H. B. de Oliveira

Stopping a Viscous Fluid by a Feedback Dissipative Field: Thermal Effects without Phase Changing

We show how the action on two simultaneous effects (a suitable coupling about velocity and temperature and a low range of temperature but upper that the phase changing one) may be responsible of stopping a viscous fluid without any changing phase. Our model involves a system, on an unbounded pipe, given by the planar stationary Navier-Stokes equation perturbed with a sublinear term f(x,θ, u) coupled with a stationary (and possibly nonlinear) advection diffusion equation for the temperature θ.

The Navier–Stokes problem modified by an absorption term

It is well known that for the classical Navier–Stokes problem the best one can obtain is some decays in time of power type. With this in mind, we consider in this work, the classical Navier–Stokes problem modified by introducing, in the momentum equation, the absorption term |u|σ−2 u, where σ > 1. For the obtained problem, we prove the existence of weak solutions for any dimension N ≥ 2 and its uniqueness for N = 2. Then we prove that, for zero body forces, the weak solutions extinct in a finite time if 1 2. Finally, we prove that for a general non-zero body force, the weak solutions exponentially decay in time for any σ > 1.It is well known that for the classical Navier–Stokes problem the best one can obtain is some decays in time of power type. With this in mind, we consider in this work, the classical Navier–Stokes problem modified by introducing, in the momentum equation, the absorption term |u|σ−2 u, where σ > 1. For the obtained problem, we prove the existence of weak solutions for any dimension N ≥ 2 and its uniqueness for N = 2. Then we prove that, for zero body forces, the weak solutions extinct in a finite time if 1 < σ < 2 and exponentially decay in time if σ = 2. In the special case of a suitable force field which vanishes at some instant, we prove that the weak solutions extinct at the same instant provided 1 < σ < 2. We also prove that for non-zero body forces decaying at a power-time rate, the solutions decay at analogous power-time rates if σ > 2. Finally, we prove that for a general non-zero body force, the weak solutions exponentially decay in time for any σ > 1.

Finite Time Localized Solutions of Fluid Problems with Anisotropic Dissipation

In this work we consider an incompressible, non-homogeneous, dilatant and viscous fluid for which the stress tensor satisfies a general non-Newtonian law. The new contribution of this work is the consideration of an anisotropic dissipative forces field which depends nonlinearly on the own velocity. We prove that, if the flow of such a fluid is generated by the initial data, then in a finite time the fluid becomes immobile. We, also, prove that, if the flow is stirred by a forces term which vanishes at some instant of time, then the fluid is still for all time grater than that and provided the intensity of the force is suitably small.

On a One-Equation Turbulent Model with Feedbacks

A one-equation turbulent model is derived in this work on the basis of the approach used for the k-epsilon model. The novelty of the model consists in the consideration of a general feedback forces field in the momentum equation and a rather general turbulent dissipation function in the equation for the turbulent kinetic energy. For the steady-state associated boundary value problem, we prove the uniqueness of weak solutions under monotonous conditions on the feedbacks and smallness conditions on the solutions to the problem. We also discuss the existence of weak solutions and issues related with the higher integrability of the solutions gradients.

On a Stochastic Coupled System of Reaction-Diffusion of Nonlocal Type

In this article we investigate the existence and uniqueness of weak solutions for a stochastic nonlinear parabolic coupled system of reaction-diffusion of nonlocal type, and with multiplicative white noise. An important result on the asymptotic behavior of the weak solutions is presented as well.