Question

For a two-dimensional incompressible flow field given by \(\vec u = A\left( {x\hat i - y\hat j} \right)\), where A > 0, which one of the following statements is FALSE?
1. It satisfies continuity equation.
2. It is unidirectional when x → 0 and y → ∞
3. Its streamlines are given by x = y .
4. It is irrotational.

Answer 1

Correct Answer - Option 3 : Its streamlines are given by x = y .

Concept:

Continuity equation for a two – dimensional flow of an incompressible fluid is:

\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\)

For the irrotational flow field:

\({\omega _z} = \frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}} = 0\)

Streamline equation:

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

Calculation:

\(\vec u = A\left( {x\hat i - y\hat j} \right)\)

u = Ax, v = -Ay

Continuity equation for a two – dimensional flow of an incompressible fluid is:

\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 0\)

\(\frac{\partial }{{\partial x}}\left( {Ax} \right) + \frac{\partial }{{\partial y}}\left( { - Ay} \right) = A - A = 0;Satisfied\)

When x → 0 and y → ∞ ⇒

\(\vec u = - Ay\hat j;unidirectional\)

For the irrotational flow field:

\({\omega _z} = \frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}} = 0\)

\(\frac{{\partial v}}{{\partial x}} - \frac{{\partial u}}{{\partial y}} = \frac{\partial }{{\partial x}}\left( { - Ay} \right) - \frac{\partial }{{\partial y}}\left( {Ax} \right) = 0;Irrotational\)

Streamline equation:

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

\(\Rightarrow \frac{{dx}}{{Ax}} = \frac{{dy}}{{ - Ay}} \Rightarrow \frac{{dx}}{x} = \frac{{dy}}{{ - y}}\)

\(\Rightarrow \ln x = - \ln y + \ln C \Rightarrow \ln x + \ln y = \ln C\)

\(\Rightarrow \ln xy = \ln C \Rightarrow xy = c\)

So, x = y is not the equation of streamline.