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On Quasi‐static Non=Newtonian Fluids with Power‐law
Author(s) -
Fuchs Martin
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(199610)19:15<1225::aid-mma827>3.0.co;2-u
Subject(s) - mathematics , differentiable function , power law , cauchy stress tensor , newtonian fluid , vector field , non newtonian fluid , combinatorics , mathematical physics , mathematical analysis , physics , geometry , classical mechanics , thermodynamics , statistics
We discuss certain classes of quasi‐static non‐Newtonian fluids for which a power‐law of the form σ D =∇ϕ(ℰ v ) holds. Here σ D is the stress deviator, v the velocity field, ℰ v its symmetric derivative and ϕ is the function \[\phi ({\cal E}v)=\frac 12\mu _\infty \left| {\cal E}v\right| ⁁2+\frac 1p\mu_0\left\{\begin{array}{c}\left( 1+\left| {\cal E}v\right| ⁁2\right) ⁁{p/2} \\\text{or} \\\left| {\cal E}v\right| ⁁p\end{array}\right\},\] ϕ(ℰ v )=1 2 μ ∞ ∣ℰ v ∣ 2 +1 p μ 0 (1+∣ℰ v ∣ 2 ) p /2 or ∣ℰ v ∣ p , μ ∞ ⩾0, μ 0 ⩾0, μ ∞ +μ 0 >0, 1< p <∞. We then prove various regularity results for the velocity field v , for example differentiability almost everywhere and local boundedness of the tensor ℰ v .