10. THE BERNOULLI EQUATION FOR STEADY IRROTATIONAL FLOW 



If the motion is steady, the constant of integration in <^ can be chosen so that dcj^/dt = 0. 

 Then Fit) reduces to a constant, and the pressure equation [9d] can be written 



/ 



dp 1 ^ 



— +-i-q^ + n = C [10a] 



P 2 



where C is a constant. Similarly, if p is uniform and constant and if the pressure at infinity 

 or on the boundary does not vary with time. Equations [9e] and [9g] become, respectively, 



P 2 



lA + kq2 + ^^C. [10c] 



P 2 



The value of C, which is not necessarily the same in these three equations, may be 

 found from the known values of the other quantities at some one point, such as a point on the 

 boundary. 



In many problems, the motion at infinity is one of uniform flow and fi = 0. Then, if U 

 is the particle velocity and p^ the pressure at infinity, 



i^oc 1 



C = + - {/2 



P 2 



and from Equation [10b], for incompressible fluid, 



P-?'cc=|p(f^^-?') [lOd] 



It is important to note that the pressure difference, p - p^, at any point depends only upon the 

 relative motion between the fluid at the point and the fluid at infinity; in particular, it remains 

 the same if a frame of reference is substituted relative to which V is zero. It is physically 

 obvious that p — p^ cannot be altered by a mere change of the frame of reference; and it is 

 easily verified that the resulting changes in t/and q are such that the difference [/^ - ?^ 

 remains unchanged. 



These are various forms of what is commonly called the Bernoulli equation for irrotation- 

 al motion. For any type of steady flow, whether irrotational or not, equations identical in form 

 can be obtained for any one streamline, but in general the constant may vary from one stream- 

 line to another. In irrotational flow the constant C has the same value for all streamlines. 



The Bernoulli equation holds throughout any region, large or small, throughout which 

 the motion happens to be irrotational. The region may even surround one or more cylinders 

 about which there is circulation; irrotationality in the neighborhood of each point of the region 

 is all that is required. 



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