Question

The stream function of a two-dimensional flow field is given as \( \psi = 2xy + 2y + 2x \). The coordinates of two points \( P \) and \( Q \) in the flow field are \( (1, 2) \) and \( (2, 5) \) respectively. The magnitude of flow discharge between the streamlines passing through \( P \) and \( Q \) is ________ (answer in integer).

Hint:

In 2D incompressible flow, the stream function \( \psi \) is constant along a streamline. The discharge between two streamlines is given by the absolute difference in their stream function values.

Answer 1

In a 2D incompressible flow, the difference in stream function values between two points gives the volumetric flow rate (discharge) per unit depth between those streamlines: \[ Q = |\psi_Q - \psi_P| \] Compute \( \psi \) at point \( P(1, 2) \): \[ \psi_P = 2(1)(2) + 2(2) + 2(1) = 4 + 4 + 2 = 10 \] Compute \( \psi \) at point \( Q(2, 5) \): \[ \psi_Q = 2(2)(5) + 2(5) + 2(2) = 20 + 10 + 4 = 34 \] Then, the discharge between the streamlines is: \[ Q = |34 - 10| = 24 \]