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৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

পরীক্ষা৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]তারিখতারিখ অনির্ধারিতসময়45 minutes
মোট প্রশ্ন৪০
সিলেবাস
Exam - 15 Topics: Hydrodynamics (a) Sources, sinks and doublets. Complex potential and complex velocity, stagnation points, Complex potential due to a source and a doublet. Circulation and vorticity, relation between circulation and vorticity. Kelvin’s Circulation theorem. [Source: Class - 10 and Relevant Books]
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৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১] · তারিখ অনির্ধারিত · ৪০ প্রশ্ন

১.
The velocity potential of a source of strength mmm in polar coordinates is:
  1. ক
    Φ = - m lnr 
  2. খ
    Φ = m lnr
  3. গ
    Φ = m/r
  4. ঘ
    Φ = m/r2
ব্যাখ্যা

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 qr​=m​/r.
The negative form −m lnr corresponds to a sink, not a source.

২.
For a sink of strength m, the radial velocity at distance r is
  1. ক
    m/r
  2. খ
    m/r2
  3. গ
    -m/r
  4. ঘ
    0
ব্যাখ্যা

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

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

৩.
The stream function of a doublet of strength µ is:
  1. ক
    Ψ = - µcosθ​/r 
  2. খ
    Ψ = µcosθ​/r
  3. গ
    Ψ =- µsinθ​/r
  4. ঘ
    Ψ = µsinθ​/r
ব্যাখ্যা

​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.

৪.
The velocity components of a doublet of strength µ are:
  1. ক
    qr​ = - µcosθ​/r2, qθ ​= µsinθ/r2 ​
  2. খ
    qr​ = µcosθ​/r2, qθ​ = µsinθ/r2​
  3. গ
    qr​=- µcosθ​/r2, qθ = - µsinθ/r2​
  4. ঘ
    qr​=−µ​/r, qθ ​= 0​
ব্যাখ্যা

৫.
A source of strength m=5 has stream function Ψ= mθ. Find the streamline passing through the point (x, y) =(1,√3​)
  1. ক
     θ = 30°
  2. খ
    θ = 45°
  3. গ
     θ = 60°
  4. ঘ
    θ = 90°
ব্যাখ্যা

৬.
A source of strength m=20 is at the origin. Find velocity magnitude at (x,y)=(0,4).
  1. ক
    4
  2. খ
    5
  3. গ
    2
  4. ঘ
    10
ব্যাখ্যা

৭.
A sink at the origin creates velocity at a point on x-axis. The velocity is directed:
  1. ক
    Radially outward from origin
  2. খ
    Radially inward to origin
  3. গ
    Tangential to circle r=constant
  4. ঘ
    None of these
ব্যাখ্যা

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

৮.
For a source located at the origin, the velocity vector at any point is:
  1. ক
    Tangential to a circle
  2. খ
    Along the radial line outward
  3. গ
    Along the radial line inward
  4. ঘ
    At 45° to the radial line
ব্যাখ্যা

৯.
At θ =180°, the radial velocity for a doublet of strength µ is:
  1. ক
    Zero
  2. খ
    Positive outward
  3. গ
    Negative inward
  4. ঘ
    It is impossible to say exactly.
ব্যাখ্যা

১০.
For a doublet,  where is the tangential velocity it zero?
  1. ক
    θ=90°,270° 
  2. খ
    θ=0°,180°
  3. গ
    θ=45°,135°
  4. ঘ
    Never Zero
ব্যাখ্যা



১১.
The strength m of a source represents:
  1. ক
    The circulation around the source
  2. খ
    The flux per unit time from the source
  3. গ
    The tangential velocity
  4. ঘ
    The vorticity of the flow
ব্যাখ্যা

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

১২.
A doublet of strength µ = 50 is at the origin. Find qθ (r, θ) = (5,0°)
  1. ক
    0
  2. খ
    2
  3. গ
    - 2
  4. ঘ
    5
ব্যাখ্যা

১৩.
A source at the origin produces a stream function Ψ = mθ . Which of the following is true? 
  1. ক
    Streamlines are circles r = constant
  2. খ
    Streamlines are straight lines through the origin
  3. গ
    Streamlines are hyperbolas
  4. ঘ
    Streamlines are parabolas
ব্যাখ্যা

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.

১৪.
Sink at origin: Φ = - m lnr. Complex velocity V(z)=? 
  1. ক
    mz
  2. খ
    -mz
  3. গ
    m/z
  4. ঘ
    -m/z
ব্যাখ্যা

১৫.
Φ =−Ux + Φsource​, source m = 10. Complex velocity V(z) = ? 
  1. ক
    U+10/z
  2. খ
    -U-10/z
  3. গ
    U+10z
  4. ঘ
    -U+10z
ব্যাখ্যা

১৬.
A source of strength m=10 is located at the origin. Find the velocity magnitude at point (x,y)=(3,4).
  1. ক
    1.2
  2. খ
    2
  3. গ
    3
  4. ঘ
    5
ব্যাখ্যা

১৭.
Doublet: Φ =-  µcosθ​/r. Find V(z). 
  1. ক
    µ/z
  2. খ
    - µ/z
  3. গ
    µ/z2
  4. ঘ
    - µ/z2
ব্যাখ্যা

১৮.
A source of strength m=10 and a doublet of strength µ=20 are placed at the origin. Where is the stagnation point located? 
  1. ক
    x = 2
  2. খ
    x = -2
  3. গ
    x = 0
  4. ঘ
    y = 2
ব্যাখ্যা

