Extended lubrication theory for generalized Couette flow through converging gaps

The lubrication theory based approximation of the Navier-Stokes equations represents a widely used approach for the computation of viscous force dominated flows through narrow channels or gaps. Its inherent neglect of advective transport becomes increasingly questionable when considering strongly converging geometries and particularly when considering heat transfer as well. The present work develops an extended lubrication theory based model, which accounts for inertia and convective heat transfer in terms of first-order perturbations. The extended model is applied to generalized Couette flow through strongly converging annular gaps typically met in wire coating dies, where the moving wall-driven motion is characterized by very high local shear rates and steep axial pressure gradients. The evaluation of the analytical solutions against numerical results from CFD simulations demonstrates the scope and the limits of the original lubrication theory approximation in describing the local flow and thermal conditions inside the gap. Including first-order convective contributions the extended model is shown to provide a significantly improved description, especially of the redistribution of the generated viscous heat. The most pronounced improvements are shown by the temperature profiles predicted at axial positions with rapidly changing cross-section for fluids with higher Prandtl numbers. The observed gain in accuracy is also reflected by very accurate predictions of the total transfer rates at the boundaries in the global balances of heat and momentum. Using the presently proposed extension the lubrication theory based model represents a reliable and computationally efficient approach for investigating the flow and heat transfer inside narrow gaps, which is well applicable to a wide range of different gap geometries.