৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

Steady flow: velocity at a point does not change with time.
Other conditions (divergence or curl zero) may or may not hold, depending on the flow type.

Uniform flow → velocity is same everywhere in space at a given instant.
It can still be steady (constant in time) or unsteady (changing in time).

Streamline: tangent to velocity vector at every instant.
Pathline: actual path traced by a particle.
Streakline: locus of all particles passing through a fixed point.
Vortex line: tangent to vorticity vector, not velocity.

Unsteady flow: velocity changes with time → local acceleration is present.
Uniform flow: velocity is the same everywhere in space → no spatial variation (∇q=0).

Streamline: tangent to velocity vector.
Pathline: actual trajectory followed by a particle.
Streakline: locus of particles passing through a fixed point.
Vortex line: tangent to vorticity vector.

Time dependence: u=3xt depends on t ⇒ Unsteady.
Spatial dependence: u∼x, v∼y ⇒ varies with position ⇒ Non-uniform.

Streamline Equation

Check each option:

(a) ∂u/∂x + ∂v/∂y = ∂(xy)/∂x + ∂(x − y)/∂y = y + (−1) = y − 1 ≠ 0
(b) ∂u/∂x + ∂v/∂y = ∂(y²)/∂x + ∂(x²)/∂y = 0 + 0 = 0
(c) ∂u/∂x + ∂v/∂y = ∂(e^x)/∂x + ∂(e^y)/∂y = e^x + e^y ≠ 0

(a) ∂v/∂x − ∂u/∂y = (−1) − (1) = −2 ≠ 0 → rotational
(b) ∂v/∂x − ∂u/∂y = 0 − 0 = 0 → irrotational
(c) ∂v/∂x − ∂u/∂y = (2x) − (2y) ≠ 0 → rotational
(d) ∂v/∂x − ∂u/∂y = (1) − (2y) ≠ 0 → rotational

Unsteady flow → velocity depends explicitly on time t.

(a) No t → steady
(b) u=t depends on time → unsteady
(c) No t→ steady

Uniform flow → velocity independent of spatial coordinates.

(a) Constant → uniform
(b) Depends on x → non-uniform
(c) Zero velocity (constant) → uniform

Unsteady: u = t → depends on time
Non-uniform: v = x depends on position
Rotational check:

→ rotational

So it is rotational, unsteady, and non-uniform.

For incompressible flow → A+B=0.

So the flow is rotational.

Euler’s Equation

Body force must be conservative (F=−∇ϕ) for Bernoulli.

 Local acceleration is the rate of change of velocity at a fixed point.
∂q/∂t
In Y direction
∂v/∂t

Body force includes external forces like gravity acting on the fluid particle
F represent Body force.
For x Component F_x

Bernoulli’s equation represents conservation of mechanical energy along a streamline:

Pressure head+Velocity head+Potential head (Ω)=constant
Here:

p/ρ→ pressure head
q²/2 → velocity head
Ω→ potential head

Bernoulli’s equation states that for steady, incompressible, inviscid flow, the sum of:
Pressure head p/ρ+Velocity head q²/2+Potential head Ω is constant along a streamline.

This represents conservation of mechanical energy of the fluid particle.

This is the Bernoulli equation representing the conservation of mechanical energy along a streamline.

ω=(0,2,0)