}} } \right], $$

(57)

$$ \left[ {R_{3} } \right]_{j - 1/2}^{n - 1} = - h_{j} \left[ {\frac{1}{\Pr }\left( {\frac{{p_{j}^{{}} - p_{j - 1}^{{}} }}{{h_{j} }}} \right) + 2\left( {F_{j - 1/2}^{{}} p_{j - 1/2}^{{}} } \right) - \left( {u_{j - 1/2}^{{}} H_{j - 1/2}^{{}} } \right)} \right], $$

(58)

$\left[ {R_{1} } \right]_{j - 1/2}^{n - 1}$, $\left[ {R_{2} } \right]_{j - 1/2}^{n - 1}$ and $\left[ {R_{3} } \right]_{j - 1/2}^{n - 1}$ involve only know quantities if we assume that the solution is known for $y = y^{n - 1}$. To linearize the nonlinear system of equations (A26) to (A32) using Newton’s method, we introduce the following iterates:

$$ \begin{aligned} F_{j}^{{\left( {i + 1} \right)}} = F_{j}^{\left( i \right)} + \delta F_{j}^{\left( i \right)} ,\,\,\,\,\,\,\,\,\,\,\,u_{j}^{{\left( {i + 1} \right)}} = u_{j}^{\left( i \right)} + \delta u_{j}^{\left( i \right)} ,\,\,\,\,\,\,\,\,\,\,\,v_{j}^{{\left( {i + 1} \right)}} = v_{j}^{\left( i \right)} + \delta v_{j}^{\left( i \right)} , \hfill \\ G_{j}^{{\left( {i + 1} \right)}} = G_{j}^{\left( i \right)} + \delta G_{j}^{\left( i \right)} ,\,\,\,\,\,\,\,\,\,\,\,t_{j}^{{\left( {i + 1} \right)}} = t_{j}^{\left( i \right)} + \delta t_{j}^{\left( i \right)} , \hfill \\ H_{j}^{{\left( {i + 1} \right)}} = H_{j}^{\left( i \right)} + \delta H_{j}^{\left( i \right)} ,\,\,\,\,\,\,\,\,\,p_{j}^{{\left( {i + 1} \right)}} = p_{j}^{\left( i \right)} + \delta p_{j}^{\left( i \right)} . \hfill \\ \end{aligned} $$

(59)

Then we substitute these expressions into equations (A26)–(A32) except for the term, $y^{n - 1}$, and this yields:

$$ \left( {F_{j}^{\left( i \right)} + \delta F_{j}^{\left( i \right)} } \right) - \left( {F_{j - 1}^{\left( i \right)} + \delta F_{j - 1}^{\left( i \right)} } \right) - \frac{{h_{j} }}{2}\left( {u_{j}^{\left( i \right)} + \delta u_{j}^{\left( i \right)} + u_{j - 1}^{\left( i \right)} + \delta u_{j - 1}^{\left( i \right)} } \right) = 0, $$

(60)

$$ \left( {u_{j}^{\left( i \right)} + \delta u_{j}^{\left( i \right)} } \right) - \left( {u_{j - 1}^{\left( i \right)} + \delta u_{j - 1}^{\left( i \right)} } \right) - \frac{{h_{j} }}{2}\left( {v_{j}^{\left( i \right)} + \delta v_{j}^{\left( i \right)} + v_{j - 1}^{\left( i \right)} + \delta v_{j - 1}^{\left( i \right)} } \right) = 0, $$

(61)

$$ \left( {G_{j}^{\left( i \right)} + \delta G_{j}^{\left( i \right)} } \right) - \left( {G_{j - 1}^{\left( i \right)} + \delta g_{j - 1}^{\left( i \right)} } \right) - \frac{{h_{j} }}{2}\left( {t_{j}^{\left( i \right)} + \delta t_{j}^{\left( i \right)} + t_{j - 1}^{\left( i \right)} + \delta t_{j - 1}^{\left( i \right)} } \right) = 0, $$

(62)

$$ \left( {H_{j}^{\left( i \right)} + \delta H_{j}^{\left( i \right)} } \right) - \left( {H_{j - 1}^{\left( i \right)} + \delta H_{j - 1}^{\left( i \right)} } \right) - \frac{{h_{j} }}{2}\left( {p_{j}^{\left( i \right)} + \delta p_{j}^{\left( i \right)} + p_{j - 1}^{\left( i \right)} + \delta p_{j - 1}^{\left( i \right)} } \right) = 0, $$

