By B. H. Chirgwin
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Extra resources for Elementary Classical Hydrodynamics: The Commonwealth and International Library: Mathematics Division
3 later) that the kinetic energy of the fluid is finite. Mathematical investigation of the irrotational motion of an incompressible inviscid fluid in given circumstances, then, reduces to finding a solution > of Laplace's equation subject to certain boundary conditions on d(j)/dv; similar problems occur in other contexts such as electrostatics. e. there is only one irrotational flow pattern which satisfies the given boundary conditions. To prove the uniqueness theorem of hydrodynamics we assume that there are two velocity potential functions
5) In this case the equation of motion becomes and we deduce that where F(t) is an arbitrary function of time only. The important point is that, at any instant, the left-hand side of eqn. 6) has the same value at all points of the region of irrotational motion, and not merely along a streamline. 1 SOME GENERAL THEOREMS 37 3. e. w = f - ^ + F - f - ^ v 2 = constant. 7) For an incompressible fluid Q is constant and 1 iv = — + V+— v2 = constant. 7a) hold throughout the region of irro tational motion for all times.
For lines inclined at other angles the flow has intermediate values. This concept of "flow" is given a precise measure by defining (v*e) / to be the flow along a straight line of length / in a direction e drawn in a fluid having uniform velocity v. 12) JA where C is the flow between A and B along the path of the lineintegral. If the path chosen is a closed loop r, and A = B, then the flow around the loop is called the circulation and is given by C = (Cv-ds. ) Example. If the streamlines in a fluid are circles around which the velocity is uniform, the circulation around a circle of radius a is C = 2nav = 2na2o) where co is the angular velocity of a particle of the fluid about the centre of the circle.