Influence of activation energy and applied magnetic field on triple-diffusive quadratic mixed convective nanoliquid flow about a slender cylinder

Abstract

In many industrial processes such as paper manufacturing, optical fibre, nanowires, and coating, the process design engineers are concerned with efficient heat and mass transfer rates near bounding surfaces of the fluid machinery. The prime objective of the study is to analyse the effects of a binary chemical reaction and Arrhenius activation energy in a quadratic combined convective magneto nanofluid flow about a moving slender cylinder. In addition, the study comprises activation energies and binary chemical reactions for species diffusion, namely liquid hydrogen and oxygen diffusions, which are often employed as control mechanisms for efficient heating and cooling processes. The highly coupled nonlinear partial differential equations (NPDEs) with boundary constraints have been used to model the flow problem, which are then converted into a dimensionless set of equations by utilizing non-similar transformations. Further, the obtained set of NPDEs would be linearized via the quasilinearization technique and then numerically solved by the implicit finite difference method. The study’s interesting and important results are that the rising activation energy values increase the species concentration distributions and decrease the same for chemical reaction parameters. The augmenting values of the quadratic convection and thermophoresis characteristics enhance the nanoliquid’s velocity and temperature, respectively.

Appendix

$$ \begin{gathered} \begin{array}{*{20}c} {A_{1}^{i} = \frac{{\xi + f + \xi f_{\xi } }}{{(1 + \xi \eta )}};} & {\,A_{2}^{i} = - \frac{{(\xi F_{\xi } + 8\text{Re} M^{2} )}}{{(1 + \xi \eta )}};} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {A_{3}^{i} = - \frac{{\xi F}}{{(1 + \xi \eta )}};} & {A_{4}^{i} \, = \,\frac{{8Ri\,(1 + 2\gamma G)}}{{(1 + \xi \eta \,)}};} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {A_{5}^{i} = \frac{{8RiNc_{1} (1 + 2\beta _{{c1}} H)}}{{(1 + \xi \eta \,)}}\,;} & {A_{6}^{i} = \frac{{8RiNc_{2} (1 + 2\beta _{{c2}} P)}}{{(1 + \xi \eta \,)}}\,;} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {A_{7}^{i} = - \frac{{8RiNr}}{{(1 + \xi \,\eta )}}\,;} & {\,A_{8}^{i} \, = \,\,\, - \frac{1}{{(1 + \xi \,\eta )}}(8M^{2} \text{Re} \, + \xi F_{\xi } F + 8Ri(\gamma G^{2} + Nc_{1} \beta _{{c1}} H^{2} + Nc_{2} \beta _{{c2}} R^{2} ))\,;} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {B_{1}^{i} \, = \,\,\frac{\xi }{{(1\, + \,\xi \eta )}} + \,\,\frac{{(f\, + \,\xi \,f_{\xi } )}}{{(1\, + \,\xi \,\eta )}}\Pr + \,2\Pr Nt\,G_{\eta } + \Pr Nb\,S_{\eta } \,;\,\,} & {B_{2}^{i} \, = \,\, - \frac{{\xi \Pr F}}{{(1\, + \,\xi \,\eta )}}\,;\,} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {B_{3}^{i} \, = \, - \left\{ {\frac{{\,\xi Pr\,G_{\xi } \, + 4\Pr \text{Re} \,EcM^{2} (1 - F/2)}}{{(1 + \xi \eta )}}} \right\}\,;} & {B_{4}^{i} = \Pr Nb\,G_{\eta } \,;\,} \\ \end{array} \hfill \\ B_{5}^{i} \, = \,\,\Pr Nt\,G_{\eta }^{2} + \Pr Nb\,S_{\eta } G_{\eta } - \frac{{\xi \Pr \,G_{\xi } F}}{{(1 + \xi \eta )}} - \frac{{4\Pr \text{Re} EcM^{2} (1 - F^{2} /4)}}{{(1 + \xi \eta )}}; \hfill \\ \begin{array}{*{20}c} {C_{1}^{i} = \frac{\xi }{{(\,1\, + \,\xi \,\eta )}} + \frac{{Sc_{1} (f + \,\xi \,f_{\xi } )}}{{(\,1\, + \,\xi \,\eta )}};} & {C_{2}^{i} = - \frac{{4\text{Re} \,Kc_{1} Sc_{1} }}{{(\,1\, + \,\xi \,\eta )}}(1 + \lambda G)^{{n_{{\,1}} }} e^{{ - E\,_{1} /(1 + \lambda G)}} ;} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {C_{3}^{i} = - \,\,\frac{{\xi \,Sc_{1} }}{{(\,1\, + \,\xi \,\eta )}}\,\,F\,;} & {C_{4}^{i} = - \frac{{\xi Sc_{1} }}{{\left( {1 + \xi \eta } \right)}}H_{\xi } ;} \\ \end{array} \hfill \\ C_{5}^{i} = C_{4}^{i} F - \frac{{4\text{Re} \,Kc_{1} Sc_{1} }}{{(\,1\, + \,\xi \,\eta )}}(1 + \lambda G)^{{n_{{\,1}} }} e^{{ - E\,_{1} /(1 + \lambda G)}} \left[ {E_{1} + n_{{\,1}} (1 + \lambda G)} \right]; \hfill \\ \begin{array}{*{20}c} {D_{1}^{i} = \frac{\xi }{{(\,1\, + \,\xi \,\eta )}} + \frac{{Sc_{2} (f + \,\xi \,f_{\xi } )}}{{(\,1\, + \,\xi \,\eta )}};} & {D_{2}^{i} = - \frac{{4\text{Re} \,Kc_{2} Sc_{2} }}{{(\,1\, + \,\xi \,\eta )}}(1 + \lambda G)^{{n_{{\,2}} }} e^{{ - E\,_{2} /(1 + \lambda G)}} ;} \\ \end{array} \hfill \\ D_{3}^{i} = - \,\,\frac{{\xi \,Sc_{1} }}{{(\,1\, + \,\xi \,\eta )}}\,\,F\,; \hfill \\ \begin{array}{*{20}c} {D_{4}^{i} = - \frac{{\xi Sc_{1} }}{{\left( {1 + \xi \eta } \right)}}H_{\xi } ;} & {D_{5}^{i} = D_{4}^{i} F - \frac{{4\text{Re} \,Kc_{2} Sc_{2} }}{{(\,1\, + \,\xi \,\eta )}}(1 + \lambda G)^{{n_{{\,2}} }} e^{{ - E_{2} /(1 + \lambda G)}} \left[ {E_{2} + n_{{\,2}} (1 + \lambda G)} \right];} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {E_{1}^{i} = \,\,\frac{\xi }{{(1\, + \,\xi \eta )}} + \frac{{Le\,(f + \,\,\,\xi \,f_{\xi } )}}{{(1\, + \,\xi \eta )}};} & {E_{2}^{i} = - \,\,\frac{{\xi Le}}{{(1 + \xi \eta )}}\,\,F\,;} & {E_{3}^{i} = - \,\,\,\frac{{\xi \,Le}}{{(1 + \xi \eta )}}S_{\xi } \,;} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {E_{4}^{i} = \frac{{Nt}}{{Nb}}\frac{\xi }{{(1 + \xi \eta )}};} & {E_{5}^{i} = \frac{{Nt}}{{Nb}};} & {E_{6}^{i} = F_{3}^{i} F.} \\ \end{array} \hfill \\ \end{gathered} $$