160 



J. C. Decius. 



[J. F. 1. 



S. APPLICATION TO HYDRODYNAMICS. 



Consider the dimensional composition of the variables appearing in 

 the analytical description of fluid flow. The Navier-Stokes equation 

 and the equation of continuity contain variables of the following dimen- 

 sional types: v = velocity, p = pressure, R = any length, / = time, 

 p = density, m = viscosity. If the external forces are due to gravity, 

 the acceleration of gravity, g, is included. In addition, there exists an 

 equation of state which expresses p as a function of p alone in case the 

 flow is assumed to be isothermal or adiabatic. The implication of this 

 last condition is important in the consideration of possible scaling solu- 

 tions: although the specific form of the equation of state is unknown, it 

 must be expressible in terms of variables, such as the bulk modulus, 

 whose columns in A are linear combinations of the columns in A repre- 

 senting p and p. In order to maintain, temporarily, the generality of 

 the discussion, it will be supposed that the density is given by the series: 



P = Z piipy 



(5.1) 



with a S CO. 



The A matrix is then as follows: 



A model experiment will now be considered in which R plays the role 

 of the deliberately scaled variables: n^ = 1. Then the criterion of 

 Eq. 4.1 requires rj/ = r/ -+- 1. If the medium is unchanged with scale, 

 in the general case the rank of (p,, n) is 3 so that, since r^f also equals 3, 

 scaling is impossible. Even for an incompressible fluid (p( = except 

 for i = 0) a scaling solution with fixed g is impossible if ix is fixed. The 

 idealizations which lead to the familiar scaling approximations in terms 

 of the Froude, Reynolds, and Mach numbers are described in Table I. 

 In each case the set of variables designated as "ignored" are shown to be 



Table I. 

 Scaling Laws for Hydrodynamics. 



