3. This problem is a special case of a model for mountain winds driven by natural convection that was solved by Prandtl. The mountain is represented by an infinite vertical flat plate in a fluid whose dimensionless temperature T varies uniformly with height x:T=x. Here is a dimensionless temperature gradient. The dimensionless temperature Tw of the plate is maintained at a constant value 1 above the ambient temperature. You have to determine the velocity, temperature, and pressure that arise in this situation. a) Define a dimensionless temperature =Tx which is the difference between the fluid temperature, T, at any point (x,y) and the temperature at infinity at the same level x. For large Grashof numbers, Gr, the solution far from the mountain is u=0,=0. Determine the pressure, p(x,y) for this solution. Why is this solution not valid over all space? b) Next consider the natural convection boundary layer that forms near the plate. Write the steady dimensionless boundary layer equations for the x-component of velocity, u, and using the stretched boundary layer variables Y=Gr1/4yV=Gr1/4v. These equations admit a parallel flow solution, u=u(Y),V=0,=(Y). Obtain the simplified equations for this case. What are the boundary conditions for u and ? c) Show that the equations and boundary conditions are satisfied by the solutions u=2l21exp(lY)sin(lY),=exp(lY)cos(lY)wherel=(4Pr)1/4 d) For such a parallel flow solution how does the boundary layer thickness depend on x ? How does it depend on the parameters Gr and ? 3. This problem is a special case of a model for mountain winds driven by natural convection that was solved by Prandtl. The mountain is represented by an infinite vertical flat plate in a fluid whose dimensionless temperature T varies uniformly with height x:T=x. Here is a dimensionless temperature gradient. The dimensionless temperature Tw of the plate is maintained at a constant value 1 above the ambient temperature. You have to determine the velocity, temperature, and pressure that arise in this situation. a) Define a dimensionless temperature =Tx which is the difference between the fluid temperature, T, at any point (x,y) and the temperature at infinity at the same level x. For large Grashof numbers, Gr, the solution far from the mountain is u=0,=0. Determine the pressure, p(x,y) for this solution. Why is this solution not valid over all space? b) Next consider the natural convection boundary layer that forms near the plate. Write the steady dimensionless boundary layer equations for the x-component of velocity, u, and using the stretched boundary layer variables Y=Gr1/4yV=Gr1/4v. These equations admit a parallel flow solution, u=u(Y),V=0,=(Y). Obtain the simplified equations for this case. What are the boundary conditions for u and ? c) Show that the equations and boundary conditions are satisfied by the solutions u=2l21exp(lY)sin(lY),=exp(lY)cos(lY)wherel=(4Pr)1/4 d) For such a parallel flow solution how does the boundary layer thickness depend on x ? How does it depend on the parameters Gr and
