The effect of locally induced flow structure on global heat transfer for plane laminar shear flow

Heat transfer in a plane laminar shear flow configuration consisting of two infinitely long plates orientated parallel to each other is investigated theoretically. The upper plate, which is planar, drives the flow; the lower one, which is fixed, has a regular sinusoidally varying profile. A closed form analytical solution for velocity, based on lubrication theory, together with a semi–analytic one for temperature, from application of Ritz’s direct method, is derived for creeping flow. In addition, detailed numerical solutions are obtained from a finite element formulation of the weak form of the governing equations for mass, momentum and energy (temperature) conservation, enabling the effects of inertia to be explored. It is shown that changes in the mean plate separation, that is the geometry, and the level of inertia present affect the local hydrodynamic flow structure in the form of kinematically and inertially induced eddies, respectively. These in turn impact on the local ”laminar thermal mixing”, and consequently enhance the global heat transfer. Results are reported for a wide range of Pecl´et, Reynolds and Nusselt numbers with agreement between the two methods of solution, for the case of creeping flow, found to be extremely good. The key flow features that emerge are: (i) For creeping flow and varying Pecl´et number, the thermal field is asymmetric for all values of the Pecl´et number other than the limiting conditions of zero and infinity, at which extremes the corresponding thermal field is symmetric. In the limit of infinite Pecl´et number the eddy becomes a basin of fluid at uniform temperature. (ii) Global heat transfer in the case of creeping flow, expressed in terms of the Nusselt number, for a given Pecl´et number increases as the mean plate separation decreases, that is as the local kinematically induced eddy structure becomes more pronounced. (iii) There exists a subtle inter–play between variations in the mean plate separation and the level of inertia imposed, in that both influence the presence or otherwise of eddies. Starting from a creeping flow condition the introduction of inertia can in addition both enlarge and skew an existing eddy. When this information is condensed to a series of Nusselt number curves the indication is that it should be possible, from a practical standpoint, to find a critical mean plate separation, for a given Pecl´et number, for which local inertially influenced eddy effects on the global heat transfer are at a minimum.