By A. J. Chorin, J. E. Marsden (auth.)
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Additional info for A Mathematical Introduction to Fluid Mechanics
V is small" is not a physically . f Ista t tl . lS sma 11 " meanlng u emen un ess some sca 1 lng lS is a meaningful statement. 51 As with incompressible ideal flow, the pressure p sible viscous flow is determined through the equation in incompresdiv u = o. We now shall explore the role of the pressure in incompressible flow in more depth. Let D be a region in (the plane or in) space with smooth boundary aD. Theorem. Any vector field We claim the following decomposition theorem. 6) has zero divergence and is parallel to Proof.
Thus we see that of momentum across the boundary of it approximates in a reasonable way Wt • ~ modifies the transport We will choose (J = so that the transport of momentum by molecular motion. 1) acting on should be a linear function of n. S In fact if one merely assumes it is an arbitrary continuous function of ~, then using balance of 45 momentum, one can actually prove it is linear in is called Cauahy's Theorem. Our assumptions on 1. ~ n. This result * ~ are as follows. e. and is ~ is related to ~ by some VU linear transformation.
If we think of viscous flow as ideal flow with viscous effects added on, it is not unreasonable to assume that p is still a function of p. A note of caution is appropriate here. The expressions for p(p) used in practical situations are often borrowed from the science of equilibrium thermodynamics. 1)) is identical to science. Not all quantities called p p p as defined here as defined in that other are equal. The use of expres- sions from equilibrium thermodynamics requires an additional physical justification, which is indeed often available, but which must not be forgotten.
A Mathematical Introduction to Fluid Mechanics by A. J. Chorin, J. E. Marsden (auth.)