J. W. Davis — Motion of Compressible Fluids. 107 



Aet. XI. — On the Motion of Compressible Fluids ; by 



J. WOODBRTDGE DaYIS. 



In the case of steady motion of a perfect fluid, the Eulerian 

 differential equations of motion can be reduced to one equa- 

 tion in which the variation takes place along a stream-line. 

 The integral of this is 



/ 



* = _o_£ + c* (i) 



Applied to each elemental tube of flow, that is, to a tube of 

 variable infinitesimal thickness whose boundary is composed 

 of stream-lines, the equation of continuity becomes 



<rqp = M.f (2) 



p, p, q, cr, XI, are the pressure, density, velocity, area of cross- 

 section, and potential of extraneous forces per unit mass, at 

 the variable distance s measured along the tube; C is the con- 

 stant total energy per unit mass carried along the tube ; M is 

 the constant quantity of fluid that passes any section in unit 

 time. /?, a, M, are necessarily positive in a physical problem : 

 we may always assume that the direction of the motion is posi- 

 tive ; and this we have done in forming equation (2). Hence 

 q also will always be positive, if the motion is physically real. 



When the fluid is compressible, and the pressure is a single- 

 valued function of density only, such that, for all positive 

 values of density, the density increases with increase of pres- 

 sure, and diminishes with relief of pressure, and the ratio of 

 the increment of pressure to the increment of density either 

 remains constant or increases with compression, we may write 



n, ^ dp . . . d 2 p . 

 p =f(p), -y-is positive; -y^- is zero, or is positive. (3) 

 up dp 



From (1), (3), 



'dp 



H/ 



= + |0-O-i-J 



From (2), (4), 



,*{c- -f} = ^ ( 5) 



cr is a single-valued function of s ; and so is XI for. the 

 forces of nature. But q is apparently not single-valued. At 

 any point, it is to be found in terms of a and XI, by the solu- 

 tion of equation (5), which is equivalent to finding the inter- 



* Lamb's Hydrodynamics, p. 23, eq. (3). 

 f L. c. p. 23, Art. 23. 



Am. Jour. Sci. — Fourth Series, Yol. XII, No. 68.— August, 1801. 



