By S. Friedlander, D. Serre

This is often the fourth quantity in a chain of survey articles protecting many facets of mathematical fluid dynamics, an important resource of open mathematical difficulties and fascinating physics.

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**Extra info for Handbook of Mathematical Fluid Dynamics, Volume 4**

**Example text**

As mentioned earlier, this theory is known to be a special case of the more general theories that are now available and, as such, the predictions based on this simple theory are of interest. Only the symmetric part of T and the de via to ric part of M are determined by the constitutive equations given in Eqs. 4). The trace of M, trM = mrr = m and the skew-symmetric part TA of T are not determined by the constitutive equations. However, TA can be determined from the equations of motion, once m is known.

The effects of couple stresses will be larger for smaller values of h and will decrease with the increase in h. The increase in h need only be continued until a limiting value of JL has been obtained. The wall shear T w will be the value of fyx at the walls. From Eqs. I!.. 23) which is the same as the distribution for the nonpolar case. The expression for the wall shear is therefore T w = h dp/ dx, which is the same as in the nonpolar case. Note that even though the through flow is affected by couple stresses.

2aIL 89 D where D, Yo, Y\, Y2, Zo, Z\ and Z2 are given by Eq. 14), and Thus, this flow is affected by couple stresses for both boundary conditions A and B. 7 Poiseuille Flow Through Circular Pipes Consider the steady flow of an incompressible fluid through a circular pipe of radius R, due to an axial pressure gradient. 1) (r) where x is measured along the centerline of the pipe. 2) - -1 -a [r -au] . p.. =.. 7) where It = "fI'I/-t and Co(rl t) = A3Io(rl J) Also, by using tlx = /-ta ula r and t~ = + A4Ko(rl n.