Transport Properties

4.2.1 Introduction

The molecular transport of the three entities momentum, energy, and mass basically obeys similar laws, which combine linearly the gradients of flow velocity, temperature and species concentration with the coefficients of the respective transport properties: viscosity, thermal conductivity, diffusivity, see, e. g., [2]. Fluids, which can be described in this way are called “Newtonian fluids”.

The basic formulation for molecular momentum transport is Newton’s law of friction

(4.9)

Here Tyx is the shear stress exerted on a fluid surface in x-direction by the y-gradient of the velocity u in that direction. The coefficient p is the fluid viscosity.

Similarly, Fourier’s law of heat conduction reads:

The heat flux in the y-direction, qy, is proportional to the temperature gradient in that direction. The coefficient к is the thermal conductivity. If thermo-chemical non-equilibrium effects are present in the flow, heat is trans­ported also by mass diffusion, Section 4.3.2. Note that always the transport is in the negative y-direction.

This also holds for the molecular transport of mass, which is described by Fick’s first law

jAy = —jBy = – pDAB^j~. (4.11)

The diffusion mass flux jAy is the flux of the species A in y-direction relative to the bulk velocity v in this direction. It is proportional to the mass-fraction gradient d^A/dy in that direction. Dab = DBa is the mass diffusivity in a binary system with the species A and B. The reader is referred to, e. g., [2] for equivalent forms of Fick’s first law. Besides the concentration- driven diffusion also pressure-, and temperature-gradient driven diffusion can occur, Sub-Section 4.3.3. Molecular transport of mass occurs in flows with thermo-chemical non-equilibrium, but also in flows with mixing processes, for instance in propulsion devices.

The transport properties viscosity p, thermal conductivity к, and mass diffusivity DaB of a gas are basically functions only of the temperature. We give in the following sub-sections relations of different degree of accuracy for the determination of transport properties of air. Emphasis is put on simple power-law approximations for the viscosity and for the thermal conductivity. They can be used for quick estimates, and are also used for the basic analytical
considerations throughout the book. Of course they are valid only below approximately T = 2,000 K, i. e., for not or only weakly dissociated air. Models of high-temperature transport properties are considered in Sub-Section 4.2.5.