Consider a pipe in a cylindrical coordinate. The flow is a fully developed laminar flow and the temperature field is T(x, r, θ). At a certain cross section x_0, the mean temperature is T_m and the mean velocity is u_m (To calculate T_m or u_m, integrate the temperature or velocity over the whole cross section and divide the result by the area). A conjecture is that u*partial T/partial x=u_m*dT/dx, where partial denotes the partial derivative. This seems reasonable since the heat transfer should be easier where fluid flows faster, which leads to a smaller gradient. But how can I prove it strictly?
A small reason can be given as follows:
Given that flow is fully developed & laminar .
If it is a incompressible Newtonian flow then it's density and dynamic viscosity is constant.
Dynamic Viscosity depends on temperature rather than pressure.
Consequently we may conclude that there is no change in temperature.
( I request Dr. Hong Zhu and other research scholars to kindly comment whether my answer is right or wrong)
There is no generalized argument regarding the exact correlation between fluid flow rate and temperature gradient.
If we apply the Chilton-Colburn analogy for the flow in a tube, then we will get a relationship as:
(Sh/Nu)^3 = (Sc/Pr),
where Sh is the Sherwood number, Nu is the Nusselt number, Sc is the Schmidt number, and Pr is the Prandtl number.
Further, in the context of heat transfer, Pr = Pe/Re, where Pe is the Peclet number and Re is the Reynolds number.
So Re = (Pe/Sc)*(Sh/Nu)^3, it means the correlation is going to depend on various phenomena, whether the flow is influenced by convective (advective) forces or by viscous forces, the type of fluid and its rheology, mode of heat transfer etc.
You can find the answer on your question in the treatise of Schlichting, chapter Thermal boundary layer.
Sorry about the question. I found just now that I had misunderstood the textbook.
What I really want to prove is that partial T/partial x=dT/dx. If a dimensionless temperature Theta is defined as λ(T-T_m)/(q_W*R), it yields that partial Theta/partial x=0 due to the fully developed condition, which follows that partial T/partial x=dT/dx.
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