Diffusion of liquid hydrogen in time-dependent MHD mixed convective flow

Abstract

An innovative study on liquid hydrogen diffusion in time-dependent mixed convection flow is carried out in the presence of magnetic field effects. In fact, this is the first approach to analyze such flow problems, and also this is the first research paper to study the non-uniform heat sink/source and nonlinear chemical reaction in the presence of liquid hydrogen diffusion. Initially, the governing equations are reduced to dimensionless form by using non-similar transformations and are linearized by applying quasilinearization technique. Then, the finite difference approximation is utilized to discretize the resulting equations. The mixed convection is analyzed along with exponentially stretching surface through various graphs on profiles as well as gradients. The results display that the non-uniform heat source parameter increases the fluid velocity as well as temperature, and the magnetic parameter reduces the friction at the wall. Specifically, the skin friction coefficient decreases about 40% in the presence of magnetic field. The mass transfer rate increases for high-order chemical reaction and for destructive chemical reaction rate. The mass transfer rate is found to be high for the diffusion of liquid nitrogen than that for the diffusion of liquid hydrogen. In fact, the mass transfer rate increases about 22% for the diffusion of liquid nitrogen. This study can assist the design engineers who are working in pertain to the diffusion of liquid gases in mixed convection regimes. Also, the obtained data in the present study can be more useful for future investigations about time-dependent mixed convection nanofluid flow problems.

Appendix

The coefficients in Eqs. (14)–(16) are defined as:

$$ J_{1}^{{\text{i}}} = \phi \left\{ {\left( {1 + \xi } \right)\frac{f}{2} + \xi f_{\xi } + \xi \tau \left( {f_{\tau } + \phi^{ - 1} \phi_{\tau } f} \right)} \right\}; $$

$$ J_{2}^{{\text{i}}} = - \xi \phi \left( {2\left( {1 + \tau \phi^{ - 1} \phi_{\tau } } \right)P + P_{\xi } + \tau P_{\tau } } \right) - \xi {\text{Re}}\;\phi^{ - 1} \phi_{\tau } - \xi M^{2} {\text{Re}}; $$

$$ J_{3}^{{\text{i}}} = - \,\,\xi \,\phi \,\,P\,; $$

$$ J_{4}^{{\text{i}}} = - \xi ({\text{Re}} + \phi \tau P); $$

$$ J_{5}^{{\text{i}}} = \xi \phi^{ - 1} \lambda ; $$

$$ J_{6}^{{\text{i}}} = - \xi \phi^{ - 1} \lambda N; $$

$$ J_{7}^{{\text{i}}} = - \xi \phi \left( {P + P_{\xi } + \tau P_{\tau } } \right)P - \xi \tau \phi_{\tau } \left( {P^{2} + \varepsilon^{2} } \right) - \xi \varepsilon \left( {\varepsilon \phi + {\text{Re}}\phi^{ - 1} \phi_{\tau } } \right) - \xi \varepsilon M^{2} {\text{Re}}; $$

$$ K_{1}^{{\text{i}}} = \phi { \Pr }\left\{ {(1 + \xi )\frac{f}{2} + \xi f_{\xi } + \xi \tau \left( {\phi^{ - 1} \phi_{\tau } f + f_{\tau } } \right)} \right\}; $$

$$ K_{2}^{{\text{i}}} = \phi (B^{*} - 2\xi { \Pr }P); $$

$$ K_{3}^{{\text{i}}} = - \xi \phi { \Pr }P; $$

$$ K_{4}^{{\text{i}}} = - \xi { \Pr }({\text{Re}} + \phi \tau P); $$

$$ K_{5}^{{\text{i}}} = - \xi \phi { \Pr }\left[ {2Q + Q_{\xi } + \tau Q_{\tau } + 2\phi M^{2} {\text{ReEc}}\left( {\varepsilon - P} \right)} \right] + A^{*} \phi ; $$

$$ K_{6}^{{\text{i}}} = - \xi \phi { \Pr }\left[ {2Q + Q_{\xi } + \tau Q_{\tau } } \right]P - \xi \phi^{2} M^{2} {\text{RePrEc}}\left( {\varepsilon^{2} - P^{2} } \right); $$

$$ L_{1}^{{\text{i}}} = \phi {\text{Sc}}\left\{ {(1 + \xi )\frac{f}{2} + \xi f_{\xi } + \xi \tau \left( {f_{\tau } + \phi^{ - 1} \phi_{\tau } f} \right)} \right\}; $$

$$ L_{2}^{{\text{i}}} = - \xi {\text{Sc}}\left\{ {e^{(2n - 3)\xi } {\text{Re}}\Delta nR^{n - 1} + 2\phi P} \right\}; $$

$$ L_{3}^{{\text{i}}} = - \xi \phi {\text{Sc}}P; $$

$$ L_{4}^{{\text{i}}} = - \xi Sc\left( {{\text{Re}} + \phi \tau P} \right); $$

$$ L_{5}^{{\text{i}}} = - \xi \phi {\text{Sc}}\left( {2R + R_{\xi } + \tau R_{\tau } } \right); $$

$$ L_{6}^{{\text{i}}} = - \xi \phi {\text{Sc}}\left( {2R + R_{\xi } + \tau R_{\tau } } \right)P + \xi e^{(2n - 3)\xi } {\text{Re}}\Delta {\text{Sc}}\left( {R^{n} - nR^{n - 1} } \right). $$