Boundary-layer physics

We recall from fluid dynamics that we expect boundary-layers to occur within the fluid close to solid surfaces at high Reynolds numbers.  The current interest where the Grashof number is large is analogous.  In a momentum boundary-layer the inertia and viscous forces are about the same order of magnitude.  The governing equation thus contains the dominant inertia and viscous terms.  As the Reynolds number is increased, the boundary layer gets thinner as δ/L ~ 1/Re^1/2.  This scaling for this can be determined from the form of the equations with out solving any differential equations.  The drag on the plate can be determined to order one accuracy simply by knowing the boundary layer thickness, δ, and calculating τ = μ du/dy =~ μ U/δ.  

You may not be familiar with heat transfer boundary layer behavior, but a lot of similarities occur.  First now the conduction and convection terms are the same order of magnitude.  Thus convective heat transfer involves both conduction and convection.  You will notice below that the convection-conduction boundary-layer equation for energy, written in terms of T, looks quite similar to the flow direction boundary-layer equation for momentum.  The primary difference is that the energy equation does not have a term like dp/dx, unless there is a heat source within the boundary layer.  This is a source term and in the case of fluid flow is the reason the fluid is flowing (a momentum source).  

Another similarity between the present problem and fluid flow boundary layers is that we can get the basic scaling of the solution from the form of the equations.  Thus once we have the equations, can figure out how to nondimensionalize them and finally turn the PDE's into ODE's, we will know how the Nusselt number scales with Grashof number without solving the equations!

Strictly speaking boundary-layer theory is the first order result of some clever "perturbation" expansions of the differential equations.  Leal shows this in his book, ( L. G. Leal  (1992) Laminar Flow and Convective Transport Processes,  Butterworth.)  He also shows that for a flow around a sphere, or heat transfer from a sphere, the governing equations at the lowest order (which is what we are solving) are the same for a sphere or a flat plate.  The main reason for this is because the layer is very thin compared to the radius of the sphere.  Thus "the earth is flat"! -- or at least curvature does not matter.  

If we derive the continuity, momentum and energy equations for flow past a heated flat plate we get