Free stream 8 (x) T Free stream Thermal Velocity boundary layer boundary layer (a) (b) A differential energy analysis of this flow under common assumptions reduces the energy equation to du where T = T(r, y) is the temperature of the fluid in the boundary layer, u is the velocity parallel to the plate, v is the velocity perpendicular to the plate, a is the thermal diffusivity of the fluid, v is the kinematic viscosity, and c, is the specific heat of the fluid at constant pressure. a) Find the base dimensions, in [MLT0), where 0 is temperature, of a and , using the given equation. Recall that v has SI units of [m2s-). Do not simply use the internet, textbooks, or your prior knowledge to determine the dimensions of these variables. b) It is common to "shift" both the fluid temperature in the boundary layer, T, and the far-field fluid temperature, To, by considering these temperatures relative to the surface temperature of the plate, T,. This effectively reduces these 3 variables to 2 variables: x= T– T, and X = T-T,. Also notice that v and c, only show up as a ratio. It must then be the case that any solution to this problem can be written in terms of this ratio, and therefore we can treat v/c, as a single variable. Knowing this, show that 7 dimensionless variables are needed to fully describe the temperature of the flowing fluid within the boundary layer for a plate of length L. Neither 8(x) nor d(x) need to be in your variable list. c) Using L, u, and X, as repeating variables, find a suitable set of dimensionless variables to describe this problem. Write your result as T* = f(r", y", u", v*, A", B") where T* is the dimensionless shifted temperature; z*, y*, u*, and u* are relatively straightforward dimensionless variables; A* is a dimensionless variable the includes the thermal diffusivity; and B* is a dimensionless variable that includes the specific heat capacity. d) Nondimensionalize the differential equation using your results from part (c). e) The so-called "viscous dissipation" term (the second term on the RHS of the governing differential equation) is often small compared to the other terms, and is therefore frequently dropped from the differential equation. Under this approximation, show that the dimensionless differential equation can be written as +v*. Re,Pr dy-2 where Re, = "L is the familiar Reynolds number, and Pr = 4 is an important convection heat transfer parameter called the Prandtl number. This is the most common form of the dimensionless differential equation used to study this general class of convection problem.