Concept of Continuum - contd from previous slide

• One of the factors considered important in determining the validity of continuum model is molecular density. It is the distance between the molecules which is characterised by mean free path ( λ ). It is calculated by finding statistical average distance the molecules travel between two successive collisions. If the mean free path is very small as compared with some characteristic length in the flow domain (i.e., the molecular density is very high) then the gas can be treated as a continuous medium. If the mean free path is large in comparison to some characteristic length, the gas cannot be considered continuous and it should be analysed by the molecular theory.
  
  
• A dimensionless parameter known as Knudsen number, K _n = λ / L, where λ is the mean free path and L is the characteristic length. It describes the degree of departure from continuum.

  Usually when K _n> 0.01, the concept of continuum does not hold good.

  Beyond this critical range of Knudsen number, the flows are known as

  slip flow (0.01 < K _n < 0.1),

  transition flow (0.1 < K _n < 10) and

  free-molecule flow (K_n > 10).

However, for the flow regimes considered in this course , K _n is always less than 0.01 and it is usual to say that the fluid is a continuum.

Other factor which checks the validity of continuum is the elapsed time between collisions. The time should be small enough so that the random statistical description of molecular activity holds good.


In continuum approach, fluid properties such as density, viscosity, thermal conductivity, temperature, etc. can be expressed as continuous functions of space and time.