Question

A two dimensional flow has velocities in x and y directions given by u = 2 xyt and v = -y²t, where t denotes time. The equation for streamline passing through x = 1, y = 1 is
1. x²y = 1
2. x²y² = 1
3. x/y² = 1
4. xy² = 1

Answer 1

Correct Answer - Option 4 : xy² = 1

Concept: 

A streamline is an imaginary curve drawn in space such that tangent drawn to it at any point will give the velocity of fluid-particle at a given instant of time. 

The equation of streamline for 2-D flow is given as:

\( \frac{{{\rm{dx}}}}{{\rm{u}}} = \frac{{{\rm{dy}}}}{{\rm{v}}}\)

where u and v are the x and y components of the velocity.

Calculation:

Given:

u = 2xyt, v = -y²t

\(\frac{{dx}}{{2xyt}} = \frac{{dy}}{{-y^2t}} \Rightarrow \smallint \frac{{dx}}{x} = - 2\smallint \frac{{dy}}{y}\)

In x = -2In y + ln C

ln x + ln y² = ln C

ln xy² = ln C

xy² = C

When equation for streamline passing through x = 1, y = 1 then C = 1

∴ xy2 = 1 is the equation of streamline passing through x = 1, y = 1.