(63)

$$ \begin{aligned} \left[ {\left( {v_{j}^{(i)} + \delta v_{j}^{(i)} } \right) - \left( {v_{j - 1}^{(i)} + \delta v_{j - 1}^{(i)} } \right)} \right] + \frac{{h_{j} }}{2}\left[ {\left( {F_{j}^{(i)} + \delta F_{j}^{(i)} + F_{j - 1}^{(i)} + \delta F_{j - 1}^{(i)} } \right)\left( {v_{j}^{(i)} + \delta v_{j}^{(i)} + v_{j - 1}^{(i)} + \delta v_{j - 1}^{(i)} } \right)} \right] \hfill \\ \,\,\,\,\,\,\,\, - \frac{{h_{j} }}{4}\left( {u_{j}^{(i)} + \delta u_{j}^{(i)} + u_{j - 1}^{(i)} + \delta u_{j - 1}^{(i)} } \right)^{2} - \frac{{h_{j} }}{2}\left( {\frac{1}{{\kappa_{x} }} + Nm} \right)\left( {u_{j}^{(i)} + \delta u_{j}^{(i)} + u_{j - 1}^{(i)} + \delta u_{j - 1}^{(i)} } \right) \hfill \\ \,\,\,\,\,\,\, + \frac{{\varepsilon h_{j} }}{4}\left( {G_{j}^{(i)} + \delta G_{j}^{(i)} + G_{j - 1}^{(i)} + \delta G_{j - 1}^{(i)} } \right)^{2} + \frac{{h_{j} }}{2}\left( {H_{j}^{(i)} + \delta H_{j}^{(i)} + H_{j - 1}^{(i)} + \delta H_{j - 1}^{(i)} } \right) = \left[ {R_{1} } \right]_{j - 1/2}^{n - 1} , \hfill \\ \end{aligned} $$

(64)

$$ \begin{aligned} \left[ {\left( {t_{j}^{(i)} + \delta t_{j}^{(i)} } \right) - \left( {t_{j - 1}^{(i)} + \delta t_{j - 1}^{(i)} } \right)} \right] + \frac{{h_{j} }}{2}\left[ {\left( {F_{j}^{(i)} + \delta F_{j}^{(i)} + F_{j - 1}^{(i)} + \delta F_{j - 1}^{(i)} } \right)\left( {t_{j}^{(i)} + \delta t_{j}^{(i)} + t_{j - 1}^{(i)} + \delta t_{j - 1}^{(i)} } \right)} \right] \hfill \\ \,\,\,\,\,\,\, - \frac{{h_{j} }}{2}\left[ {\left( {u_{j}^{(i)} + \delta u_{j}^{(i)} + u_{j - 1}^{(i)} + \delta u_{j - 1}^{(i)} } \right)\left( {G_{j}^{(i)} + \delta G_{j}^{(i)} + G_{j - 1}^{(i)} + \delta G_{j - 1}^{(i)} } \right)} \right] \hfill \\ \,\,\,\,\,\,\,\, - \left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)\frac{{h_{j} }}{2}\left( {G_{j}^{(i)} + \delta G_{j}^{(i)} + G_{j - 1}^{(i)} + \delta G_{j - 1}^{(i)} } \right) = \left[ {R_{2} } \right]_{j - 1/2}^{n - 1} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( { 4 9 {\text{f}}} \right) \hfill \\ \end{aligned} $$

(65)

$$ \begin{aligned} \frac{1}{\Pr }\left[ {\left( {p_{j}^{(i)} + \delta p_{j}^{(i)} } \right) - \left( {p_{j - 1}^{(i)} + \delta p_{j - 1}^{(i)} } \right)} \right] + \frac{{h_{j} }}{2}\left[ {\left( {F_{j}^{(i)} + \delta F_{j}^{(i)} + F_{j - 1}^{(i)} + \delta F_{j - 1}^{(i)} } \right)\left( {p_{j}^{(i)} + \delta p_{j}^{(i)} + p_{j - 1}^{(i)} + \delta p_{j - 1}^{(i)} } \right)} \right] \hfill \\ \,\,\,\,\,\,\, - \frac{{h_{j} }}{4}\left[ {\left( {u_{j}^{(i)} + \delta u_{j}^{(i)} + u_{j - 1}^{(i)} + \delta u_{j - 1}^{(i)} } \right)\left( {H_{j}^{(i)} + \delta H_{j}^{(i)} + H_{j - 1}^{(i)} + \delta H_{j - 1}^{(i)} } \right)} \right]\,\, = \left[ {R_{3} } \right]_{j - 1/2}^{n - 1} . \hfill \\ \end{aligned} $$

