.In the problems such as rarefied gas flow (as encountered in flights into the upper reaches of the
atmosphere ) , we must abandon the concept of a continuum in favour of microscopic and statistical
point of view.
As a consequence of the continuum assumption, each fluid property is assumed to have a definite value
at every point in the space .Thus fluid properties such as density , temperature , velocity and so on are
considered to be continuous functions of position and time .
Consider a region of fluid as shown in fig 1.5. We are interested in determining the density at
. Thus the mean density V would be given by ρ=
. In general, this will
not be the value of the density at point ‘c’ . To determine the density at point ‘c’, we must select a small
, surrounding point ‘c’ and determine the ratio
and allowing the volume to shrink
continuously in size.
Assuming that volume
is initially relatively larger (but still small compared with volume , V) a
typical plot might appear as shown in fig 1.5 (b) . When
becomes so small that it contains only a
small number of molecules , it becomes impossible to fix a definite value for
; the value will vary
erratically as molecules cross into and out of the volume. Thus there is a lower limiting value of
ꞌ . The density at a point is thus defined as
Since point ‘c’ was arbitrary , the density at any other point in the fluid could be determined in a like
manner. If density determinations were made simultaneously at an infinite number of points in the fluid ,