Numerical study of chemical reaction effects in magnetohydrodynamic Oldroyd-B: oblique stagnation flow with a non-Fourier heat flux model

Abstract

Reactive magnetohydrodynamic flows arise in many areas of nuclear reactor transport. Working fluids in such systems may be either Newtonian or non-Newtonian. Motivated by these applications, in the current study, a mathematical model is developed for electrically conducting viscoelastic oblique flow impinging on stretching wall under transverse magnetic field. A non-Fourier Cattaneo–Christov model is employed to simulate thermal relaxation effects which cannot be simulated with the classical Fourier heat conduction approach. The Oldroyd-B non-Newtonian model is employed which allows relaxation and retardation effects to be included. A convective boundary condition is imposed at the wall invoking Biot number effects. The fluid is assumed to be chemically reactive and both homogeneous–heterogeneous reactions are studied. The conservation equations for mass, momentum, energy and species (concentration) are altered with applicable similarity variables and the emerging strongly coupled, nonlinear non-dimensional boundary value problem is solved with robust well-tested Runge–Kutta–Fehlberg numerical quadrature and a shooting technique with tolerance level of 10⁻⁴. Validation with the Adomian decomposition method is included. The influence of selected thermal (Biot number, Prandtl number), viscoelastic hydrodynamic (Deborah relaxation number), Schmidt number, magnetic parameter and chemical reaction parameters, on velocity, temperature and concentration distributions are plotted for fixed values of geometric (stretching rate, obliqueness) and thermal relaxation parameter. Wall heat transfer rate (local heat flux) and wall species transfer rate (local mass flux) are also computed and it is observed that local mass flux increases with strength of heterogeneous reactions whereas it decreases with strength of homogeneous reactions. The results provide interesting insights into certain nuclear reactor transport phenomena and furthermore a benchmark for more general CFD simulations.

Graphical abstract

Appendix

Introducing Eq. (31) into Eqs. (23–30) and eliminating the pressure term we have:

