৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

For a source at the origin, fluid flows radially outward.
The potential function in polar coordinates is defined as

ϕ=m lnr
This comes from integrating the radial velocity q_r​=m​/r.
The negative form −m lnr corresponds to a sink, not a source.

A sink is the opposite of a source: fluid flows radially inward.
The radial velocity is obtained from the potential:

q_r​=∂ϕ​/∂r=∂​(−m lnr)/∂r=−m/r​
The negative sign shows inward motion.

​A doublet is the limiting case of a source–sink pair of equal strength placed very close together.
Its stream function is derived as:

Ψ =−μsinθ​/r
The cosine form appears in the velocity potential, not in the stream function.

Sink always draws fluid towards the origin (radially inward).

Source strength = total discharge (volume flux) per unit time.

Streamlines are given by ψ=constant.
For a source:

ψ=mθ=constant          ⇒θ=constant
θ=constant are straight radial lines from the origin (like rays).
If ψ=f(r), the streamlines would have been circles; but here it depends only on θ.

A stagnation point is where the fluid comes to rest, i.e., velocity is zero.

Stagnation points always lie on a stagnation streamline, which is a streamline of constant ψ.
This streamline separates incoming fluid from flow redirected around the stagnation region.
The actual value of ψ depends on the flow configuration, not necessarily zero.

Vorticity is the curl of the velocity field: ω=∇×q.
Physically, it measures how fast a fluid particle is spinning per unit area, not the total rotation of the entire flow.

Zero circulation everywhere indicates no rotation of fluid particles.
Therefore, the flow is irrotational, even if it is moving

By definition, stagnation points are points where flow velocity is zero.
This is independent of the type of flow; the fluid temporarily comes to rest at that point.

Since, Irrotational flow → Γ=0 at the start, it remains zero always → still irrotational.

Direct statement of theorem → circulation does not change with time for a material loop.

Viscosity introduces shear stresses → circulation may change. The theorem only holds for inviscid flows

If Γ=0 at the start, it remains 0 always → fluid remains irrotational.

ω_z​=2, A=π(2²)=4π.
Γ=ω_z​A=2⋅4π=8π.