Hydrodynamic flows and heat transfer in slightly noncylindrical channels

Abstract

Viscous incompressible laminar flow and heat transfer in channels with a small arbitrary deviation from a cylindrical surface are examined. A linear system of equations and boundary conditions for the disturbed dynamic and thermal fields, obtained by linearizing the complete system of Navier-Stokes equations with respect to the solution for developed flows in cylindrical tubes of arbitrary cross section, is presented. In the important practical case in which the perturbations of the channel surface are concentrated on an interval of finite length it is shown that the integral dynamic and thermal characteristics of the channel can be found without solving the three-dimensional equations by going over to effective two-dimensional boundary-value problems which are fundamentally no more difficult to solve than those for developed flows. Extensions of the theory to flows with low-efficiency power sources are given. Applications to plane channels and circular tubes with deformed surfaces are considered. Among the numerous applications requiring information about the integral characteristics of flows in channels whose initially cylindrical surface is slighty deformed, we note the problem of heat transfer intensification by slightly deforming the tube surface with careful estimation of the accompanying increase in resistance [1] and the calculation of the resistance of capillaries and biological transport systems in the form of tubes and channels when the walls are deformed [2]. Below we consider laminar flow in channels with deformed walls. Whereas for the first problem this class of flows is only one of those possible (in general it is necessary to analyze the transition, turbulence and flow separation effects), in the second case, which is characterized by low Reynolds numbers, the laminar flow model is perfectly adequate.