Heat Transfer in Channel. Account of Dissipation

Let us consider the problem of dissipative heating of a fluid in a plane channel of width 2h with isothermal walls on which the same constant temperature is 

Mass and Heat Transfer Under Complicating Factors 

If the fluid temperature at the entry cross-section is equal to the wall temperature Ts, then, along some initial part of the tube, the fluid is gradually heated by internal friction until a balance is achieved between the heat withdrawal through the walls and the dissipative heat release. In the region where such an equilibrium is established, the fluid temperature does not vary along the channel, that is, the temperature field is stabilized (provided that the velocity profile is also stabilized). In what follows we just study this thermally and hydrodynamically stabilized flow. 

Following [208, 276], we assume that the heat release does not affect the physical properties of the fluid (i.e., viscosity, density, and thermal conductivity coefficient are temperature independent). In this case, the velocity profile can be found independently of the heat problem (for laminar flow, see Subsection 1.5-2). 

The equation for the temperature distribution in the region of heat stabilization can be obtained from the equation of heat transfer in Supplement 6 (where the temperature depends only on the transverse coordinate Y and the convective terms are equal to zero, since Vx = V(Y) and Vy = Vz = 0). This equation has the form 

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