(66)

Next, we drop the terms that are quadratic in the following: $\left( {\delta F_{j}^{(i)} ,\,\delta u_{j}^{(i)} ,\,\delta v_{j}^{(i)} ,\,\delta G_{j}^{(i)} ,\,\delta t_{j}^{(i)} ,\,\delta H_{j}^{(i)} ,\,\delta p_{j}^{(i)} } \right)$. We also drop the superscript I for simplicity. After some algebraic manipulations, the following linear tridiagonal system of equations is obtained:

$$ \delta F_{j}^{{}} - \delta F_{j - 1}^{{}} - \frac{{h_{j} }}{2}\left( {\delta u_{j}^{{}} + \delta u_{j - 1}^{{}} } \right) = (r_{1} )_{j - 1/2} , $$

(67)

$$ \delta u_{j}^{{}} - \delta u_{j - 1}^{{}} - \frac{{h_{j} }}{2}\left( {\delta v_{j}^{{}} + \delta v_{j - 1}^{{}} } \right) = (r_{2} )_{j - 1/2} , $$

(68)

$$ \delta G_{j}^{{}} - \delta G_{j - 1}^{{}} - \frac{{h_{j} }}{2}\left( {\delta t_{j}^{{}} + \delta t_{j - 1}^{{}} } \right) = (r_{3} )_{j - 1/2} , $$

(69)

$$ \delta H_{j}^{{}} - \delta H_{j - 1}^{{}} - \frac{{h_{j} }}{2}\left( {\delta t_{j}^{{}} + \delta t_{j - 1}^{{}} } \right) = (r_{4} )_{j - 1/2} , $$

(70)

$$ \begin{aligned} (a_{1} )_{j} \delta v_{j}^{{}} + (a_{2} )_{j} \delta v_{j - 1}^{{}} + (a_{3} )_{j} \delta F_{j}^{{}} + (a_{4} )_{j} \delta F_{j - 1}^{{}} + (a_{5} )_{j} \delta u_{j}^{{}} + (a_{6} )_{j} \delta u_{j - 1}^{{}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + (a_{7} )_{j} \delta G_{j}^{{}} + (a_{8} )_{j} \delta G_{j - 1}^{{}} + (a_{9} )_{j} \delta H_{j}^{{}} + (a_{10} )_{j} \delta H_{j - 1}^{{}} = (r_{5} )_{j - 1/2} , \hfill \\ \end{aligned} $$

(71)

$$ \begin{aligned} (b_{1} )_{j} \delta t_{j}^{{}} + (b_{2} )_{j} \delta t_{j - 1}^{{}} + (b_{3} )_{j} \delta F_{j}^{{}} + (b_{4} )_{j} \delta F_{j - 1}^{{}} + (b_{5} )_{j} \delta u_{j}^{{}} + (b_{6} )_{j} \delta u_{j - 1}^{{}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + (b_{7} )_{j} \delta G_{j}^{{}} + (b_{8} )_{j} \delta G_{j - 1}^{{}} = (r_{6} )_{j - 1/2} , \hfill \\ \end{aligned} $$

(72)

$$ \begin{aligned} (c_{1} )_{j} \delta p_{j}^{{}} + (c_{2} )_{j} \delta p_{j - 1}^{{}} + (c_{3} )_{j} \delta F_{j}^{{}} + (c_{4} )_{j} \delta F_{j - 1}^{{}} + (c_{5} )_{j} \delta u_{j}^{{}} + (c_{6} )_{j} \delta u_{j - 1}^{{}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + (c_{7} )_{j} \delta H_{j}^{{}} + (c_{8} )_{j} \delta H_{j - 1}^{{}} = (r_{7} )_{j - 1/2} , \hfill \\ \end{aligned} $$

