Complex fluids: Modeling and Algorithms (Mathématiques et by Pierre Saramito

By Pierre Saramito

This publication provides a accomplished evaluation of the modeling of advanced fluids, together with many universal ingredients, comparable to toothpaste, hair gel, mayonnaise, liquid foam, cement and blood, which can't be defined through Navier-Stokes equations. It additionally deals an updated mathematical and numerical research of the corresponding equations, in addition to a number of sensible numerical algorithms and software program options for the approximation of the ideas. It discusses business (molten plastics, forming process), geophysical (mud flows, volcanic lava, glaciers and snow avalanches), and organic (blood flows, tissues) modeling purposes. This ebook is a worthy source for undergraduate scholars and researchers in utilized arithmetic, mechanical engineering and physics.

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In the three dimensional case, the situation is different. The computing time of the factorization is O(n 2 ), which is much more time consuming and the resolution O(n 4/3 ) is also not advantageous. Later, we will study more efficient iterative solvers. The forthcoming Sect. 12 presents the resolution of the Stokes and Navier–Stokes problems using the Rheolef library [284, 286] that implements all these efficient algorithms for sparse matrices. But beforehand, we have to study the evaluation of the right-handsides of the Stokes subproblems during the resolution of the Navier–Stokes equation with the method of characteristics.

The problem counts seven physical parameters: η, ω1 , ω2 , r1 , r2 , ρ and g. Let W = ω1 − ω2 and Σ = η(ω1 − ω2 ) be the characteristic angular velocity and stress, respectively. 30). 33). Note that the computation of ω˜ is still decoupled to that of p, ˜ which can be computed in a second time. 33) for computing the angular velocity involves only one dimensionless number, the geometry confinement. After integration, we get the following explicit expression of the angular velocity u θ (r ) = ω2 r + (ω1 − ω2 ) r 1− r1 r2 2 r1 r 2 − r1 r2 2 and the computation of the pressure is similar as in the previous case.

4 (equivalence norms on H01 ) The previous result is not obvious as 2D(u) L 2 (Ω) involves only the symmetric part of the velocity gradient while u H 1 (Ω) involves the full gradient. )1/2 is a norm in H01 equivalent to . H 1 (Ω) , ∀u ∈ H01 (Ω) 3 H 1 (Ω) . 10 (existence and uniqueness of velocity for the Stokes problem) Assume that the boundary ∂Ω and the data uΓ and f are smooth enough. 27) and that the 3 right-hand-side f ∈ H −1 (Ω) . 47) admits a unique solution u ∈ K (uΓ ). Proof A direct proof can be found in [119, p.

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