The classical streamfunction

Properties of the simple streamfunction. Let us consider first the two-dimensional, steady, constant density flow of a liquid in homogeneous, isotropic media. The Darcy flow, in this case, satisfies Laplace s equation 

Equation 4-4 suggests that we can define a function T (X, Y) such that 

Equations 4-5 and 4-6, after all, are simply relationships that introduce no additional assumptions for example, substitution in Equation 4-4 yields a trivial 0 = 0. However, the function T, or streamfunction, possesses interesting properties. 

let us consider the kinematic definition of a streamline. A streamline is a flow trajectory across which fluid motion is absent fluid moves tangentially to it. Thus, its local slope must be equal to the ratio of the vertical to the horizontal velocities, 

If we now substitute, initially Equations 4-2 and 4-3, and then Equations 4-5 and 4-6 into Equation 4-7, we obtain successively 

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When m = 0, as for liquid flows satisfying the linear pressure equation, our P(x,y) reduces to the classical streamfunction. But the concept applies equally to steady, nonlinear gas flows, and similar properties for T are obtained. To see this, let us divide Equation 9-23 by Equation 9-24 that is,... 

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