The Effect of Viscous Dissipation on a Simple Shear Flow

THE EFFECT OF VISCOUS DISSIPATION ON A SIMPLE SHEAR FLOW 

We have seen, in the previous chapter, that the effects of viscous dissipation will be small whenever the Brinkman number is small. This is often the case, and it is common practice in 

In this section, we return to the analysis of simple, unidirectional shear flow that was considered in Section B of Chap. 3, but instead of neglecting viscous dissipation altogether, we consider its influence when the Brinkman number is small, but nonzero. The starting point is Eqs. (3-34) and (3-35), which are reproduced here for convenience  

If Br = 0, as assumed in Chap. 3, the temperature is constant across the gap (i.e., 0 = 0 for all y) and the material coefficients JT and k are both equal to one. For small, but nonzero, Br, on the other hand, 0 will vary across the gap, and we must take account of the corresponding changes in the material coefficients. For this purpose, it is sufficient to approximate the viscosity and thermal conductivity for small changes in the temperature relative to the value To that exists in the absence of dissipation. Hence, we approximate Jl and A in the forms 

The coefficients a and ft are material constants for the specific fluid. It will be noted that (4-40) and (4-41) are written in a form suitable for a liquid because it is assumed that the viscosity decreases as the temperature increases. 

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C. The Effect of Viscous Dissipation on a Simple Shear Flow... 

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