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Role of Induced Magnetic Field on Hydromagnetic Mixed Convection Flow in Vertical Microannulus in Existence of Radial Magnetic Field

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Abstract

The present study presents the convective flow of an electrically conducting fluid in vertical microannulus formed by two concentric cylinders in the presence of imposed radial magnetic field and induced magnetic field. The governing equations of the motion are a set of simultaneous ordinary differential equations, and their exact solutions in dimensionless form have been obtained for the velocity field, the induced magnetic field and the temperature field. The expressions for the induced current density and skin friction have also been obtained. Results for various flows characteristic are presented through graphs and tables delineating the effect of various parameters characterizing the flow. Furthermore, it is observed that the reverse flow near the inner surface of outer cylinder is controlled by selecting suitable values of Knudsen number, fluid–wall interaction parameter and radius ratio.

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Abbreviations

\( C_{\rho 0} \) :

Specific heat at constant pressure

\( \ln \) :

Fluid–wall interaction parameter, \( {{\beta_{t} } \mathord{\left/ {\vphantom {{\beta_{t} } {\beta_{\nu } }}} \right. \kern-0pt} {\beta_{\nu } }} \)

\( g \) :

Gravitational acceleration

Gr :

Grashof number

\( Gr/Re \) :

Mixed convection parameter

\( k_{1} \) :

Radius of the inner cylinder

\( k_{2} \) :

Radius of the outer cylinder

\( Kn \) :

Knudsen number, \( {\lambda \mathord{\left/ {\vphantom {\lambda w}} \right. \kern-0pt} w} \)

\( M \) :

Magnetic field parameter

\( q \) :

Volume flow rate

\( Q \) :

Dimensionless volume flow rate

\( Pr \) :

Prandtl number

\( r^{'}\) :

Dimensional radial coordinate

\( R \) :

Dimensionless radial coordinate

\( \mathop R\limits^{ \wedge } \) :

Specific gas constant

Re :

Reynolds number

\( T \) :

Temperature of fluid

\( T_{0} \) :

Reference temperature

\( T_{1} \) :

Temperature at outer surface of the inner cylinder

\( u \) :

Axial velocity

\( u_{\text{m}} \) :

Mean velocity

U :

Dimensionless axial velocity

\( w \) :

Dimensional gap between the cylinders

\( \sigma_{t} ,\sigma_{v} \) :

Thermal and tangential momentum accommodation coefficients, respectively

\( {{{\text{d}}P^{'}} \mathord{\left/ {\vphantom {{{\text{d}}P^{'}} {{\text{d}}Z^{'}}}} \right. \kern-0pt} {{\text{d}}Z^{'}}} \) :

Pressure gradient along the axis of the microannulus

\( {{{\text{d}}P} \mathord{\left/ {\vphantom {{{\text{d}}P} {{\text{d}}Z}}} \right. \kern-0pt} {{\text{d}}Z}} \) :

Dimensionless pressure gradient along the axis of the microannulus

\({Z^{'}}, {r^{'}}\) :

Axial and radial coordinates, respectively

\( Z,R \) :

Dimensionless axial and radial coordinates, respectively

\( \alpha \) :

Thermal diffusivity

\( \beta_{0} \) :

Coefficient of thermal expansion

\( \beta_{t} ,\beta_{v} \) :

Dimensionless variables

γ :

Ratio of specific heats

\( \mu_{0} \) :

Dynamic viscosity

\( \theta \) :

Dimensionless temperature

ρ 0 :

Density

\( \nu \) :

Fluid kinematic viscosity \( \left( {{{\mu_{0} } \mathord{\left/ {\vphantom {{\mu_{0} } {\rho_{0} }}} \right. \kern-0pt} {\rho_{0} }}} \right) \)

η :

Ratio of radii \( \left( {{{k_{1} } \mathord{\left/ {\vphantom {{k_{1} } {k_{2} }}} \right. \kern-0pt} {k_{2} }}} \right) \)

\( \lambda \) :

Molecular mean free path

\( k_{0} \) :

Thermal conductivity

τ:

Skin friction

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Authors and Affiliations

  1. Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria

    Basant K. Jha

  2. Department of Mathematics, Bingham University, Abuja, Nigeria

    Babatunde Aina

Authors
  1. Basant K. Jha
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  2. Babatunde Aina
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Correspondence to Babatunde Aina.

