Some Remarks on the Navier-Stokes Equations with a Pressure-Dependent Viscosity.

Incompressible Poiseuille flows of Newtonian liquids with a pressure-dependent viscosity

“…Mathematical issues arising in the case of incompressible Newtonian or non-Newtonian flows with a pressure-dependent viscosity have been addressed by Renardy [22], Gazzola [23], and Málek et al [24,25]. The existence of flows of fluids with pressuredependent viscosity and the associated assumptions have been discussed by Bulíček et al [26].…”

Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid

“…Perhaps the most comprehensive cycle of recent works considering flows of such fluids both mathematically from the point of view of existence and wellposedness and numerically are by Hron, Málek, Rajagopal and collaborators [1,[8][9][10][11][12]. Earlier studies include [13][14][15], to name a few. It is argued in these references that even though the fluid can exhibit a strong dependence of viscosity on pressure its compressibility frequently is much weaker.…”

On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate

“…Let us remark that, to our knowledge, there is no large-time or large-data existence theory available if the viscosity depends only on the pressure. In [16] local existence and uniqueness results are obtained by assuming that the viscosity satisfies ν(p)/p → 0, p → +∞, which contradicts the experimental results available, while also requiring an additional condition on eigenvalues of D(v) in terms of ∂ν/∂p. In [17,18], a short time existence of a smooth solution for smooth small data is established under very restrictive conditions.…”

“…In Section 3, we show that in the case of inner flows, i.e. under the assumption (5), the Hopf's extension that meets (16) and (17) exists even for ϕ large. Thus, for r ∈ 9 existence result for inner flows.…”

“…In the following theorem, the existence result is provided without requiring (5), but the conditions (15), (16) and (17) concerning the boundary data and its extension are a priori assumed. We prove the theorem in Section 2.…”