The product of slope of equipotential line and the slope of streamline at the point of intersection is equal to _______.

Concept:

Streamline: It is an imaginary curve drawn in space such that tangent drawn to it at any point will give the velocity of that fluid particle at a given instant of time. A line along which stream function (ψ) is constant is known as streamline.

Equipotential line: A line along which velocity potential function (ϕ) is constant is known as the equipotential line.

\({\left( {\frac{{dy}}{{dx}}} \right)_ϕ } \times {\left( {\frac{{dy}}{{dx}}} \right)_ψ } = - 1\)

Slope of equipotential Line × Slope of stream function = -1

They are orthogonal to each line other.

For a streamline, \(ψ(x,y)=constant\) and the differential of ψ  is zero.

\(dψ=\frac{\partialψ}{\partial x}dx+\frac{\partialψ}{\partial y}dy\)

\(dψ=-vdx+udy\)

\((\frac{{\partial y}}{{\partial x}} )_{ψ=const}= \frac{v}{u}\)

For an equipotential line, \(ϕ(x,y)=constant\) and the differential of ϕ is zero.

\(dϕ=\frac{\partialϕ}{\partial x}dx+\frac{\partialϕ}{\partial y}dy\)

\(dϕ=udx+vdy\)

\((\frac{{\partial y}}{{\partial x}} )_{ϕ=const}= -\frac{u}{v}\)

\((\frac{{\partial y}}{{\partial x}} )_{ψ=const}=- \frac{1}{(\frac{{\partial y}}{{\partial x}} )_{ϕ=const}}\)