Influence of eddies on heat transfer in Couette flow over undulated substrates

In Couette flows over undulated substrates eddies can be generated under creeping flow conditions. In contrast to free surface flows on undulated substrates even smooth bottom undulations allow for eddy generation due to the kinematical constraints. The subject of our paper is how these flow patterns interact with the temperature field in non–isothermal flows. Our analysis of the thermo–mechanical coupling is focused on the two dominant effects, namely convection and thermoviscosity, whereas dissipation heat, buoyancy and temperature–dependence of the remaining material parameters are neglected. We solve the problem in two steps: First, the influence of the eddies on the convective heat transfer is considered by solving the heat conduction equation with convection. For the velocity field we take the solution resulting analytically from Reynolds' lubrication approximation for the isothermal flow. The thermoviscous feedback of the resulting temperature field to the flow is considered in forthcoming papers. For the construction of the solution an analytical approach based on a nonorthogonal series representation of the fundamental fields and a variational formulation of the field equations is used. The results are visualised and the physical effects they reveal are discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)