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On Variational Models for Quasi‐static Bingham Fluids
Author(s) -
Fuchs Martin,
Grotowski Joseph F.,
Reuling Jürgen
Publication year - 1996
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(199608)19:12<991::aid-mma810>3.0.co;2-r
Subject(s) - mathematics , combinatorics , smoothness , mathematical physics , weak solution , boundary (topology) , energy (signal processing) , mathematical analysis , geometry , statistics
We study a quasi‐static incompressible flow of Bingham type with constituent law \[ \begin{array}{ll} T = p\left| {\cal E}u\right| ⁁{p‐2}{\cal E}u+\beta \frac{{\cal E}u}{\left| {\cal E}u\right| } & \text{if }{\cal E}u\neq 0, \\ \left| T\right| \leq \beta & \text{if }{\cal E}u = 0, \end{array} \] T = p ∣ℰ u ∣ p ‐2 ℰ u +β ℰ u ∣ℰ u ∣ if ℰ u ≠0, ∣ T ∣⩽β if ℰ u = 0, where p ≥2 and β>0. Here ℰ u denotes the strain velocity and T the corresponding stress. The problem admits a variational formulation in the sense that the velocity field u minimizes the energy I ( u ) = ∫ Ω ∣ℰ u ∣ p +β∣ℰ u ∣d x in the space { v ∈ H 1, p (Ω,ℝ n ): div v = 0} subject to appropriate boundary conditions. We then show smoothness of u on the set { x ∈Ω: ℰ u ≠0}.