Axisymmetric Poiseuille Flow with Temperature-Dependent Viscosity under Pressure and Temperature Gradients

Abstract

The study of steady-state axisymmetric Poiseuille flow of a Newtonian fluid induced by streamwise pressure and temperature gradients in the case of the dynamic viscosity coefficient dependent on the temperature is reduced to finding solutions to a three-parameter boundary-value problem for a third-order ordinary differential equation. In the domain of the parameter space corresponding to negative axial temperature gradients, there exist two branches of solutions describing flows accompanied by heat removal from the fluid. When the branches meet, they form a boundary in the phase space beyond which no solutions to the Poiseuille-type problem exist. One of the branches can be continued into the domain of non-negative values of the streamwise temperature gradient and contains an isothermal Poiseuille solution. Along this branch, curve of the flow rate as a function of the dimensionless axial temperature gradient has a minimum in the domain of positive values of the latter. In this part of the parameter space, the heat exchange regime with the external medium depends on the relation between all three dimensionless numbers of the problem. The heat exchange regime affects the nature of flow, slowing down the flow near the rigid wall during heat transfer, and forming a more filled velocity profile when heat is absorbed by fluid.