Planar Unsteady Flows of Incompressible Heat Conducting Shear Thinning Fluids

It is evident that behaviour of many fluids varies with varying temperature. We present mathematical properties of unsteady planar flows of shear thinning incompressible fluids where the change of the temperature influences the viscosity, which is at the same time shear thinning (power law fluids with power law index r − 2 < 0). Contrary of [2] (where a threedimensional flow is studied for 9/5 < r < 2) and many other sources of recent years we will deal only with planar flows for 3/2 < r < 2 with Navier’s slip boundary conditions. At first we formulate the system of equations governing such flows and then we give the definition of a weak solution and existence theorem for the power index r > 3/2 with a sketch of proof. Finally, we present several observations arising from numerical experiments for two variants of the equation, that represent balance of energy. Introduction The fact that viscosity and thermal conductivity depend upon the temperature, is known for many years. The dependency of these quantities on temperature and pressure was deeply studied in the twentieth century and their investigation continues up to these days. There is a couple of models describing how the viscosity and thermal conductivity change with temperature. In general, the viscosity μ is decreasing with increasing temperature θ. In some models suggested in literature the viscosity depending on the temperature vanishes as the temperature tends to infinity. Such behaviour is described for example by Reynold’s model (see [3]) in the form μ(θ) = μ0 exp(−mθ), where μ0 and m are positive constants. (1) In some other cases — materials with glass transition temperature — like Williams-Landel-Ferry model, the viscosity does not drop bellow some small quantity μ(θ) = μ0 exp ( −m1(θ − θr) m2 + (θ − θr) ) , where μ0, m1, m2 and θr are empiric parameters. (2) In a last few decades papers dealing with systems where viscosity depends on the shear rate, pressure and temperature (or some combinations of them) appear more and more frequently. Nice brief survey of theoretical results can be found in [2]. In this article we shall consider the model for planar flow of incompressible fluid with viscosity depending on shear-rate and temperature for power law index r < 2 which does not seem to be treated in literature. We shall formulate the problem, bring in the theorem about the existence of a weak solution and outline a sketch of proof. Formulation of the problem We are interested in unsteady inner flow in a bounded open set Ω ⊂ R with the boundary ∂Ω over time interval (0, T ). The state of the incompressible fluid in the domain Q = (0, T )× Ω is described in terms of the velocity field v, the pressure p (the mean normal stress) and the temperature θ by a system of partial differential equations that are a consequence of the balance of mass, balance of linear and angular momentum and balance of energy. The balance of mass for homogeneous and incompressible fluid reduces to the divergence of the velocity being zero. The balance of angular momentum will be met automatically by virtue of the form chosen for the Cauchy stress T: T = ̺(−pI+ S) , where ̺ is the constant density, S being symmetric. (3) 24 WDS'08 Proceedings of Contributed Papers, Part III, 24–31, 2008. ISBN 978-80-7378-067-8 © MATFYZPRESS