If the pressure in the cavity or bubble is p^, then p^ = p at r = /? or 



"■ R dt \ dt I 2 \dt I 



[120h] 



If Pj^ is known as a function of ff, this differential equation determines /? as a function of 

 the time. 



The kinetic energy of the fluid is 



T = — p \ q/- (i77T^) dr = 27rpA 



2 I I ,2 



JR 



dr A^ ^/dRV , , 



Inp — - = lTjpR^\-^-\ [120i] 



\dt I 



from [120d]. (See Reference 1, Articles 56, 91a; Reference 2, Section 15.20.) 



121. POINT SOURCE IN A UNIFORM STREAM 



Let the flow due to a point source be superposed upon a uniform streaming motion. 

 Take the origin at the source and the a;-axis parallel to the flow at infinity. Since the motion 

 is then axisymmetric about the a;-axis, it suffices to take as a second coordinate the distance 

 'cS from the axis, and to study the flow in a single plane; see Figure 201. 



U 



Figure 201 — A point source at 0, in a uniform stream. See Section 121. 

 The potential and stream functions can be written, from Equations [119a, b] and [119f], 



cf,^ V L^ — y 4, = v{~ ^2 ^ 62 Ij^ j.^Jl 



[121a, b,c] 



Here 6 is a positive constant and U denotes the velocity at infinity, taken positive when 

 directed toward negative x. The x and sr components of velocity are, from [118h,i], 



305 



