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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 
qis 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
 

  
 

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