Prove that the difference of values of the steam function at the two points represents the flux of the fluid across any join the two points.

Steps to Solve

1. Understand the Steam Function The steam function $\psi$ in fluid dynamics is used to describe the flow of incompressible fluids. It is defined such that the velocity components of the fluid can be derived from it. Specifically, for a two-dimensional flow, the velocity components can be expressed as: $$ u = \frac{\partial \psi}{\partial y} $$ $$ v = -\frac{\partial \psi}{\partial x} $$

2. Define the Points and the Join Consider two points in the fluid flow: point 1 and point 2, with steam function values $\psi_1$ and $\psi_2$, respectively. We want to analyze the relationship between the difference in these values and the fluid flux across a join (a line) between these two points.

3. Calculate the Flux Across the Join The flux $Q$ across the join can be determined by integrating the fluid flow across the area defined by the two points. This is given by: $$ Q = \int_{A} \left( u , dy + v , dx \right) $$ Substituting the velocity components $u$ and $v$, we have: $$ Q = \int_{A} \left( \frac{\partial \psi}{\partial y} dy - \frac{\partial \psi}{\partial x} dx \right) $$

4. Evaluate the Integral Using the property of line integrals, we can express the net flux across the join in terms of the steam function: $$ Q = \psi_1 - \psi_2 $$ This shows that the flux across the join is directly proportional to the difference in values of the steam function at the two points.

5. Final Proof Statement Thus, we can conclude that: $$ \psi_1 - \psi_2 = Q $$ indicating that the difference in the steam function values indeed represents the flux of the fluid across any join between those two points.