The Entrainment Theorem for the Darcy free convection over a permeable vertical plate

The Cheng–Minkowycz model of the Darcy free convection boundary layer flow over a permeable vertical plate with prescribed power-law temperature distribution Tw(x)=T∞+A⋅xλ and an applied lateral mass flux is revisited in this paper. The relationship between the wall heat flux and the entrainment velocity (the similar transversal velocity at the outer edge of the boundary layer) as function of the mass transfer parameter fw is examined analytically by using the Merkin transformation method. It is shown that at the value of fw where the Nusselt number becomes zero and changes sign, the entrainment velocity passes through its minimum value (Entrainment Theorem). The converse statement is also true, and holds for all the surface temperature distributions with power-law exponent in the range −1<λ<0. It also applies to the Darcy free convection over a permeable vertical plate with exponential temperature distribution when the effect of viscous dissipation is significant.