By Pierre Saramito
This booklet provides a entire review of the modeling of advanced fluids, together with many universal elements, corresponding to toothpaste, hair gel, mayonnaise, liquid foam, cement and blood, which can't be defined by means of Navier-Stokes equations. It additionally deals an updated mathematical and numerical research of the corresponding equations, in addition to numerous functional numerical algorithms and software program ideas for the approximation of the strategies. It discusses commercial (molten plastics, forming process), geophysical (mud flows, volcanic lava, glaciers and snow avalanches), and organic (blood flows, tissues) modeling purposes. This booklet is a necessary source for undergraduate scholars and researchers in utilized arithmetic, mechanical engineering and physics.
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Extra resources for Complex fluids: Modeling and Algorithms
2 (symmetric part of vector gradient norm inequality) 2D(u) c0 u L 2 (Ω) H 1 (Ω) , ∀u ∈ H01 (Ω) 3 where c0 > 0 is a constant depending only upon Ω. Proof This result follows from the Korn inequality, which is a classical result of functional analysis. g. [86, p. 115] or [119, p. 86]. 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 .
Then, at each time step, successive systems could be solved for different right-hand-sides. Thus, the direct method is advantageous in the two-dimensional case. 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.
Moreover, the coefficient factor depends only upon r1 , r2 and z 0 that are related to the dimension of the geometry, and this apparatus is an efficient method for measuring the viscosity. The case ω1 > 0 and ω2 = 0 was used in 1923 by Taylor  to explore the limit angular velocity for the stable laminar solution. For sufficiently large Reynolds numbers, the solution develops three dimensional patterns. This limit is well-known today and the mathematical properties of the Navier–Stokes equations at large Reynolds numbers has been systematically explored for this particular flow in 1994 in the beautiful book by Chossat and Iooss .