On the Mathematical Analysis of Thick Fluids
- 58 Downloads
In chemical engineering models, shear-thickening or dilatant fluids converge in the limit case to a class of incompressible fluids with a maximum admissible shear rate, the so-called thick fluids. These non-Newtonian fluids can be obtained, in particular, as the power limit of the Ostwald–de Waele fluids, and can be described as a new class of evolution variational inequalities, in which the shear rate is bounded by a positive constant or, more generally, by a bounded positive function. It is established the existence, uniqueness, and the continuous dependence of solutions to this general class of thick fluids with variable threshold on the absolute value of the deformation rate tensor, the solutions of which belong to a time dependent convex set. For sufficiently large viscosity, the asymptotic stabilization toward a unique steady state is also proved.
KeywordsShear Rate Weak Solution Variational Inequality Continuous Dependence Quasivariational Inequality
Unable to display preview. Download preview PDF.
- 3.T. Bhattacharya, E. DiBenedetto, and J. J. Manfredi, “Limits as p→∞ of Δp u p = f and related extremal problems,” Rend. Sem. Mat. Univ. Politec. Torino, 47, 15–68 (1989). Special Issue (1991).Google Scholar
- 5.R. P. Chhabra and J. F. Richardson, Non-Newtonian Flow and Applied Rheology: Engineering Applications (second ed.), Butterworth-Heinemann, Oxford (2008).Google Scholar
- 6.J. C. De los Reyes and G. Stadler, “A nonsmooth model for discontinuous shear thickening fluids: analysis and numerical solution,” UT Austin ICES, Report 12-42 (2012), to appear in: Interfaces and Free Boundaries, 16 (2014).Google Scholar
- 9.M. Fuchs and G. Seregin, “Variational methods for problems from plasticity theory and for generalized Newtonian fluids,” Lect. Notes Math., 1749 (2000).Google Scholar
- 10.C. Gerhardt, “On the existence and uniqueness of a warpening function in the elastic-plastic torsion of a cylindrical bar with a multiply connected cross section,” Lect. Notes Math., 503 (1976).Google Scholar
- 12.A. Haraux, “Nonlinear evolution equations. Global behavior of solutions,” Lect. Notes Math., 841 (1981).Google Scholar
- 18.J. Malek and K. R. Rajagopal, “Mathematical issues concerning the Navier–Stokes equations and some of its generalizations,” In: Evolutionary equations, Vol. II, Handb. Diff. Eqs., Elsevier/North-Holland, Amsterdam (2005), pp. 371–459.Google Scholar
- 19.J. Mewis and N. J. Wagner, Colloidal Suspension Rheology, Cambridge University Press, Cambridge (2012).Google Scholar
- 20.F. Miranda and, J. F. Rodrigues, “On a variational inequality for incompressible non-Newtonian thick flows,” to appear.Google Scholar
- 22.L. Nirenberg, “An extended interpolation inequality,” Ann. Scuola Norm. Pisa Cl. Sci., 3rd serie, 20, No. 4, 733–737 (1966).Google Scholar