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Appendix

Appendix

$$ \begin{array}{*{20}l} {D_{1} = \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{1} }}{{\left( {4 - \eta^{2} } \right)}}\left[ {\frac{{4\eta^{2} }}{{\left( {4 - \eta^{2} } \right)}} - \eta^{2} \ln \left( \eta \right)} \right] - \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{0} \eta^{2} }}{{\left( {4 - \eta^{2} } \right)}},} \hfill \\ {D_{2} = \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{1} }}{{\left( {4 - \eta^{2} } \right)}}\left[ {\frac{8\eta }{{\left( {4 - \eta^{2} } \right)}} - 2\eta \ln \left( \eta \right) - \eta } \right] - \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{2A_{0} \eta }}{{\left( {4 - \eta^{2} } \right)}},} \hfill \\ {D_{3} = \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{1} }}{{\left( {4 - \eta^{2} } \right)}}\frac{4}{{\left( {4 - \eta^{2} } \right)}} - \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{0} }}{{\left( {4 - \eta^{2} } \right)}},\quad D_{5} = \frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{1} }}{{\left( {4 - \eta^{2} } \right)}},} \hfill \\ {D_{4} = \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{1} }}{{\left( {4 - \eta^{2} } \right)}}\left[ {\frac{8}{{\left( {4 - \eta^{2} } \right)}} - 1} \right] - \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{2A_{0} }}{{\left( {4 - \eta^{2} } \right)}},} \hfill \\ {D_{6} = \left( \eta \right)^{\delta } - \beta_{v} Kn\left( {1 - \eta } \right)\delta \left( \eta \right)^{\delta - 1} ,\quad D_{7} = \left( \eta \right)^{ - \delta } + \beta_{v} Kn\left( {1 - \eta } \right)\delta \left( \eta \right)^{ - \delta - 1} ,} \hfill \\ {D_{8} = \eta^{2} D_{5} - \beta_{v} Kn\left( {1 - \eta } \right)2\eta D_{5} ,\quad D_{9} = \beta_{v} Kn\left( {1 - \eta } \right)D_{2} - D_{1} ,\quad D_{10} = 1 + \beta_{v} Kn\left( {1 - \eta } \right)\delta ,} \hfill \\ {D_{11} = 1 - \beta_{v} Kn\left( {1 - \eta } \right)\delta ,\quad D_{12} = D_{5} + \beta_{v} Kn\left( {1 - \eta } \right)2D_{5} ,\quad D_{13} = - \beta_{v} Kn\left( {1 - \eta } \right)D_{4} - D_{3} ,} \hfill \\ {D_{14} = D_{6} D_{11} - D_{7} D_{10} ,\quad D_{15} = D_{8} D_{11} - D_{7} D_{12} ,\quad D_{16} = D_{9} D_{11} - D_{7} D_{13} ,} \hfill \\ {D_{17} = D_{7} \left( {1 - \eta } \right) - \left( {1 - \eta } \right)D_{11} ,\quad D_{18} = {{D_{16} } \mathord{\left/ {\vphantom {{D_{16} } {D_{14} }}} \right. \kern-0pt} {D_{14} }},\quad D_{19} = {{D_{17} } \mathord{\left/ {\vphantom {{D_{17} } {D_{14} }}} \right. \kern-0pt} {D_{14} }},\quad D_{20} = {{D_{15} } \mathord{\left/ {\vphantom {{D_{15} } {D_{14} }}} \right. \kern-0pt} {D_{14} }},} \hfill \\ {D_{21} = D_{7} D_{10} - D_{6} D_{11} ,\quad D_{22} = D_{8} D_{10} - D_{6} D_{12} ,\quad D_{23} = D_{9} D_{10} - D_{6} D_{13} ,} \hfill \\ {D_{24} = D_{6} \left( {1 - \eta } \right) - \left( {1 - \eta } \right)D_{10} ,\quad D_{25} = {{D_{23} } \mathord{\left/ {\vphantom {{D_{23} } {D_{21} }}} \right. \kern-0pt} {D_{21} }},\quad D_{26} = {{D_{24} } \mathord{\left/ {\vphantom {{D_{24} } {D_{21} }}} \right. \kern-0pt} {D_{21} }},\quad D_{27} = {{D_{22} } \mathord{\left/ {\vphantom {{D_{22} } {D_{21} }}} \right. \kern-0pt} {D_{21} }},} \hfill \\ {D_{28} = \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{1} }}{{\left( {4 - \eta^{2} } \right)}}\left[ {\frac{{2\eta^{2} }}{{\left( {4 - \eta^{2} } \right)}} - \frac{{\eta^{2} \ln \left( \eta \right)}}{2} + \frac{{\eta^{2} }}{4}} \right] - \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{0} \eta^{2} }}{{2\left( {4 - \eta^{2} } \right)}},} \hfill \\ {D_{29} = \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{1} }}{{\left( {4 - \eta^{2} } \right)}}\left[ {\frac{2}{{\left( {4 - \eta^{2} } \right)}} + \frac{1}{4}} \right] - \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{0} }}{{\left( {4 - \eta^{2} } \right)}},\quad D_{30} = 1 - \eta^{\delta } ,\quad D_{31} = \eta^{ - \delta } - 1,} \hfill \\ {D_{32} = \delta D_{29} - \delta D_{28} ,\quad D_{33} = \frac{{\delta D_{5} }}{2} - \frac{{\delta \eta^{2} D_{5} }}{2},\quad D_{34} = D_{19} D_{30} + D_{26} D_{31} ,} \hfill \\ {D_{35} = D_{33} - D_{20} D_{30} - D_{27} D_{31} ,} \hfill \\ {D_{36} = D_{18} D_{30} + D_{25} D_{31} + D_{32} ,\quad D_{37} = - {{D_{35} } \mathord{\left/ {\vphantom {{D_{35} } {D_{34} }}} \right. \kern-0pt} {D_{34} }},\quad D_{37} = - {{D_{35} } \mathord{\left/ {\vphantom {{D_{35} } {D_{34} }}} \right. \kern-0pt} {D_{34} }},\quad D_{39} = {1 \mathord{\left/ {\vphantom {1 {\delta + 2,}}} \right. \kern-0pt} {\delta + 2,}}} \hfill \\ {D_{40} = {{ - 1} \mathord{\left/ {\vphantom {{ - 1} {\delta + 2,}}} \right. \kern-0pt} {\delta + 2,}}\quad D_{41} = \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{0} }}{{4\left( {4 - \delta^{2} } \right)}},\quad D_{42} = \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} }}\frac{{A_{1} }}{{4\left( {4 - \delta^{2} } \right)}},} \hfill \\ {D_{43} = \frac{Gr}{Re}\frac{1}{{\left( {1 - \eta } \right)^{2} 4\left( {4 - \delta^{2} } \right)}},\quad D_{44} = D_{39} - D_{39} \left( \eta \right)^{\delta + 2} ,\quad D_{45} = D_{40} - D_{40} \left( \eta \right)^{ - \delta + 2} ,} \hfill \\ {D_{46} = \frac{1 - \eta }{2} - \frac{{\eta^{2} - \eta^{3} }}{2},\quad D_{47} = \eta^{4} - 1,\quad D_{48} = \frac{1}{{4 - \delta^{2} }} + \frac{1}{16} - \frac{{\eta^{2} }}{{4 - \delta^{2} }} + \frac{{\eta^{4} \ln \left( \eta \right)}}{4} - \frac{{\eta^{4} }}{16},} \hfill \\ {D_{49} = D_{43} - D_{43} \left( \eta \right)^{4} ,\quad D_{50} = \frac{{1 - \eta^{2} }}{2},\quad D_{51} = D_{37} D_{19} - D_{20} ,\quad D_{52} = D_{38} D_{19} + D_{18} ,} \hfill \\ \begin{aligned} D_{53} = D_{37} D_{26} - D_{27} ,\quad D_{54} = D_{38} D_{26} + D_{25} ,\quad D_{55} = D_{51} D_{44} + D_{53} D_{45} + D_{37} D_{46} + D_{49} , \hfill \\ D_{56} = D_{52} D_{44} + D_{54} D_{45} + D_{38} D_{46} + D_{41} D_{47} + D_{42} D_{48} , \hfill \\ \end{aligned} \hfill \\ \end{array} $$

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Jha, B.K., Aina, B. Role of Induced Magnetic Field on Hydromagnetic Mixed Convection Flow in Vertical Microannulus in Existence of Radial Magnetic Field. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 90, 259–269 (2020). https://doi.org/10.1007/s40010-018-0586-3

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