Two-dimensional Poiseuille flow of a Newtonian fluid with temperature-dependent viscosity

Abstract

The problem of a two-dimensional Poiseuille flow of a Newtonian fluid with a viscosity exponentially dependent on temperature under the influence of longitudinal pressure and temperature gradients is reduced to a two-parameter boundary value problem for a third-order ordinary differential equation with respect to a dimensionless unknown characterizing the change in temperature across the channel. In the range of negative values of the dimensionless longitudinal temperature gradient, the solution has two branches. By closing, the branches form a boundary on the parameter plane beyond which Poiseuille-type flows do not exist. The solutions belonging to different branches differ in the flow rate and intensity of heat transfer by the liquid through solid boundaries. The latter circumstance entails deceleration of the flow in the near-wall areas. In the region of positive values of the dimensionless longitudinal temperature gradient, the solution is unique. With an increase in the mentioned parameter, the heat flow to the liquid from the walls increases, which entails the formation of a more bulged velocity profile compared to the isothermal case. At sufficiently large values of the longitudinal temperature gradient, there appears a ‘stick’ flow with a uniform velocity distribution everywhere except for narrow layers near solid boundaries. It is shown that the rate is proportional to the ratio of the densities of the transverse and longitudinal heat flows. Explicit expressions for the pressure and shear stress fields are given. The properties of monotonicity of the velocity and temperature profiles as functions of the transverse coordinate have been studied. It is shown that, regardless of the values of the dimensionless parameters of the problem, the velocity has a positive maximum on the central axis of the channel and monotonically decreases to zero at the solid boundary.