(73)

where:

$$ \begin{aligned} (a_{1} )_{j} = 1 + h_{j} F_{j - 1/2} ,\,\,\,\,\,\,(a_{2} )_{j} = - 1 + h_{j} F_{j - 1/2} , \hfill \\ (a_{3} )_{j} = h_{j} v_{j - 1/2} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(a_{4} )_{j} = (a_{3} )_{j} , \hfill \\ (a_{5} )_{j} = h_{j} \left[ {u_{j - 1/2} - \frac{1}{2}\left( {\frac{1}{{\kappa_{x} }} + Nm} \right)} \right],\,\,\,\,\,\,\,\,\,\,\,\,\,(a_{6} )_{j} = (a_{5} )_{j} ,(a_{7} )_{j} = \varepsilon h_{j} G_{j - 1/2} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(a_{8} )_{j} = (a_{7} )_{j} ,(a_{9} )_{j} = \frac{1}{2}h_{j} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(a_{10} )_{j} = (a_{9} )_{j} , \hfill \\ \end{aligned} $$

(74)

$$ \begin{aligned} (b_{1} )_{j} &= 1 + h_{j} F_{j - 1/2} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(b_{2} )_{j} = - 1 + h_{j} F_{j - 1/2} , \hfill \\ (b_{3} )_{j} &= h_{j} t_{j - 1/2} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(b_{4} )_{j} = (b_{3} )_{j} , \hfill \\ (b_{5} )_{j} &= - h_{j} G_{j - 1/2} ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(b_{6} )_{j} = (b_{5} )_{j} , \hfill \\ (b_{7} )_{j} &= h_{j} \left[ { - u_{j - 1/2} - \frac{1}{2}\left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)} \right],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(b_{8} )_{j} = (b_{7} )_{j} , \hfill \\ \end{aligned} $$

(75)

$$ \begin{aligned} (c_{1} )_{j} &= \frac{1}{\Pr } + h_{j} F_{j - 1/2} ,\,\quad (c_{2} )_{j} = - \frac{1}{\Pr } + h_{j} F_{j - 1/2} , \hfill \\ (c_{3} )_{j} &= h_{j} p_{j - 1/2} ,\,\quad (c_{4} )_{j} = (c_{3} )_{j} , \hfill \\ (c_{5} )_{j} &= - \frac{{h_{j} }}{2}H_{j - 1/2} ,\,\,\quad (c_{6} )_{j} = (c_{5} )_{j} ,\quad (c_{7} )_{j} = - \frac{{h_{j} }}{2}u_{j - 1/2} ,\,\,\quad \,(c_{8} )_{j} = (c_{7} )_{j} , \hfill \\ \end{aligned} $$

(76)

$$ \begin{aligned} \left( {r_{1} } \right)_{j - 1/2} &= F_{j - 1} - F_{j} + h_{j} u_{j - 1/2} ,\,\,\,\left( {r_{2} } \right)_{j - 1/2} &= u_{j - 1} - u_{j} + h_{j} v_{j - 1/2} , \hfill \\ \left( {r_{3} } \right)_{j - 1/2} &= G_{j - 1} - G_{j} + h_{j} t_{j - 1/2} ,\,\,\,\left( {r_{4} } \right)_{j - 1/2} = H_{j - 1} - H_{j} + h_{j} p_{j - 1/2} , \hfill \\ \left( {r_{5} } \right)_{j - 1/2} &= \left( {v_{j - 1} - v_{j} } \right) - 2h_{j} F_{j - 1/2} v_{j - 1/2} + h_{j} u_{j - 1/2}^{2} + \left( {\frac{1}{{\kappa_{x} }} + Nm} \right)h_{j} u_{j - 1/2} \hfill \\ &\quad - \varepsilon h_{j} G_{{_{j - 1/2} }}^{2} - h_{j} H_{j - 1/2} + \left[ {R_{1} } \right]_{j - 1/2}^{n - 1} , \hfill \\ \left( {r_{6} } \right)_{j - 1/2} &= \left( {t_{j - 1} - t_{j} } \right) - 2h_{j} F_{j - 1/2} t_{j - 1/2} + 2h_{j} u_{j - 1/2} G_{j - 1/2} + h_{j} \left( {\frac{1}{{\kappa_{\theta } }} + Nm} \right)G_{j - 1/2} + \left[ {R_{2} } \right]_{j - 1/2}^{n - 1} , \hfill \\ \left( {r_{7} } \right)_{j - 1/2} &= \frac{1}{\Pr }\left( {p_{j - 1} - p_{j} } \right) - 2h_{j} F_{j - 1/2} p_{j - 1/2} + h_{j} u_{j - 1/2} H_{j - 1/2} + \left[ {R_{3} } \right]_{j - 1/2}^{n - 1} . \hfill \\ \end{aligned} $$

