By A. J. Chorin, J. E. Marsden (auth.)

ISBN-10: 0387904069

ISBN-13: 9780387904061

ISBN-10: 1468400827

ISBN-13: 9781468400823

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**Extra resources for A Mathematical Introduction to Fluid Mechanics**

**Example text**

Since ~ is orthogonal to grad p, we get ~ E .. 10 ) dt Ekinetic where Iz~1 2 = Z~-Z~ = IZul 222 + IZvl . + Izwl . 2). 1 we noted that ideal flow in a channel leads to unreasonable results. We now reconsider this example with viscous effects. Example. 3-3. y y. l""'Ll,///////////~//// , flow direction Y = I I , I ~ : I I PI pressur~ a' \ \ \ \ Ifixed wall I , , x~a\ pressu~e = P2 I \ \ \ \ \ \ \ \ \ \ ~=L \ \ \ \ • x fixed wall Figure 1. 3-3 Let us seek a solution for which a function of x, ~Cx,y) PI = pCa), with = Cu(x,y),a) P2 = peL) and fluid is "pushed" in the positive x-direction.

E. "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.

2) We can rewrite this by putting all the ~. trace in one term: is the first aoeffiaient of visaosity and where ~ 1 =A+3l1 is the seaond aoeffiaient of visaosity. 3) + ~)~ az is the Laplacian of u. 3) completely describes the flow of a viscous fluid. d 0 47 where v = and where is the coefficient of kinematia visaosity. p These equations must be supplemented by boundary conditions. e. fluid does not cross the boundary, but may move tangentially to the boundary. For the Navier-Stokes equations, the extra term number of derivatives of ~ vLlu involved from one to two.

### A Mathematical Introduction to Fluid Mechanics by A. J. Chorin, J. E. Marsden (auth.)

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