Question: Consider a flat plate surrounded by a fluid at rest (at rest outside the boundary layer) and oriented vertically to a gravity field of strength g. If the plate is heated to a temperature above that of the fluid, the fluid immediately adjacent to the plate will be heated, its density will decrease below that of the surrounding fluid, the resulting buoyancy force will put the fluid in motion, and a free-convection boundary layer will form. If the thermal expansion coefficient for the fluid is defined as
β = -1(∂ρ/∂T)p
ρ
develop the applicable momentum integral equation of the boundary layer under conditions where the density variation through the boundary layer is small relative to the free-fluid density.