By J.T. Oden

ISBN-10: 3540071695

ISBN-13: 9783540071693

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**Example text**

Then u is the gradient of a scalar function p. PROOF. Apply the lemma u = u1 + Vp. 11) Since Vp is orthogonal to all divergence-free vector fields satisfying the boundary conditions, it follows that u, must enjoy the same property. Hence u;dx=0. 2 and introduce a tool useful in many problems arising from the study of two-dimensional irrotational incompressible flows: the complex formalism. First, we write F by means of quantities appearing in the Euler equation. 1) aB where n is the external normal to B.

53) where a is a constant. 54) is verified, so that such flows are stationary. The existence of the Beltrami 48 1. 53) after taking the curl: -Au = a2u. 55) must be completed by the boundary conditions on u. There is a wide literature on the subject. See, for instance, [Dri 91] and references quoted therein. 1 (Liouville Theorem) Theorem. Let s1,(x) be a flow and let u((b,(x), t) be the vector field defined as u(m,(x), t) = d ,(x). 1) Then the following two statements are equivalent: (i) 4),(x) is incompressible (ii) V u = 0.

Observe that, in general, E is a function of space and time. We can compute the time evolution of the internal energy density to obtain a fifth equation expressing the energy balance. It is a challenging, but very difficult problem, to make rigorous the above considerations. What is necessary is a good control of the long-time behavior of Hamiltonian systems, of which very little is known. However, there are results concerning the hydrodynamical behavior of stochastic systems (see, for instance, [DeP 91]).

### Computational Mechanics by J.T. Oden

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