Continuum Hypothesis

We know very well that all matter is made up of molecules, which are in random motion. Any fluid we consider has molecules bombarding each other and the boundaries, i.e. the walls of the container. There is no guarantee whatever that molecules are present at that point at a given instant of time. But still we say that fluid velocity at a point is so many meters per second or that density is so many kgs per square meter. Where is the justification for this? Of course, we can say that we define density or velocity at point in an average sense. That is as an average of velocities (or densities) of the molecules that pass through a small volume surrounding that point. The size of this small volume has to meet with certain criteria. It must be smaller than the physical dimensions of the region under consideration like the wing of an aircraft or the pipe in a hydraulic system. At the same time it must be sufficiently large to accommodate a large number of molecules to make any averaging meaningful. It seems that there is a lower limit to the size of this volume.

Figure 1.3: Definition of Density

 

The existence of this limit is established by considering the definition of density as mass per unit volume ( $\delta m/ \delta \forall$). Consider a small volume $\delta \forall$ around the point P (Fig. 1.3) within the region of interest, R. Let us calculate density at P by considering different sizes of $\delta \forall$. Values of density so calculated are plotted in the same figure. It is clear that the size $\delta \forall$ has an enormous influence on the calculated value of density. Too small a $\delta \forall$, the value of calculated density fluctuates because the number of molecules within $\delta \forall$ is varying significantly with time. Too big a $\delta \forall$ might mean that density itself is varying significantly within the region of interest. As seemed before it is clear that there is a limit $\delta \forall_0$ below which molecular variations assume importance and above which one finds a macroscopic variation of density within the region. Therefore it appears that density is best defined as a limit -

$$\rho = \lim_{\delta \forall \rightarrow \delta \forall_0} {\delta m \over \delta \forall}$$ (1.1)

At Standard Temperature and Pressure conditions (STP) the limit ( $\delta \forall_0$)is around $10^{-9}$ $mm^{3}$ and air at this tiny volume has about $3$x$10^{7}$ number of molecules. This is a large enough number to give a constant value of density despite the rigorous molecular motion within it. For many of the applications in Fluid Mechanics, this volume is smaller than the overall dimensions of the regions of interest considered such as an aeroplane, wing of an aeroplane, ship or the parts of an engine etc. These considerations do not hold good when we go to greater altitudes. For example, at an altitude of 130 km the molecular mean free path is about 10.2 m and there are only $1.6$x$10^{17}$ molecules in a cubic meter of air(Molecular Mean Path, $\lambda$ is defined as the average distance a molecule has to travel before it collides with another molecule. At STP conditions its value is $6\times 10^{-8}m$). Under these conditions it becomes necessary to consider the effect of every molecule or groups of molecules, as in calculations concerning re-entry vehicles. That branch of fluid mechanics is called Rarefied Gasdynamics.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney