৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

For 2D flow, Ψ is constant along a streamline.
The curves Ψ = constant represent streamlines.

Definition of stream function ensures incompressibility: ∇·q = 0.
Relation: u = ∂Ψ∂y, v = −∂Ψ/∂x

u = ∂Ψ/∂y = ∂(xy)/∂y = x, v= −∂Ψ/∂x = −∂(xy)/∂x = −y

Stream function  is constant along stream line.
Ψ = constant → x² + y² = constant → circles around origin

A stream function ensures continuity in 2D incompressible flow, but irrotationality requires ∇ × q = 0.

Ψ = const is always a streamline.

Stream function Ψ → continuity satisfied (incompressible)
Velocity potential φ → irrotational condition satisfied.
So both together ⇒ incompressible & irrotational.

These are the Cauchy–Riemann equations in fluid flow form.

ক) Yes, because continuity holds → Continuity is necessary but not sufficient.×
খ) Yes, because Cauchy–Riemann equations are satisfied →  Correct √
গ) No, because ∇×q≠0 → vorticity is 0 ×
ঘ) No, because flow is compressible → the flow is incompressible ×

w(z) is the complex potential, widely used in potential flow theory.

q_r​ = -cosθ, q_θ ​= sinθ.

Check with Ψ  =-rsinθ:

q_{r ​}= (1/r) ∂Ψ/∂θ = (1/r)(-rcos⁡θ) = -cos⁡θ, 
q_θ = -∂Ψ /∂r = sin⁡θ
Matches the velocity field.

Flow is irrotational but velocities vary with position → this is a pure extensional (strain) flow.

Flow is irrotational → velocity potential exists.

In 2D incompressible flow, $\Delta \Psi =\text{flow rate per unit depth}$ between two streamlines.

Continuity condition for incompressible flow.

At (x,y)=(2,1): u=−3, v=−6