The effect of chemical reaction on thermo-solutal magneto-convection under non-equilibrium temperature conditions

Abstract

The onset of convection in a porous medium saturated by the Oldroyd-type viscoelastic fluid, heated and salted from below, is investigated by incorporating the effects of chemical reactions on boundaries and externally imposed magnetic fields with non-equilibrium temperature conditions. The normal mode technique is used to perform linear stability analysis. The whole study is divided into two parts: (i) the parametric study for stability analysis in the oscillatory case and (ii) the stability analysis with the comparative study between different boundary conditions for controlling parameters for the limited case (in the stationary case, the viscoelastic effect is missing). These boundary surfaces are: (a) realistic bounding surfaces (i.e., rigid–free and free–rigid (R/R, R/F and F/R)) and (b) non-realistic bounding surface, i.e., free–free (F/F). For studying the viscoelastic fluid behavior, i.e., effect of the viscoelastic parameters (i.e., relaxation parameter $({\lambda }_{1})$) and retardation parameter $({\lambda }_{2})$, we discussed the oscillatory state on free–free boundary surfaces. To analyze the consequences of different controlling parameters, numerical computation has been performed, and the results are illustrated in graphical form. For oscillatory convection, the minimum of the critical Rayleigh number drops as the relaxation parameter (${\lambda }_{1})$, solute Rayleigh number (Ra_S) and Lewis number (Le) increase while it increases as the retardation parameter (${\lambda }_{2})$, Chadrashekhar number (Q) and interphase heat transfer coefficient ( $\tau )$ increase. In comparative study for stationary convection, the graphs demonstrate that while the critical Rayleigh number reduces as the value of Damk $\ddot{o}$ hler number ($\chi )$ grows, it increases along with increase in Chandrashekhar number (Q), interphase heat transfer coefficient ($\tau )$ and Lewis number (Le).

Appendix-I

$${p}_{1}=\alpha \gamma {\tau }^{2}, {p}_{2}={\alpha }^{2}, {p}_{3}={\left({{\delta }_{4}}^{2}+\gamma \tau \right)}^{2},{p}_{4}=Q{P}_{m}{{\delta }_{1}}^{4}{\pi }^{2},{p}_{5}={\left({P}_{m}{{\delta }_{1}}^{2}\right)}^{2},{p}_{6}=\frac{1}{{\varepsilon }^{2}}, {p}_{7}={{\delta }_{2}}^{2},{p}_{8}={\lambda }_{1}{\lambda }_{2}{{\delta }_{2}}^{2}, {p}_{9}={{\lambda }_{1}}^{2},{p}_{10}=\frac{{{\delta }_{1}}^{2}}{Va}, {p}_{11}=\frac{Q{P}_{m}{{\delta }_{1}}^{2}{\pi }^{2}}{\varepsilon }, {p}_{12}=\left({\lambda }_{2}-{\lambda }_{1}\right){{\delta }_{2}}^{2},{p}_{13}=\frac{{{\delta }_{3}}^{2}+\tau }{{a}^{2}}, {p}_{14}=\frac{\gamma {\tau }^{2}\left({{\delta }_{4}}^{2}+\gamma \tau \right)}{{a}^{2}},$$

$$\begin{aligned} p_{{15}} = & Ra_{S} ,p_{{16 = }} \left( {\delta _{5} ^{2} + \chi } \right)^{2} , \\ p_{{17}} = & \chi + \delta _{3} ^{2} + \tau ,p_{{18}} = Ra_{S} \left( {\delta _{5} ^{2} + \chi } \right), \\ m_{1} = & p_{6} p_{8} \left( {p_{1} + p_{3} } \right),m_{2} = p_{2} p_{6} p_{8} , \\ m_{3} = & \left( {p_{4} p_{9} + p_{6} p_{7} + p_{5} p_{8} + p_{6} p_{8} p_{{18}} - p_{9} p_{6} p_{{18}} } \right)\left( {p_{1} + p_{3} } \right), \\ m_{4} = & p_{2} \left( {p_{4} p_{9} + p_{6} p_{7} + p_{5} p_{8} + p_{6} p_{8} p_{{16}} - p_{6} p_{9} p_{{18}} } \right), \\ m_{5} = & \left( {p_{1} + p_{3} } \right)\left( {p_{4} p_{9} p_{{16}} + p_{4} p_{{16}} + p_{5} p_{7} + p_{6} p_{7} p_{{16}} + p_{5} p_{8} p_{{16}} - p_{5} p_{9} p_{{18}} - p_{6} p_{{18}} } \right), \\ m_{6} = & p_{2} \left( {p_{4} p_{9} p_{{16}} + p_{4} p_{{16}} + p_{5} p_{7} + p_{6} p_{7} p_{{16}} + p_{5} p_{8} p_{{16}} - p_{5} p_{9} p_{{18}} - p_{6} p_{{18}} } \right), \\ m_{7} = & \left( {p_{1} + p_{3} } \right)\left( {p_{5} p_{7} p_{{16}} - p_{5} p_{{16}} } \right),m_{8} = p_{2} \left( {p_{5} p_{7} p_{{16}} - p_{5} p_{{16}} } \right), \\ m_{9} = & p_{2} p_{{13}} p_{6} p_{9} p_{{10}} ,m_{{10}} = \left( {p_{3} p_{{13}} - p_{{14}} } \right)p_{6} p_{9} p_{{10}} , \\ m_{{11}} = & p_{2} p_{{13}} \left( {p_{5} p_{9} p_{{10}} + p_{6} p_{{10}} - p_{9} p_{{11}} + p_{6} p_{{12}} } \right), \\ m_{{12}} = & \left( {p_{3} p_{{13}} - p_{{14}} } \right)\left( {p_{5} p_{9} p_{{10}} + p_{6} p_{{10}} - p_{9} p_{{11}} + p_{6} p_{{12}} } \right), \\ m_{{13}} = & p_{2} p_{{13}} \left( {p_{5} p_{{10}} - p_{{11}} + p_{5} p_{{12}} } \right), \\ m_{{14}} = & \left( {p_{3} p_{{13}} - p_{{14}} } \right)\left( {p_{5} p_{{10}} - p_{{11}} + p_{5} p_{{12}} } \right), \\ m_{{15}} = & p_{2} p_{{15}} p_{{17}} ,m_{{16}} = p_{3} p_{{15}} p_{{17}} - p_{{14}} p_{{15}} a^{2} . \\ \end{aligned}$$