Herschel–bulkley Fluids: Existence and Regularity of Steady Flows

Abstract

The equations for steady flows of Herschel–Bulkley fluids are considered and the existence of a weak solution is proved for the Dirichlet boundary-value problem. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous constitutive relation between the Cauchy stress and the symmetric part of the velocity gradient. Such a fluid stiffens if its local stresses do not exceed τ*, and it behaves like a non-Newtonian fluid otherwise. We address here a class of nonlinear fluids which includes shear-thinning p-law fluids with 9/5 < p ≤ 2. The flow equations are formulated in the stress-velocity setting (cf. Ref. 25). Our approach is different from that of Duvaut–Lions (cf. Ref. 10) developed for classical Bingham visco-plastic materials. We do not apply the variational inequality but make use of an approximation of the Herschel–Bulkley fluid with a generalized Newtonian fluid with a continuous constitutive law.

On Flows of Bingham-Type Fluids with Threshold Slippage

“…Starting with the pioneering works by Mosolov and Miasnikov [6] and Duvaut and Lions [7], a large number of mathematicians have worked on the theoretical analysis of Bingham fluids and other similar viscoplastic media (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein).…”

Navier–Stokes Equations with Shear Thinning Viscosity. Regularity up to the Boundary

“…Concerning local regularity, or existence and local regularity for boundary value problems (including the space-periodic case) under the assumptions p < 2 and n ≥ 3, we refer the reader to [1], [2], [?] [11], [12], [15], [17], [18], [20], [25], [29] and references therein. References [30] and [31] concern the study of electrorheological fluids, and reference [16] an Euler scheme for Newtonian fluids.…”

Navier–Stokes Equations with Shear-Thickening Viscosity. Regularity up to the Boundary

“…Without any claim of completeness, in addition to the articles already quoted, we would like to mention the following articles related to the problems treated in this paper: [1], [4], [9], [10], [11], [12], [14], [15], [24], [28], [29], [30], [31], [36], [37], [38], [39], and all the relevant references therein.…”