$$ \begin{aligned} & 2\frac{{\partial^{2} \zeta }}{{\partial x_{1}^{2} }}\frac{{\partial^{2} \zeta }}{{\partial x_{1} \partial x_{2} }} + \frac{\partial \zeta }{{\partial x_{2} }}\left( {\frac{{\partial^{3} \zeta }}{{\partial x_{1}^{3} }} + \frac{{\partial^{3} \zeta }}{{\partial x_{1} \partial x_{2}^{2} }}} \right) + \frac{\partial \zeta }{{\partial x_{1} }}\left( {\frac{{\partial^{3} \zeta }}{{\partial x_{2} \partial x_{1}^{2} }} - \frac{{\partial^{3} \zeta }}{{\partial x_{2}^{3} }}} \right) - \nabla^{4} \zeta + \beta_{1} \left\{ {2\frac{{\partial^{3} \zeta }}{{\partial x_{2} \partial x_{1}^{2} }}\left( {\frac{\partial \zeta }{{\partial x_{2} }}\frac{{\partial^{2} \zeta }}{{\partial x_{2}^{2} }} - \frac{\partial \zeta }{{\partial x_{2} }}\frac{{\partial^{2} \zeta }}{{\partial x_{1}^{2} }} - \frac{\partial \zeta }{{\partial x_{1} }}\frac{{\partial^{2} \zeta }}{{\partial x_{1} \partial x_{2} }}} \right) + 2\frac{{\partial^{3} \zeta }}{{\partial x_{1} \partial x_{2}^{2} }}\left( {\frac{\partial \zeta }{{\partial x_{1} }}\frac{{\partial^{2} \zeta }}{{\partial x_{1}^{2} }} - \frac{\partial \zeta }{{\partial x_{1} }}\frac{{\partial^{2} \zeta }}{{\partial x_{2}^{2} }} - \frac{\partial \zeta }{{\partial x_{2} }}\frac{{\partial^{2} \zeta }}{{\partial x_{1} \partial x_{2} }}} \right) + \left( {\frac{\partial \zeta }{{\partial x_{1} }}} \right)^{2} \left( {\frac{{\partial^{4} \zeta }}{{\partial x_{2}^{4} }} + \frac{{\partial^{4} \zeta }}{{\partial x_{1}^{2} \partial x_{2}^{2} }}} \right) + \left( {\frac{\partial \zeta }{{\partial x_{2} }}} \right)^{2} \left( {\frac{{\partial^{4} \zeta }}{{\partial x_{1}^{4} }} + \frac{{\partial^{4} \zeta }}{{\partial x_{1}^{2} \partial x_{2}^{2} }}} \right) + 2\frac{\partial \zeta }{{\partial x_{1} }}\left( {\frac{{\partial^{3} \zeta }}{{\partial x_{2}^{3} }}\frac{{\partial^{2} \zeta }}{{\partial x_{1} \partial x_{2} }} - \frac{\partial \zeta }{{\partial x_{2} }}\frac{{\partial^{4} \zeta }}{{\partial x_{2} \partial x_{1}^{3} }} - \frac{\partial \zeta }{{\partial x_{2} }}\frac{{\partial^{4} \zeta }}{{\partial x_{1} \partial x_{2}^{3} }}} \right)} \right\} - \beta_{2} \left\{ {2\left( {\frac{{\partial^{2} \zeta }}{{\partial x_{2}^{2} }} - \frac{{\partial^{2} \zeta }}{{\partial x_{1}^{2} }}} \right)\left( {\frac{{\partial^{4} \zeta }}{{\partial x_{1} \partial x_{2}^{3} }} + \frac{{\partial^{4} \zeta }}{{\partial x_{2} \partial x_{1}^{3} }}} \right) - \frac{\partial \zeta }{{\partial x_{1} }}\left( {\frac{{\partial^{5} \zeta }}{{\partial x_{2}^{5} }} + \frac{{\partial^{5} \zeta }}{{\partial x_{2} \partial x_{1}^{4} }} + 2\frac{{\partial^{5} \zeta }}{{\partial x_{1}^{2} \partial x_{2}^{3} }}} \right) + \frac{\partial \zeta }{{\partial x_{2} }}\left( {\frac{{\partial^{5} \zeta }}{{\partial x_{1}^{5} }} + \frac{{\partial^{5} \zeta }}{{\partial x_{1} \partial x_{2}^{4} }} + 2\frac{{\partial^{5} \zeta }}{{\partial x_{1}^{3} \partial x_{2}^{2} }}} \right) - 2\frac{{\partial^{2} \zeta }}{{\partial x_{1} \partial x_{2} }}\left( {\frac{{\partial^{4} \zeta }}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + \frac{{\partial^{4} \zeta }}{{\partial x_{2}^{4} }}} \right) - 2\frac{{\partial^{3} \zeta }}{{\partial x_{2} \partial x_{1}^{2} }}\left( {\frac{{\partial^{3} \zeta }}{{\partial x_{1}^{3} }} + \frac{{\partial^{3} \zeta }}{{\partial x_{1} \partial x_{2}^{2} }} + \frac{\partial \zeta }{{\partial x_{1} }}\frac{{\partial^{5} \zeta }}{{\partial x_{2} \partial x_{1}^{4} }}} \right)} \right\} + M\left\{ {\frac{{\partial^{2} \zeta }}{{\partial x_{2}^{2} }} - \beta_{1} \left( {\frac{{\partial^{2} \zeta }}{{\partial x_{2}^{2} }}\frac{{\partial^{2} \zeta }}{{\partial x_{1} \partial x_{2} }} + \frac{\partial \zeta }{{\partial x_{1} }}\frac{{\partial^{3} \zeta }}{{\partial x_{2}^{3} }}} \right)} \right\} \\ & - \lambda \left\{ {\frac{\partial T}{{\partial x_{2} }} + \beta_{1} \left( {\frac{{\partial^{2} \zeta }}{{\partial x_{2}^{2} }}\frac{\partial T}{{\partial x_{2} }} + \frac{\partial \zeta }{{\partial x_{2} }}\frac{{\partial^{2} T}}{{\partial x_{1} \partial x_{2} }} - 2\frac{{\partial^{2} \zeta }}{{\partial x_{1} \partial x_{2} }}\frac{\partial T}{{\partial x_{2} }} - \frac{{\partial^{2} T}}{{\partial x_{2}^{2} }}\frac{\partial \zeta }{{\partial x_{1} }} - T\frac{{\partial^{3} \zeta }}{{\partial x_{1} \partial x_{2}^{2} }}} \right)} \right\} = 0, \\ \end{aligned} $$

