7. THE LAPLACE EQUATION 



In an incompressible fluid of uniform density the velocity potential satisfies a very 

 simple differential equation. If the velocity potential (/> is substituted from Equations [6b, c, d] 

 into the equation of continuity, Equation [2b], there results 



^^,^_!^,^.0 [7a] 



dx'^ dy^ dz^ 



This equation must be satisfied at every point throughout the region in which irrotational flow 

 exists. It is known as the Laplace equation and is often written in the symbolic form^ d, = 

 where ^^ stands for the differential operator 



dx'^ dy^ ds^ 



The Laplace equation is encountered in many other brandies of physics, such as 

 electricity, heat flow, and elasticity, and the properties of its solutions are well known.^^' ^•^' ^^ 

 Because of the linearity of the Laplace Equation, its solutions possess the following useful 

 properties. If (^ is a solution, so is Ccp where C is any constant. If and ch^ are two solu- 

 tions, (f)^ + c/), is another solution; the particle velocity corresponding to 0, + (^^ ^^ ^"Y point 

 is the vector sum of the velocities corresponding to 0. and to oS,. These statements may be 

 verified by substitution in the equation. Finally, if 0, is a solution, so is 02 = dcfi./dx or 

 d4>^/ dy or c?0j/ dz; for, after substituting 0j for in Equation [7a], differentiating witli 

 respect to x, for example, and changing the order of integration, 



^501^^ ^ + ii ^ = 



Thus d<h^/dx is another solution of the Laplace equation. 



The problem of determining the motion of a frictionless, incompressible fluid under 

 given conditions thus reduces to the problem of solving the Laplace equation subject to 

 certain boundary conditions. Any solution of the equation represents a possible type of 

 irrotational flow in which the components of the velocity are given in terms of by Equations 

 [6b, c, d]. Since the density of the fluid does not occur either in the Laplace equation or in 

 Equation [4a] expressing the usual boundary condition, each type of flow can exist in a fluid 

 of any density. So long as the motion is not too rapid, gases as well as liquids can be assumed 

 to move approximately as if they were incompressible. 



Once the velocity at each point is known, the distribution of pressure may be found 

 from the pressure equation to be obtained presently. s. 



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