In a particular application involving fluid flow at a rate m` through a circular tube of length L and diameter D, the surface heat flux is known to have a sinusoidal variation with x, which is of the form q″s(x) = q″ₛ,ₘ sin(πx/L). The maximum flux, q″ₛ,ₘ, is a known constant, and the fluid enters the tube at a known temperature, Tₘ,ᵢ.
Assuming the convection coefficient to be constant, how do the mean temperature of the fluid and the surface temperature vary with x? i. e. determine the equation for Tₘ(x) and Tₛ(x).