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Effect of local thermal non-equilibrium (LTNE) on onset of Brinkman-Bénard convection and on heat transport is investigated. Rigid-rigid and free-free, isothermal boundaries are considered for investigation. The assumption of LTNE leads to an 'advanced onset' situation compared to that predicted by the local thermal equilibrium (LTE) assumption. This results in the 'enhanced heat transport' situation in the problem. Asymptotic analysis for small and large values of inter-phase heat transfer coefficient is also carried out on critical Rayleigh number, critical wave number and Nusselt number. In respect of boundary influences on onset and heat transport, it is found that classical results hold even under the LTNE assumption. The other parameters' influences on onset and heat transport are qualitatively similar in LTNE and LTE cases. Keywords Brinkman-Bénard convection • Porous media • Rigid-rigid • Free-free • Monodisperse • LTNE • Nonlinear List of symbols Latin symbols A,B,C Amplitudes of linear regime (m) c p Specific heat at constant pressure (Jkg −1 K −1) d Channel depth (m) D, E Amplitudes of nonlinear regime (m) g Acceleration due to gravity (ms −2) h Inter-phase heat transfer coefficient (Wm −2 K −1

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The effect of local thermal nonequilibrium (LTNE) on the entropy generation and heat transfer characteristics in the magnetohydrodynamic flow of a couple‐stress fluid through a high‐porosity vertical channel is studied numerically using the higher‐order Galerkin technique. The Boussinesq approximation is assumed to be valid and the porous medium is considered to be isotropic and homogeneous. Two energy equations are considered one each for solid and fluid phases. The term involving the heat transfer coefficient in both equations renders them mutually coupled. Thermal radiation and an internal heat source are considered only in the fluid phase. The influence of inverse Darcy number, Hartmann number, couple‐stress fluid parameter, Grashof number, thermal radiation parameter, and interphase heat transfer coefficient on velocity and temperature profiles is depicted graphically and discussed. The entropy generation, friction factor, and Nusselt number are determined, and outcomes are presented via plots. The effect of LTNE on the temperature profile is found to cease when the value of the interphase heat transfer coefficient is high, and in this case, we get the temperature profiles of fluid and solid phases are uniform. The physical significance of LTNE is discussed in detail for different parameters' values. It is found that heat transport and friction drag are maximum in the case of LTNE and minimum in the case of local thermal equilibrium. We observe that LTNE opposes the irreversibility of the system. The corresponding results of a fluid‐saturated densely packed porous medium can be obtained as a limiting case of the current study.