For laminar flow in a tube with a constant wall heat flux, it can be shown that the
microscopic energy equation can be reduced to the following form where qy is the
heat flux in the y direction
In the above equation, v0 is the fluid velocity, which is assumed to be linear and
constant, R is the tube radius and α is the thermal diffusivity.
(a) Transform the above equation to an equivalent form by introducing the
following dimensionless variables
(b) Next, introduce a new independent variable called a similarity variable into the
equation derived in the previous step defined as
Show that the following transformed ordinary differential equation emerges from this
analysis.
(c) Suppose that the boundary conditions are
Using these boundary conditions, show that the solution to the transformed ODE
shown in step (b) is:
here x is a dummy variable for integration and Γ is the gamma function. Refer to
Chapter 4 in the text for the definition of the gamma function.
(d) Starting with the definition for the heat flux
Show that the temperature profile can be obtained by integrating the heat flux
Then, show that the following dimensionless form can be derived from the above
relationship
