Continuum Hypothesis and Transport Phenomena

Continuum Hypothesis 

A fluid, or any other substance for that matter, is composed of a large number of molecules in constant motion and undergoing collisions with each other. Matter is therefore discontinuous or discrete at microscopic scales. In principle, it is possible to study the mechanics of a fluid by studying the motion of the molecules themselves, as is done in kinetic theory or statistical mechanics. However, we are generally interested in the gross behaviour of the fluid, that is, in the average manifestation of the molecular motion. For example, forces are exerted on the boundaries of a container due to the constant bombardment of the molecules; the statistical average of this force per unit area is called pressure, a macroscopic property. So long as we are not interested in the mechanism of the origin of pressure, we can ignore the molecular motion and think of pressure as simply “force per unit area.” 

It is thus possible to ignore the discrete molecular structure of matter and replace it by a continuous distribution, called a continuum. For the continuum or macroscopic approach to be valid, the size of the flow system (characterized, for example, by the size of the body around which flow is taking place) must be much larger than the mean free path or the molecules. For ordinary cases, this is not a great restriction, since the mean free path is usually very small. For example, the mean free path for standard atmospheric air is ≈ 5 x 10^-8 m. In special situations, the mean free path of the molecules can be quite large and the continuum approach breaks down. In the upper altitudes of the atmosphere, for example, the mean free path of the molecules may be of the order of a matter, a kinetic theory approach is necessary for studying the dynamics of these rarefied gases. 

Transport Phenomena 

Consider a surface area AB within a mixture of two gases, say nitrogen and oxygen (Fig. 1), and assume that the concentration C of nitrogen (kilograms of nitrogen per cubic metre of mixture) varies across AB. 

Fig. 1 Mass flux qm due to concentration variation C(y) across AB

Random migration of molecules across AB in both directions will result in a net flux or nitrogen across AB, from the region  of higher C towards the region of lower C. Experiments show that, to a good approximation, the flux of one constituent in a mixture is proportional to its concentration gradient and it is given by 

                                       qm = - km Δ C                                  (1) 

Here the vector qm is the mass flux (kg m^-2 s^-1) of the constituent, Δ C is the concentration gradient of that constituent, and km is a constant of proportionality that depends on the particular pair of constituents in the mixture and the thermodynamic state. For example, km, for diffusion of nitrogen in a mixture with oxygen is different than km, for diffusion of nitrogen in a mixture with carbon dioxide. The linear relation (1) for mass diffusion is generally known as Fick's law. 

Relations like these are based on empirical evidence and are called phenomenological laws. Statistical mechanics can sometimes be used to derive such laws, but only for simple situations. The analogous relation for heat transport due to temperature gradient is Fourier's law and it is given by 

                                         q = - k ΔT                                         (2) 

where q is the heat flux (J m^-2 s^-1), ΔT is the temperature gradient, and k is the thermal conductivity of the material.

Fig. 2 Shear stress ‘τ’ on surface AB. Diffusion tends to decrease velocity gradients, so that the continuous line tends toward the dashed line

Next, consider the effect of velocity gradient du/dy (Fig. 2). It is clear that the macroscopic fluid velocity ‘u’ will tend to become uniform due to the random motion of the molecules, because of intermolecular collisions and the consequent exchange of molecular momentum. Imagine two railroad trains traveling on parallel . tracks at different speeds and workers shovelling coal from one train to the other. On the average, the impact of particles of coal going horn the slower to the faster train will tend to slow down the faster train, and similarly the coal going from the faster to the slower train will tend to speed up the latter. The net effect is a tendency to equalize the speeds of the two trains. An analogous process takes place in the fluid flow problem of Fig. 2. The velocity distribution here tends toward the dashed line, which can be described by saying that the x-momentum (determined by its “concentration” u) is being transferred downward. Such a momentum flux is equivalent to the existence of a shear stress in the fluid, just as the drag experienced by the two trains results from the momentum exchange through the transfer or coal particles. The fluid above AB tends to push the fluid underneath forward, whereas the fluid below AB tends to drag the upper fluid backward. Experiments show that the magnitude of the shear stress ‘τ’ along a surface such as AB is, to a good approximation, related to the velocity gradient by the linear relation 

which is called Newton’s law of friction. Here, the constant of proportionality ′µ′ (whose unit is kg m^-1 s^-l) is known as the dynamic viscosity, which is a strong function of temperature T. For ideal gases the random thermal speed is roughly proportional to √𝑇, the momentum transport, and consequently µ, also vary approximately as √𝑇. For liquids, on the other hand, the shear stress is caused more by the intermolecular cohesive forces than by the thermal motion of the molecules. These cohesive forces, and consequently µ for a liquid, decrease with temperature.