Download Biofluid Mechanics by Jagan N. Mazumdar PDF
By Jagan N. Mazumdar
Offers such themes as: components of body structure of the circulatory approach; blood rheology; homes of flowing blood; versions for blood flows, pulsatile move and family among pulsatile strain and circulation, and others.
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Sample text
Hamiltonian structure for the Euler equations 25 of generalized solutions was proved by J. Leray (in the 1930s) and E. Hopf (in 1950/51). Uniqueness for this wide class of solutions is still unknown. The global existence and uniqueness theorems of generalized and classical solutions of the 2-D Navier–Stokes equation were proved by Ladyzhenskaya and her successors (see [Lad]). §6. Hamiltonian structure for the Euler equations Recall the coadjoint representation of an arbitrary Lie group G. It turns out that the coadjoint orbits are always even-dimensional.
The commutant of the Lie algebra S Vect(M) of the group S Diff(M) is the Lie algebra S0 Vect(M) [Arn7]. The quotient space g/[g, g] of a Lie algebra g by its commutant subalgebra [g, g] is called the one-dimensional homology (with coefficients in numbers) of the Lie algebra g. Thus the one-dimensional homology of the Lie algebra of divergencefree vector fields on M n is naturally isomorphic to the De Rham cohomology group H n−1 (M n , R) ker(d : n−1 → n )/ Im(d : n−2 → n−1 ) (for a manifold with boundary the (n − 1)-forms have to vanish on the boundary).
B). In the next section we will apply the Euler theorems to the (infinite-dimensional) group of volume-preserving diffeomorphisms [Arn4, 16]. Note that the analogy between the Euler equations for ideal hydrodynamics and for a rigid body was pointed out by Moreau in [Mor1]. §5. Applications to hydrodynamics 19 Figure 6. Trajectories of the Euler equation on an energy level surface. §5. Applications to hydrodynamics According to the principle of least action, motions of an ideal (incompressible, inviscid) fluid in a Riemannian manifold M are geodesics of a right-invariant metric on the Lie group S Diff(M).