(77)

To complete the system (A44)–(A50), we recall the boundary conditions (A24), which can be satisfied exactly with no iteration. Therefore to maintain these correct values in all the iterates, we take:

$$ \delta F_{0}^{{}} = 0,\,\,\,\,\,\delta u_{0}^{{}} = 0,\,\,\,\,\,\delta G_{0}^{{}} = 0,\,\,\,\,\,\delta H_{0}^{{}} = 0,\,\,\,\,\,\delta u_{J}^{{}} = 0,\,\,\,\,\,\delta G_{J}^{{}} = 0,\,\,\,\,\,\,\delta H_{J}^{{}} = 0. $$

(78)

Phase d) block-tridiagonal elimination of linear Keller algebraic equations

The linear system (A44)–(A50) can now be solved by the block-elimination method. The linearized difference equations of this system have a block-tridiagonal structure. Commonly, the block-tridiagonal structure consists of variables or constants; however, here, an interesting feature can be observed, that is, for the Keller Box method, it consists of block matrices. Intrinsic to the block-elimination method used in the Keller Box implicit finite difference method is the correct derivation of the elements of the block matrices from the linear system. We consider three cases, namely when j = 1, J − 1 and J. When j = 1, the linear system equations become:

$$ \delta F_{1}^{{}} - \delta F_{0}^{{}} - \frac{{h_{1} }}{2}\left( {\delta u_{1}^{{}} + \delta u_{0}^{{}} } \right) = (r_{1} )_{1 - 1/2} , $$

(79)

$$ \delta u_{1}^{{}} - \delta u_{0}^{{}} - \frac{{h_{j} }}{2}\left( {\delta v_{1}^{{}} + \delta v_{0}^{{}} } \right) = (r_{2} )_{1 - 1/2} , $$

(80)

$$ \delta G_{1}^{{}} - \delta G_{0}^{{}} - \frac{{h_{j} }}{2}\left( {\delta t_{1}^{{}} + \delta t_{0}^{{}} } \right) = (r_{3} )_{1 - 1/2} , $$

(81)

$$ \delta H_{1}^{{}} - \delta H_{0}^{{}} - \frac{{h_{j} }}{2}\left( {\delta p_{1}^{{}} + \delta p_{0}^{{}} } \right) = (r_{4} )_{1 - 1/2} , $$

(82)

$$ \begin{aligned} (a_{1} )_{1} \delta v_{1}^{{}} + (a_{2} )_{1} \delta v_{0}^{{}} + (a_{3} )_{1} \delta F_{1}^{{}} + (a_{4} )_{1} \delta F_{0}^{{}} + (a_{5} )_{1} \delta u_{1}^{{}} + (a_{6} )_{1} \delta u_{0}^{{}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + (a_{7} )_{1} \delta G_{1}^{{}} + (a_{8} )_{1} \delta G_{0}^{{}} + (a_{9} )_{1} \delta H_{1}^{{}} + (a_{10} )_{1} \delta H_{0}^{{}} = (r_{5} )_{1 - 1/2} , \hfill \\ \end{aligned} $$

(83)

$$ \begin{aligned} (b_{1} )_{1} \delta t_{1}^{{}} + (b_{2} )_{1} \delta t_{0}^{{}} + (b_{3} )_{1} \delta F_{1}^{{}} + (b_{4} )_{1} \delta F_{0}^{{}} + (b_{5} )_{1} \delta u_{1}^{{}} + (b_{6} )_{1} \delta u_{0}^{{}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + (b_{7} )_{1} \delta G_{1}^{{}} + (b_{8} )_{1} \delta G_{0}^{{}} = (r_{6} )_{1 - 1/2} , \hfill \\ \end{aligned} $$

