The Oberbeck–Boussinesq problem modified by a thermo-absorption term

Abstract

We consider the Oberbeck-Boussinesq problem with an extra coupling, establishing a suitable relation between the velocity and the temperature. Our model involves a system of equations given by the transient Navier-Stokes equations modified by introducing the thermo-absorption term. The model involves also the transient temperature equation with nonlinear diffusion. For the obtained problem, we prove the existence of weak solutions for any N 2 and its uniqueness if N = 2. Then, considering a low range of temperature, but upper than the phase changing one, we study several properties related with vanishing in time of the velocity component of the weak solutions. First, assuming the buoyancy forces field extinct after a finite time, we prove the velocity component will extinct in a later finite time, provided the thermo-absorption term is sublinear. In this case, considering a suitable buoyancy forces field which vanishes at some instant of time, we prove the velocity component extinct at the same instant. We prove also that for non-zero buoyancy forces, but decaying at a power time rate, the velocity component decay at analogous power time rates, provided the thermo-absorption term is superlinear. At last, we prove that for a general non-zero bounded buoyancy force, the velocity component exponentially decay in time whether the thermo-absorption term is sub or superlinear.

Existence for the steady problem of a mixture of two power-law fluids

“…The analysis of the problem (1.2)-(1.5) with µ 1 = 0 and µ 2 = 0 started, from the Mathematical Fluid Mechanics viewpoint, with the works on electrorheological fluids by R užička [25] and on thermorheological fluids by Antontsev et al [2,3,4]. The main existence results for the problem (1.2)-(1.5), with µ 1 = 0 and µ 2 = 0, are due to R užička [25], Huber [16] and Diening et al [12].…”

Existence of weak solutions for the generalized Navier–Stokes equations with damping

“…There, we have proved the existence of weak solutions, its uniqueness and some asymptotic properties. We carried out an analogous study in [5] for the Oberbeck-Boussinesq version of this problem, where besides the usual coupling in the buoyancy force, we have considered an extra coupling in the damping term by considering a temperature-depending function σ. In [10] the authors have proved the existence of weak and strong solutions for the Cauchy problem (1.1)- (1.4) in R 3 and with q = 2.…”