Prove that the difference of values of the steam function at the two points equals the flux of the fluid across any line joining the two points.

Steps to Solve

1. Define the Steam Function The steam function, $\psi(x, y)$, is a scalar function used in fluid dynamics that represents the flow of a fluid. The function is defined such that its contours represent streamlines of the fluid flow.

2. Establish Two Points Let’s consider two points in the fluid flow, denoted as $A(x_1, y_1)$ and $B(x_2, y_2)$.

3. Differentiate the Steam Function The difference in values of the steam function at points $A$ and $B$ is given by: $$ \Delta \psi = \psi(B) - \psi(A) = \psi(x_2, y_2) - \psi(x_1, y_1) $$

4. State the Relationship with Flux The flux of fluid across a line joining points $A$ and $B$ can be expressed through the normal component of the velocity vector, $\mathbf{v}$, across that line. The relationship can be expressed using the line integral of the velocity field: $$ \text{Flux} = \int_{A}^{B} \mathbf{v} \cdot \mathbf{n} , ds $$ Here, $\mathbf{n}$ represents the normal direction to the line, and $ds$ is the differential arc length.

5. Equate the Two Quantities For incompressible and irrotational flow, it can be shown that: $$ \Delta \psi = \int_{A}^{B} \mathbf{v} \cdot \mathbf{n} , ds $$ This means that the difference in the steam function values at two points is equal to the flux of the fluid across the line joining those two points.