For the following stream function calculate velocity at a point (1, 2)

1) ψ = 4xy

2) ψ = 3x²y - y³

Concept:

Stream Function:

It is defined as the scalar function of space and time, such that its partial derivative with respect to any direction gives the velocity component at right angles to that direction.

It is denoted by ψ and defined only for two-dimensional flow.

\(v = \frac{{\partial \psi }}{{\partial x}};u = - \frac{{\partial \psi }}{{\partial y}}\)

Properties of Stream function:

• If stream function exists, it is a possible case of fluid flow which may be rotational or irrotational
• If the stream function satisfies the Laplace equation i.e. \(\frac{{{\partial ^2}\psi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\psi }}{{\partial {y^2}}} = 0\), it is a case of irrotational flow

Calculation:

Let u and v are horizontal and vertical components of fluid velocity.

Given that, 

1) ψ = 4xy

We know that:

\({\rm{u}} = - \frac{{\partial {\rm{\psi}}}}{{\partial {\rm{y}}}} = - 4{\rm{x}}\)

And

\({\rm{v}} = + \frac{{\partial {\rm{\psi}}}}{{\partial {\rm{x}}}} = 4{\rm{y}}\)

Velocity, = ui + vj

\(\vec v\) = (-4x) i + (4y) j

Velocity at point (1, 2) i.e. at x = 1 and y = 2 is:

\(\vec v\) = - 4i + 8j

Or \(\left| {{\rm{\vec v}}} \right| = {\rm{v}} = \sqrt {{{\left( { - 4} \right)}^2} + {{\left( 8 \right)}^2}} = \sqrt{80} ~{\rm{m}}/{\rm{s}}\)

Given that, 

2) ψ = 3x2y - y3

\({\rm{u}} = - \frac{{\partial {\rm{\psi}}}}{{\partial {\rm{y}}}} = -3x^2+3y^2\)

And

\({\rm{v}} = + \frac{{\partial {\rm{\psi}}}}{{\partial {\rm{x}}}} = 6x{\rm{y}}\)

Velocity, = ui + vj

v = (-3x² + 3y²)i + (6xy)j

Velocity at point (1, 2) i.e. at x = 1 and y = 2 is:

v = -9i + 12j

\(\left| {{\rm{\vec v}}} \right| = {\rm{v}} = \sqrt {{{\left( { - 9} \right)}^2} + {{\left( 12 \right)}^2}} =15 ~{\rm{m}}/{\rm{s}}\)