Assuming a linear velocity distribution and a linear temperature distribution in the boundary layer over a flat plate, derive a relation between the thermal and hydrodynamic boundary-layer thicknesses and the Prandtl number.

Question

GIVEN

Boundary layer over a flat plate

FIND

A relation between the thermal and hydrodynamic boundary-layer thicknesses and the Prandtl number

ASSUMPTIONS

Linear velocity and temperature distributions in the boundary layers

SKETCH

Answer 1

Let Absolute viscosity of the fluid = ?

Plate surface temperature = Ts

Bulk fluid temperature = T??

Bulk fluid viscosity = U??

Density of the fluid = p?

Thermal diffusivity of the fluid = a

The linear velocity profile will be used to solve the integral momentum equation first. The integral energy equation will then be solved and combined with the momentum solution.



Substituting this into the integral momentum equation for a laminar boundary layer (Equation 5.42)



(The wall shear stress (Tw) is defined by Equation (4.2))

In this case, du/dy = constant = U?/?

Integrating



Integrating



Linear temperature profile: T = To + by

Subject to T = Ts at y = 0 ? To = Ts



Substituting this and the expression for U into the integral energy equation of the laminar boundary layer for low speed flow (Equation 5.44)



Integrating



Substituting Equation [1] into this expression