১.
The velocity of a fluid particle in 2D is given by u = 2x, v = - 2y. Which of the following is correct?
৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]
মোট প্রশ্ন৪০
সিলেবাস
Exam - 13
Topics:
Hydrodynamics
(a) Velocity and acceleration of fluid particles, Steady and unsteady flows, Uniform and
non-uniform flows, Stream lines, path lines, vortex lines and velocity potential.
Rotational and irrotational flows. Equation of continuity.
(b) Euler’s equation of motion, conservative field force. Lamb’s equations of motion.
Bernouli’s equation
[Source: Class - 09 and Relevant Books]
৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]
৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১] · তারিখ অনির্ধারিত · ৪০ প্রশ্ন
২.
In a steady flow, which of the following is true?
ব্যাখ্যা
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.
৩.
Which of the following statements is true for a uniform flow?
ব্যাখ্যা
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).
৪.
Which one of the following curves always remains tangent to the instantaneous velocity vector?
ব্যাখ্যা
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.
৫.
If the velocity potential function is Φ = x2 - y2, then the flow is:
ব্যাখ্যা
৬.
The acceleration term (q⋅∇)q is known as:
ব্যাখ্যা
৭.
In an unsteady but uniform flow, which of the following holds?
ব্যাখ্যা
Unsteady flow: velocity changes with time → local acceleration is present.
Uniform flow: velocity is the same everywhere in space → no spatial variation (∇q=0).
৮.
The vorticity vector is defined as:
৯.
Which of the following describes the pathline?
ব্যাখ্যা
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.
১০.
A 2D velocity field is given by u = y, q = - x. The vorticity is:
ব্যাখ্যা
১১.
In an unsteady 1D flow, velocity is q(x, t) = 3t + 2x. Find the total acceleration at x = 1, t = 2
ব্যাখ্যা
১২.
A stream function is given as Ψ = x2y. Find velocity components u,v.
ব্যাখ্যা
১৩.
For a velocity field: u = 3xt, v = 2y
The flow is:
The flow is:
ব্যাখ্যা
Time dependence: u=3xt depends on t ⇒ Unsteady.
Spatial dependence: u∼x, v∼y ⇒ varies with position ⇒ Non-uniform.
১৪.
The velocity field is given by: u = y, v = - x
Equation of the streamline is:
Equation of the streamline is:
ব্যাখ্যা
Streamline Equation
১৫.
Which of the following two-dimensional velocity fields satisfies the continuity equation for incompressible flow?
ব্যাখ্যা
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 = ∂(ex)/∂x + ∂(ey)/∂y = ex + ey ≠ 0
১৬.
Which of the following velocity fields represents an irrotational flow?
ব্যাখ্যা
(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
১৭.
Which of the following velocity fields represents an unsteady flow?
ব্যাখ্যা
Unsteady flow → velocity depends explicitly on time t.
(a) No t → steady
(b) u=t depends on time → unsteady
(c) No t→ steady
১৮.
Which velocity field represents a non-uniform flow?
ব্যাখ্যা
Uniform flow → velocity independent of spatial coordinates.
(a) Constant → uniform
(b) Depends on x → non-uniform
(c) Zero velocity (constant) → uniform
১৯.
For the velocity field u = t, v = x, the flow is:
ব্যাখ্যা
Unsteady: u = t → depends on time
Non-uniform: v = x depends on position
Rotational check:
→ rotational
So it is rotational, unsteady, and non-uniform.
২০.
For a flow field: u = Ax, v = By,
find the condition on constants A,B such that continuity is satisfied.
find the condition on constants A,B such that continuity is satisfied.
ব্যাখ্যা
For incompressible flow → A+B=0.
২১.
For a 3D velocity field:
u = xt, v = y, w = z2,find the x-component of acceleration of a fluid particle.
u = xt, v = y, w = z2,find the x-component of acceleration of a fluid particle.
ব্যাখ্যা
২২.
For
u = y, v = x, w = z, the equation of streamlines is:
u = y, v = x, w = z, the equation of streamlines is:
ব্যাখ্যা
২৩.
For
u = y, v = - x, w = z,
the flow is:
u = y, v = - x, w = z,
the flow is:
ব্যাখ্যা
So the flow is rotational.
২৪.
For a fluid particle with
u = x, v = - y, w = 0,F = 0, pressure p, density ρ ,what is the x-component of Euler’s equation?
u = x, v = - y, w = 0,F = 0, pressure p, density ρ ,what is the x-component of Euler’s equation?
ব্যাখ্যা
২৫.
A force field F is conservative if:
ব্যাখ্যা
২৬.
If F is conservative, Euler’s eqn becomes:
ব্যাখ্যা
২৭.
Lamb’s form of Euler’s equation is:
ব্যাখ্যা
Euler’s Equation
২৮.
Bernoulli’s equation is valid if the body force is:
ব্যাখ্যা
Body force must be conservative (F=−∇ϕ) for Bernoulli.
২৯.
In Euler’s eqn, the term -∇p/ρ represents:
ব্যাখ্যা
৩০.
Identify the local acceleration term in the y-direction:
ব্যাখ্যা
Local acceleration is the rate of change of velocity at a fixed point.
∂q/∂t
In Y direction
∂v/∂t
৩১.
Which term in Euler’s x-component represents body force?
ব্যাখ্যা
Body force includes external forces like gravity acting on the fluid particle
F represent Body force.
For x Component Fx
৩২.
For steady, incompressible, inviscid flow along a streamline, the Bernoulli equation is:
ব্যাখ্যা
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
৩৩.
Bernoulli’s equation represents conservation of mechanical energy along a streamline:
ব্যাখ্যা
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.
৩৪.
For steady, incompressible, inviscid flow, which of the following is correct along a streamline?
ব্যাখ্যা
This is the Bernoulli equation representing the conservation of mechanical energy along a streamline.
৩৫.
Fluid flows along a horizontal streamline (Ω1 = Ω2):
q1 = 3 m/s, q2= 5 m/s, p1 = 100 Pa, ρ = 1 kg/m3
Find p2
q1 = 3 m/s, q2= 5 m/s, p1 = 100 Pa, ρ = 1 kg/m3
Find p2
ব্যাখ্যা
৩৬.
A 3D velocity field of a fluid is given as:
u = 2x + t, v = - y, w = z2t
Compute the local acceleration at the point (x = 1,y = 1,z = 1) at t = 2.
u = 2x + t, v = - y, w = z2t
Compute the local acceleration at the point (x = 1,y = 1,z = 1) at t = 2.
ব্যাখ্যা
৩৭.
Velocity field:
u = z, v = 0, w = - x
Compute ω = ∇ × q
u = z, v = 0, w = - x
Compute ω = ∇ × q
ব্যাখ্যা
ω=(0,2,0)
৩৮.
Velocity field:
u = 2x, v =- y, w = 0, F = 0, ρ = 1
Compute ∂p/∂x at (x = 1,y = 2)
u = 2x, v =- y, w = 0, F = 0, ρ = 1
Compute ∂p/∂x at (x = 1,y = 2)
ব্যাখ্যা
৩৯.
A fluid particle has velocity components:
u = x2, v = y, w = 0 Compute the x-component of convective acceleration at (x = 2, y = 1).
u = x2, v = y, w = 0 Compute the x-component of convective acceleration at (x = 2, y = 1).
ব্যাখ্যা
৪০.
A fluid particle moves along a streamline with velocities q1 = 4 m/s and q2 = 6 m/s. The potential head is the same (Ω1 = Ω2). If the fluid density is ρ = 1 kg/m3, what is the pressure difference p1 - p2?