(84)

$$ \begin{aligned} (c_{1} )_{1} \delta p_{1}^{{}} + (c_{2} )_{1} \delta p_{0}^{{}} + (c_{3} )_{1} \delta F_{1}^{{}} + (c_{4} )_{1} \delta F_{0}^{{}} + (c_{5} )_{1} \delta u_{1}^{{}} + (c_{6} )_{1} \delta u_{0}^{{}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + (c_{7} )_{1} \delta H_{1}^{{}} + (c_{8} )_{1} \delta H_{0}^{{}} = (r_{7} )_{1 - 1/2} . \hfill \\ \end{aligned} $$

(85)

Designating $d_{1} = - \frac{1}{2}h_{1} ,\,\,and\,\,\delta F_{0} = 0,\,\,\,\,\delta u_{0} = 0,\,\,\,\,\,\delta G_{0} = 0,\,\,\,\,\delta H_{0} = 0$ to the corresponding matrix form we assume:

$$ \begin{aligned} \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ {d_{1} } & 0 & 0 & 0 & {d_{1} } & 0 & 0 \\ 0 & {d_{1} } & 0 & 0 & 0 & {d_{1} } & 0 \\ 0 & 0 & {d_{1} } & 0 & 0 & 0 & {d_{1} } \\ {\left( {a_{2} } \right)_{1} } & 0 & 0 & {\left( {a_{3} } \right)_{1} } & {\left( {a_{1} } \right)_{1} } & 0 & 0 \\ 0 & {\left( {b_{2} } \right)_{1} } & 0 & {\left( {b_{3} } \right)_{1} } & 0 & {\left( {b_{1} } \right)_{1} } & 0 \\ 0 & 0 & {\left( {c_{2} } \right)_{1} } & {\left( {c_{3} } \right)_{1} } & 0 & 0 & {\left( {c_{1} } \right)_{1} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\delta v_{0} } \\ {\delta t_{0} } \\ {\delta p_{0} } \\ {\delta F_{1} } \\ {\delta v_{1} } \\ {\delta t_{1} } \\ {\delta p_{1} } \\ \end{array} } \right] \hfill \\ + \left[ {\begin{array}{*{20}c} {d_{1} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ {\left( {a_{5} } \right)_{1} } & {\left( {a_{7} } \right)_{1} } & {\left( {a_{9} } \right)_{1} } & 0 & 0 & 0 & 0 \\ {\left( {b_{5} } \right)_{1} } & {\left( {b_{7} } \right)_{1} } & 0 & 0 & 0 & 0 & 0 \\ {\left( {c_{5} } \right)_{1} } & 0 & {\left( {c_{7} } \right)_{1} } & 0 & 0 & 0 & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\delta u_{1} } \\ {\delta G_{1} } \\ {\delta H_{1} } \\ {\delta F_{2} } \\ {\delta v_{2} } \\ {\delta t_{2} } \\ {\delta p_{2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {r_{1} } \right)_{1 - (1/2)} } \\ {\left( {r_{2} } \right)_{1 - (1/2)} } \\ {\left( {r_{3} } \right)_{1 - (1/2)} } \\ {\left( {r_{4} } \right)_{1 - (1/2)} } \\ {\left( {r_{5} } \right)_{1 - (1/2)} } \\ {\left( {r_{6} } \right)_{1 - (1/2)} } \\ {\left( {r_{7} } \right)_{1 - (1/2)} } \\ \end{array} } \right]. \hfill \\ \end{aligned} $$

(86)

For j = 1, we have $\left[ {A_{1} } \right]\left[ {\delta_{1} } \right] + \left[ {C_{1} } \right]\left[ {\delta_{2} } \right] = \left[ {r_{1} } \right]$. Similar procedures are followed at the different stations. Effectively, the seven linearized finite difference equations have the matrix–vector form:

$$ \varLambda \delta_{j} = \zeta_{j} , $$

(87)

where Λ = Keller coefficient matrix of order 7 × 7, δ _j = seventh-order vector for error (perturbation) quantities and ζ _j= seventh-order vector for Keller residuals. This system is then recast as an expanded matrix–vector system, viz:

$$ \varsigma_{j} \delta_{j} - \omega_{j} \delta_{j} = \zeta_{j} , $$

(88)

where now ς _j = coefficient matrix of order 7 × 7, ω_j = coefficient matrix of order 7 × 7 and ζ _j= seventh-order vector of errors (iterates) at the previous station on the grid. Finally, the complete linearized system is formulated as a block matrix system where each element in the coefficient matrix is a matrix itself.