Nusselt Numbers for Fully-Developed Flow Between Parallel Plates With One Plate Textured With Isothermal Parallel Ridges

We develop a semi-analytical solution for the Nusselt number for fully-developed flow of liquid between parallel plates, one of which is textured with isothermal parallel ridges. The opposite plate is smooth and adiabatic. The liquid is assumed to be in the Cassie state on the textured surface, on which a mixed boundary condition of no slip on the ridges and no shear along menisci applies. An existing solution for the velocity field is valid. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface. Given the nature of the isothermal boundary condition, the analysis concerns a three-dimensional developing temperature profile, and the results are obtained for a streamwise location that tends to infinity. We assume that the temperature field is governed by an infinite sum of the product of a function of the streamwise coordinate and a second function of the spanwise co-ordinates. The latter functions are eigenfunctions satisfying a two-dimensional Sturm-Liouville problem from which the eigenvalues follow. The fully-developed Nusselt number follows from the first eigenvalue.