Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Mechanical and Materials Engineering

Supervisor

Dr. Roger E. Khayat

Abstract

The effect of non-Fourier heat transfer and partial-slip boundary conditions in Rayleigh-Bénard are analyzed theoretically. Non-Fourier fluids possess a relaxation time that reflects the delay in the response of the heat flux to a change in the temperature gradient while the partial slip boundary condition assumes that the fluid velocity and temperature are not equal to that of the wall. Both non-Fourier and partial-slip effects become important when small-scale heat transfer applications are investigated such as convection around micro- and nano-devices as suggested by the extended heat transport equations from kinetic theory. Other applications are also known to exhibit one or both of these effects such as low-temperature liquids, nanofluids, granular flows, rarefied gases and polymer flows. Small scale effects are measured by the Knudsen number. From this, non-Fourier effects can be estimated, measured non-dimensionally by the Cattaneo number and modelled using the frame indifferent Cattaneo-Vernotte equation which is derived from Oldroyd’s upper-convected derivative. Linear stability of non-Fourier fluids shows that the neutral stability curve possesses a stationary Fourier branch and an oscillatory branch intersecting at some wave number, where for small relaxation time, no change in stability is expected from that of a Fourier fluid. As the relaxation time increases to a critical value, both stationary and oscillatory convection become equally probable. Past this value, oscillatory instability is expected to occur at a smaller Rayleigh number and larger wave number than for a Fourier fluid. Non-linear analysis of weakly non-Fourier fluids shows that near the onset of convection, the convective roll intensity is stronger than for a Fourier fluid. The bifurcation to convection changes from subcritical to supercritical as the Cattaneo number increases and the instability of the convection state for a non-Fourier fluid is shown to occur via a Hopf bifurcation at lower Rayleigh number and higher Nusselt number than for a Fourier fluid. When hydrodynamic slip and temperature jump boundary conditions are considered, a significant reduction in the minimum critical Rayleigh number and corresponding wave number are found. Depending on the sign used for second-order coefficients, critical conditions can be greater than or less than that for first-order boundary conditions.


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