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A presentation of a few of the elemental principles of fluid mechanics in a mathematically appealing demeanour. The textual content illustrates the actual historical past and motivation for a few buildings utilized in contemporary mathematical and numerical paintings at the Navier- Stokes equations and on hyperbolic structures, so that it will curiosity scholars during this right away attractive and tough topic. This 3rd variation encompasses a variety of updates and revisions, whereas protecting the spirit and scope of the unique booklet.
Read or Download A Mathematical Introduction to Fluid Mechanics (Texts in Applied Mathematics) (v. 4) PDF
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Additional info for A Mathematical Introduction to Fluid Mechanics (Texts in Applied Mathematics) (v. 4)
Because OxU = 0, u is only a function of y and thus, writing u(x, y) = u(y), we obtain p' 1 = -u" R Because each side depends on different variables, p' = constant, 1 Ru" = constant. 12 See, for example, S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, 1958. 3. Flow between two parallel plates; the fluid is pushed from left to right and correspondingly, P1 > P2· Integration gives ßp p(x) = Pl - yx, ßp = P1- P2, and u(y) ßp = y(l - y)R 2L. 4).
3 The Navier-Stokes Equations 35 molecular diffusion. Our opening example indicates that molecular interaction between the solid wall with zero tangential velocity (or zero average velocity on the molecular level) should impart the same condition to the immediately adjacent fluid. Another crucial feature of the boundary condition u = 0 is that it provides a mechanism by which a boundary can produce vorticity in the fluid. We shall describe this in some detail in Chapter 2. Next, weshall discuss some scaling properties of the Navier-Stokes equations with the aim of introducing a parameter (the Reynolds number) that measures the effect of viscosity on the flow.
The difference between the two-dimensional and three-dimensional conservation laws for vorticity is very important. 2. 7)' in two dimensions is a helpful tool in establishing a rigorous theory of existence and uniqueness of the Euler (and later Navier-Stokes) equations. The Iack of the same kind of conservation in three dimensions is a major obstacle to the rigorous understanding of crucial properties of the solutions of the equations of fluid dynamics. The main point here is to get existence theorems for all time.