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

Abstract

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 Peclét, 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 Peclét number, the thermal field is asymmetric for all values of the Peclét number other than the limiting conditions of zero and infinity, at which extremes the corresponding thermal field is symmetric. In the limit of infinite Peclét 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 Peclét number increases as the mean plate separation decreases, that is as the local kinematically induced eddy structure becomes more pronounced. Preprint submitted to Elsevier 11 December 2008(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 Peclét number, for which local inertially influenced eddy effects on the global heat transfer are at a minimum.

Competing Lagrangians for incompressible and compressible viscous flow

“…One of them is the use of nonlocal functionals. As an example consider the variational principle reported in [40,41] for the convection-conduction equation. Appendix B.…”

Eddy genesis and manipulation in plane laminar shear flow

“…Prior use of this 'non-orthogonal' series representation has led to rapid convergence, see for example Scholle et al [19], with the series truncated at N = 4. Furthermore, a simplified form of the no-slip conditions (7) at the lower plate results by virtue of the transformation (15).…”

“…Eddies have also been shown recently to play an important role in enhancing mass transport [18], and global heat transfer via thermal mixing [19].…”

Rayleigh–Taylor instability of viscoelastic self-rewetting film flowing down a temperature-controlled inclined substrate

“…It is supposed that the variations of the density ρ and the viscosity µ within the fluid due to temperature differences are both negligible. [43] g T s The proposed model represents an approximation to the first order in elasticity and obeys constitutive equation given by Beard and Walters [44] π i j = −pδ i j + 2µe i j − 2γ 0 δ e i j δt ,…”