Continuum Hypothesis in Fluid Mechanics The macroscopic behavior of fluids makes them appear to be continuous. However, when viewed at the microscopic scale fluids cannot be viewed as continuous. The fluid under consideration will have molecules bombarding each other. It is not possible to declare the fluid velocity at a point as there is no guarantee that the fluid molecules are present at that point at a particular instant of time.
When we calculate the fluid velocity or density at a point it implies that the value is the average fluid velocity or density of the fluid molecules passing through a small volume surrounding that point. The size of the small volume has to be smaller than the physical region under consideration. However, the size of the volume cannot be extremely small. It has to be large enough to make the averaging meaningful. Fluid can be considered as a continuous medium in situations considering the fluid properties over distances greater than the average spacing between the fluid molecules.
In situations where microscopic details of the fluid are important continuum hypothesis does not apply. An example of such a situation can be fluid flowing through a channel whose dimension is equal to the molecule size or mean free paths of the fluid molecules. For fluids, Knudsen number helps in assessing continuity approximation. If the Knudsen number is greater than or equal to one then the mean free paths of a molecule is comparable to the length scale of the problem. In such a case the continuity hypothesis is not a good assumption.