Couple stress Rayleigh–Bénard convection in a square cavity

•Consistent couple stress theory is extended to Boussinesq thermal convection problem.•Momentum balance involves a new length scale parameter and fourth order derivatives.•Fluid element bending provides an additional energy dissipation mechanism.•Critical Rayleigh number for square cavity becomes size dependent at small scales.•Moment traction versus vorticity boundary conditions significantly affect stability.