# Interdisciplinary Applied Mathematics

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Therefore, the unified model employs two empirical parameters (a1 and в) and two known parameters b = —1 and a0. Although this empiricism is not desired,    the    a    value in    Cr    varies    from    zero    in    the    slip flow regime

to an    order-one    value    of    a0    as    Kn    ^ ж.    Finally,    the    model    is adapted

to the finite aspect ratio rectangular ducts using a standard aspect ratio correction given in equation (4.35).

## 5

### Thermal Effects in Microscales

In this chapter we consider heat transfer in gas microflows. In the first section we concentrate on the thermal creep (transpiration) effects that may be important in channels with tangential temperature gradients on their surfaces. For example, a microchannel surface with a prescribed heat flux is subject to temperature variations along its surface, and this results in thermal creep flow. We analyze thermal creep with numerical simulations to demonstrate the main concept, and subsequently we describe a prototype experiment. In the second and third sections we study other temperature-induced flows and investigate the validity of the heat conduction equation in the limit Kn ^ 0. In the fourth and fifth sections we investigate the combined effects of thermal creep, heat conduction, and convection in pressure-, force-, and shear-driven channel flows.

##### 5.1 Thermal Creep (Transpiration)

It is possible to start rarefied gas flows due to tangential temperature gradients along the channel walls, where the fluid starts creeping in the direction from cold toward hot. This is the so-called thermal creep or transpiration phenomenon. We explain this counterintuitive effect with the following example: Consider two containers filled with the same gas that are kept at the same pressure

###### Pi = P‘2

but at different temperatures

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