On generalized Stokes’ and Brinkman’s equations with a pressure- and shear-dependent viscosity and drag coefficient

Large data existence theory for unsteady flows of fluids with pressure- and shear-dependent viscosities

“…Apart from optimization from below, we have also finally incorporated the value r = 2 among amenable values of the exponent r, which has only recently been achieved for the steady-state problem in [14]. The work [11] also covers the value r = 2, yet under a slightly different analogue of Assumption 2.2.…”

“…It is highly probable, however, that like in the steady case (see [14]), one may relax the condition to the point It is highly probable, however, that like in the steady case (see [14]), one may relax the condition to the point…”

“…This particular model goes back to two papers by Málek et al [23,24] and has been dealt with on multiple occasions ever since (see e.g. [5,14,17,22] and the discussion below Theorem 3.1). It has been convincingly documented in experiments that viscosity of a fluid may vary significantly with the pressure (exponentially or even more dramatically; see e.g.…”

Well productivity index for compressible fluids and gases

“…In this paper we discuss two types of fluids: slightly compressible fluid, characterized by the equation of state ρ(p) ∼ exp(γp) with very small compressibility constant (γ ∼ 10 −8 ), and strongly compressible fluid (in particular, ideal gas) characterized by the equation of state ρ(p) ∼ p, [20]. While in general porosity depends on pressure, see [21,26,8], we only consider it to be a function of spatial variable x. In case of slightly compressible fluid we studied different properties of g-Forchheimer equations in [2,4,14].…”

“…We will study equations (8) and (9) in the open domain U ⊂ R n with the C 2 boundary ∂U = Γ = Γ e ∪ Γ i . The Γ e is considered to be external impermeable boundary of the reservoir with u • N = 0, where N is the outward normal vector to Γ e .…”