১৯.
The combined complex potential of a source of strength m and a doublet of strength µ at the origin is: 
  1. ক
    w = m lnz+ µ​/z 
  2. খ
    w=m lnz - µ/z
  3. গ
    w=m lnz + µ​z
  4. ঘ
    w=m lnz - µ​/z2
ব্যাখ্যা

২০.
Stagnation points in a flow occur at points where:
  1. ক
    Velocity Potential Zero
  2. খ
    Stream Function Zero
  3. গ
    Complex Velocity Zero
  4. ঘ
    Complex Potential Zero
ব্যাখ্যা

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

২১.
A sink of strength m=8 is placed at the origin. Find the velocity at r=2
  1. ক
    4 outward
  2. খ
    4 inward
  3. গ
    16 outward
  4. ঘ
    16 inward
ব্যাখ্যা

২২.
The stagnation point lies on which line in the flow?
  1. ক
    Equipotential line Φ = 0 
  2. খ
    Ψ = constant 
  3. গ
    x-axis always
  4. ঘ
    Arbitrary circle
ব্যাখ্যা

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.

২৩.
Circulation around a closed path in an irrotational flow is:
  1. ক
    zero
  2. খ
    non-zero
  3. গ
    Infinity
  4. ঘ
    Depends on Path
ব্যাখ্যা

২৪.
Vorticity in a fluid flow represents:
  1. ক
    Total rotation along a closed loop
  2. খ
    Rotation per unit area of a fluid particle
  3. গ
    Velocity magnitude
  4. ঘ
    Pressure gradient
ব্যাখ্যা

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.

২৫.
The relation between circulation Γ around a closed loop C and vorticity ω is 
  1. ঘ
    None of these
ব্যাখ্যা

২৬.
A doublet of strength µ = 20 is placed at the origin. Find the radial velocity qr​ at point (r, θ) = (2,60°)
  1. ক
    - 2.5
  2. খ
    + 2.5
  3. গ
    + 5√3/2 
  4. ঘ
    - 5√3/2
ব্যাখ্যা

২৭.
For a solid body rotation with velocity u=−Ωy, v=Ωx over a circular area of radius R, the circulation around the boundary is: 
  1. ক
    πR2Ω
  2. খ
    2πR2Ω
  3. গ
    Ω/πR2
  4. ঘ
    0
ব্যাখ্যা

২৮.
If circulation Γ = 0 around any closed loop, the flow is: 
  1. ক
    Rotational
  2. খ
    Irrotational
  3. গ
    Compressible
  4. ঘ
    Unsteady
ব্যাখ্যা

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

২৯.
A sink of strength m produces velocity field qr​=−m/r​. What is the circulation around a circle of radius r?
  1. ক
    m
  2. খ
    -m
  3. গ
    0
  4. ঘ
    Infinite
ব্যাখ্যা

৩০.
For a doublet of strength µ=50, compute the tangential velocity at (r = 10,θ = 90°) 
  1. ক
    0
  2. খ
    0.25
  3. গ
    0.5
  4. ঘ
    1
ব্যাখ্যা

৩১.
Which relation connects circulation Γ and vorticity ωz​ in a plane area A? 
  1. ক
    Γ=ωz​ 
  2. খ
    Γ=ωz​A
  3. গ
    Γ=ωz​/A
  4. ঘ
    Γ=2ωzA​
ব্যাখ্যা

৩২.
A doublet of strength µ = 20 is placed at the origin. Find the Transverse velocity​ at point (r, θ) = (2, 30°) 
  1. ক
    −2.5
  2. খ
    2.5
  3. গ
    5√3/2 
  4. ঘ
    -5√3/2
ব্যাখ্যা

৩৩.
If zs​ is a stagnation point, what is the fluid velocity at that point?
  1. ক
     V = U∞
  2. খ
    V=∞
  3. গ
    V=m/zs
  4. ঘ
    V=0
ব্যাখ্যা

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.

৩৪.
In a 2D region of area A=5m2, the vorticity is uniform at ωz=4s−1. What is the circulation? 
  1. ক
    1.25
  2. খ
    1
  3. গ
    20
  4. ঘ
    100
ব্যাখ্যা

৩৫.
For free vortex flow with Γ= 5m2/s, what is the circulation around a closed circular path of radius 10m?
  1. ক
    0
  2. খ
    2
  3. গ
    5
  4. ঘ
    50
ব্যাখ্যা

৩৬.
A fluid is initially irrotational. According to Kelvin’s theorem, after 10 seconds, circulation around any closed material circular loop of radius 2m will be:
  1. ক
    0
  2. খ
    5
  3. গ
    10
  4. ঘ
    20
ব্যাখ্যা

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

৩৭.
In Kelvin’s theorem, the material derivative of circulation is:
    ব্যাখ্যা

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

    ৩৮.
    Which of the following would invalidate Kelvin’s circulation theorem?
    1. ক
      Inviscid flow
    2. খ
      Conservative body force
    3. গ
      Viscous effects
    4. ঘ
      Barotropic fluid assumption
    ব্যাখ্যা

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

    ৩৯.
    If a flow is initially irrotational, according to Kelvin’s theorem it will:
    1. ক
      Become rotational over time
    2. খ
      Remain irrotational
    3. গ
      Circulation will increase linearly
    4. ঘ
      Circulation will decrease exponentially
    ব্যাখ্যা

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

    ৪০.
    If vorticity is ωz​=2s−1 over a circular region of radius r=2m, the circulation is: 
    1. ক
      2π
    2. খ
      4π
    3. গ
      8π
    4. ঘ
      16π
    ব্যাখ্যা

    ωz​=2, A=π(22)=4π.
    Γ=ωz​A=2⋅4π=8π.