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Differential Analysis of Fluid Flow (continued)

Special Case of Fluid motion in the xy-plane (continued)

Streamlines are lines in the flow field that are everywhere tangent to the fluid velocities.

So along each streamline the value of ψ is a constant. Since flow never crosses

streamlines, the difference in the values of ψ between adjacent streamlines equals

the volume rate of flow per unit width, q₁ and q₂ between streamlines.

See the figure below for streamlines ψ = ψ₁, ψ = ψ₂, and ψ = ψ₃

Assume ψ₁> ψ₂ . Then q₁ as shown and q₁ = ψ₁ - ψ₂

Assume ψ₂> ψ₃ . Then q₂ as shown and q₁ = ψ₂ - ψ₃

For the case where ψ₁< ψ₂ , then q₁ = ψ₂ - ψ₁ in the flow is in the opposite direction