Magnetohydrodynamic mixed convection in a vertical rectangular duct with all walls of arbitrary conductivity

The magnetohydrodynamic (MHD) mixed convection in a rectangular cross-section of a long vertical duct is considered. The surrounding walls of the duct can be considered for a wide range of scenarios in this analytical solution, such as arbitrary conductivity, thickness and asymmetry. Analytical solutions are also obtained for various of the governing parameters: Grashof number ( $\mathop {\textit{Gr}}\nolimits$ ), Reynolds number ( $Re$ ), and Hartmann number ( $\mathop {\textit{Ha}}\nolimits$ ). Three convection states under varying ${{\mathop {\textit{Gr}}\nolimits }}/{{\mathop {\textit{Re}}\nolimits }}$ – forced convection, mixed convection and buoyancy-dominated convection – are investigated. When $ {{\mathop {\textit{Gr}}\nolimits }}/{{\mathop {\textit{Re}}\nolimits }}$ increases to a critical value $( {{\mathop {\textit{Gr}}\nolimits }}/{{\mathop {\textit{Re}}\nolimits }})_c$ , a reverse flow is observed and $({{\mathop {\textit{Gr}}\nolimits }}/{{\mathop {\textit{Re}}\nolimits }})_c$ is identified for both insulated and electrically conducting ducts. In MHD mixed convection, where $ ({{\mathop {\textit{Gr}}\nolimits }}/{{\mathop {\textit{Re}}\nolimits }}) \sim 1$ , the fully developed flow exhibits a steady velocity gradient in the core, scaling as $\sim ({{\mathop {\textit{Gr}}\nolimits }})/({2{\mathop {\textit{Ha}}\nolimits }{\mathop {\textit{Re}}\nolimits }})$ (Tagawa et al. 2002 Eur. J. Mech. B/Fluids 21 , 383–398) in the insulated scenario, and this work extends it to the electrically conducting scenario, scaling as $\sim ({{\mathop {\textit{Gr}}\nolimits }})/({2{\mathop {\textit{Re}}\nolimits }{\mathop {\textit{Ha}}\nolimits }(1 + c{\mathop {\textit{Ha}}\nolimits })})$ , where $c$ denotes the wall conductance ratio, accompanied by asymmetrical velocity jets. Effects of conductive walls on both pressure drop and flow distribution are thoroughly analysed. The pressure gradient distribution as a function of $\mathop {\textit{Ha}}\nolimits$ is given, in which the combined effect of arbitrary sidewalls and Hartmann walls on the distributions is well illustrated. The wall asymmetry has profound effects on the velocity distribution, especially for the high-velocity jet areas where Hartmann walls exert an opposite effect to that of sidewalls. The velocity magnitude is significantly larger around lower conducting sidewalls and raises questions about new potential instability schemes for high $\mathop {\textit{Re}}\nolimits$ , as discussed in previous studies (Krasnov et al. 2016 Numerical simulations of MHD flow transition…; Kinet et al. 2009 Phys. Rev. Lett. 103 , 154501).