It is possible to start rarefied gas flows with tangential temperature gradients along the channel walls, where the fluid starts creeping in the direction from cold towards hot. This is called the thermal creep (transpiration) phenomenon.
An interesting aspect of thermal creep is that it causes zero net mass flowrate in the channels where thermal creep and pressure gradient balance each other. To demonstrate this we simulated air flow in microchannels of various dimensions (corresponding to Kn=0.365, 0.122, 0.052), connecting two tanks kept at different temperatures (300 K and 400 K). The pressure in both tanks is initially atmospheric. Thermal creep effects start pumping the fluid towards the hot tank, increasing the pressure in the hot tank and lowering the pressure in the colder one. This pressure difference eventually starts flow in the middle of the channel from hot to cold (high pressure to low pressure) direction, resulting in zero average mass flowrate in the channel as the steady state is reached.
The steady state pressure distribution along the channel center and in the reservoirs normalized with atmospheric pressure (initial pressure at both tanks) is given below for three different channel sizes. It is seen that the pressure change due to thermal creep for high Kn flows is non-negligible.
The numerical experiment defined above is an unsteady process and time scales of the problem are governed by two different transient processes. The first transient process is due to the fluid starting to creep on the channel surface. As time goes on, the creeping fluid layer starts interacting with the stagnant fluid layers above it, creating a boundary layer just like in impulsively started plane wall problem (Rayleigh problem). Of course, formation of the boundary layer creates shear stresses, which activates the velocity slip mechanism. This is the initial transient process with time scale t ~ (h^2)/(kinematic viscosity). Considering microchannels with typical height of a micron, this transient is very fast.
The second time scale of the problem corresponds to the time to get from initial transients to a steady state solution where the net mass flowrate is zero. This time scale is based on the creep velocity and tank dimensions. In particular, this time scale increases as the tank size is increased. In the limit where the tanks are reservoirs of infinite dimensions, the fluid steadily creeps from cold to the hot tank, and the pressure values at the two reservoirs remains practically the same.
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