(61)

$$ P_{r} \left[ {\frac{\partial \zeta }{{\partial x_{2} }}\frac{\partial T}{{\partial x_{1} }} - \frac{\partial \zeta }{{\partial x_{1} }}\frac{\partial T}{{\partial x_{2} }} + \beta_{2} \left\{ {\left( {\frac{\partial \zeta }{{\partial x_{2} }}} \right)^{2} \frac{{\partial^{2} T}}{{\partial x_{1}^{2} }} + \frac{\partial \zeta }{{\partial x_{1} }}\frac{{\partial^{2} T}}{{\partial x_{2}^{2} }} - 2\frac{\partial \zeta }{{\partial x_{1} }}\frac{\partial \zeta }{{\partial x_{2} }}\frac{{\partial^{2} T}}{{\partial x_{1} \partial x_{2} }} + \left( {\frac{\partial \zeta }{{\partial x_{2} }}\frac{{\partial^{2} \zeta }}{{\partial x_{1} \partial x_{2} }} - \frac{\partial \zeta }{{\partial x_{1} }}\frac{{\partial^{2} \zeta }}{{\partial x_{2}^{2} }}} \right)\frac{\partial T}{{\partial x_{1} }} + \left( {\frac{\partial \zeta }{{\partial x_{1} }}\frac{{\partial^{2} \zeta }}{{\partial x_{1} \partial x_{2} }} - \frac{\partial \zeta }{{\partial x_{2} }}\frac{{\partial^{2} \zeta }}{{\partial x_{2}^{2} }}} \right)\frac{\partial T}{{\partial x_{2} }}} \right\}} \right] = \frac{{\partial^{2} T}}{{\partial x_{2}^{2} }}, $$

(62)

$$ - \frac{\partial \zeta }{{\partial x_{1} }}j^{\prime } \left( {x_{2} } \right) = \frac{1}{{S_{c} }}j^{{\prime \prime }} \left( {x_{2} } \right) - k_{1} j\left( {x_{2} } \right)s^{2} \left( {x_{2} } \right), $$

(63)

$$ - \frac{\partial \zeta }{{\partial x_{1} }}s^{\prime } \left( {x_{2} } \right) = \frac{\delta }{Sc}s^{\prime \prime } \left( {x_{2} } \right) + k_{1} j\left( {x_{2} } \right)s^{2} \left( {x_{2} } \right), $$

(64)

$$ \left. {\begin{array}{*{20}l} {\frac{\partial \zeta }{{\partial x_{2} }} = x_{1} , \frac{\partial \zeta }{{\partial x_{1} }} = 0, \frac{\partial T}{{\partial x_{2} }} = - Bi\left( {1 - T} \right),} \hfill \\ {D_{A} j^{\prime } \left( {x_{2} } \right) = k_{s} \sqrt {\frac{\nu }{c}} j\left( {x_{2} } \right), D_{B} s^{\prime } \left( {x_{2} } \right) = - k_{s} \sqrt {\frac{\nu }{c}} j\left( {x_{2} } \right),} \hfill \\ \end{array} } \right\} {\text{at}}\,x_{2} = 0, $$

(65)

$$ \frac{\partial \zeta }{{\partial x_{2} }} = \frac{a}{c}x_{1} + \gamma_{1} x_{2} , T = 0, j\left( {x_{2} } \right) \to 1,s\left( {x_{2} } \right) \to 0, \hbox{at} \quad x_{2} \to \infty . $$

(66)