Modelling of phase lagged behaviour in fluid mechanics

Abstract

There is common understanding that classical constitutive equations such as Fick’s and Fourier’s laws becomes invalid or inapplicable when the characteristics size of the domain becomes less than the mean free molecular path in heat transfer and energy transport process. As there are advancements in nanotechnology, trying to overcome these difficulties in heat transfer and energy transport is essential. Therefore, time-lagged models are proposed to overcome these difficulties and these models are investigated.

Lagging models are based on two parameters. The heat flux and temperature gradient. The consequential model of these two lags is known as dual-phase-lag. For heat transfer problems, a time lag is introduced between the onset of temperature gradient and heat flux. This leads to hyperbolic models which were illustrated by Tzou’s dual-phase-lag model of heat conduction. The model shows that thermal energy is carried by diffusion and waves. Other models such as Kulish and Novozhilov’s integral model and Guyer-Krumhansl model are also demonstrate and confirmed that the hyperbolic energy equation is valid at small scales.

A detailed background study on the classical Fourier’s law of heat conduction and Navier-Stokes equation was conducted to understand better on heat transfer and momentum transfer. As superfluid behavior was noticed during start up flows, study on superfluid was also conducted.

After analyzing the various studies and research findings, the start-up flow in a nano-channel was investigated. Numerical simulation for two main parameters, the kinematic viscosity and thermal conductivity of the fluid was carried out. The two parameters were then analysed and it was found out that they had an inverse relationship. Moreover, the inverse relationship can be related to the Prandtl number. Furthermore, superfluid effects were also observed in the results. These effects usually occur at the beginning time unit and eventually diminishes causing the Pr number to reach an equilibrium of 1.