MECH 361 Lecture Notes - Lecture 16: Conservative Vector Field, Stream Function, Velocity Potential
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Consider the irrotational flow which is given by the following velocity potential: where u(cid:0) is a constant. It is obvious that satisfies laplace equation and therefore, the flow is potential. In order for us to visualize this flow, it is important to derive an equation for the stream function x, y ) . From eq. (7. 2. 21) we have: x + constant. X = u (7. 6. 1) u = However, from eqs. (7. 6. 2) and (7. 3. 1), v = From eqs. (7. 6. 2) and (7. 3. 1) we also have, u = = u f y = u y + const. Y = u y + constant and therefore. Since all the operations involve either the differences or derivatives of and , then we can set the constant of integration equal to zero without loss of generality. Solving eq. (7. 6. 1) for x, and eq. (7. 6. 2) for y we obtain: (7. 6. 3) (7. 6. 4) (7. 6. 5) x = Both the above equations represent families of straight lines.