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৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

পরীক্ষা৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]তারিখতারিখ অনির্ধারিতসময়45 minutes
মোট প্রশ্ন৩৯
সিলেবাস
Exam - 14 Topics: Hydrodynamics (a) Motion in two-dimensions, stream function and its physical meaning, velocity in polar coordinates, relation between stream function and velocity. [Source: Class - 10 and Relevant Books]
প্রশ্ন দেখুনঅনুশীলন
ঘনত্ব
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উত্তরিতবর্তমানপুনরায় দেখুনঅসম্পূর্ণ

৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১] · তারিখ অনির্ধারিত · ৩৯ প্রশ্ন

১.
The stream function Ψ(x, y) represents:
  1. ক
    The velocity potential of the fluid
  2. খ
    Streamlines of the flow
  3. গ
    Vorticity of the flow
  4. ঘ
    Pressure distribution in the flow
ব্যাখ্যা

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

২.
In 2D Cartesian flow (x, y), the velocity components (u, v) in terms of the stream function ψ are:
    ব্যাখ্যা

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

    ৩.
    For 2D polar flow (r, θ), the velocity components (qr, qθ) in terms of stream function Ψ(r, θ) are
    1. গ
      qr​ = ∂Ψ/∂r, qθ = ∂Ψ/∂θ
    2. ঘ
      qr​ = ∂Ψ/∂θ, qθ ​= - ∂Ψ/∂r
    ব্যাখ্যা

    ৪.
    The stream function of a 2D flow is:
    Ψ = xy Find the velocity components (u, v).
    1. ক
      u = y, v = - x
    2. খ
      u = - y, v = x
    3. গ
      u = - x, v = y
    4. ঘ
      u = x, v = - y
    ব্যাখ্যা

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

    ৫.
    Stream function: Ψ = r2sin⁡θ
    Compute qr and qθ​  at r=1, θ = π/2.
    1. ক
      qr ​= - 1, qθ ​= - 2
    2. খ
      qr​ = 1, qθ ​=2
    3. গ
      qr​ = - 1, qθ​ = 2
    4. ঘ
      None of these
    ব্যাখ্যা

    ৬.
    Stream function: Ψ = x2 - y2
    Find the velocity components (u, v) at (x = 1, y = 2).
    1. ক
      u= - 2, u = - 2
    2. খ
      u = - 2, v = - 4
    3. গ
      u = - 4, v = - 2
    4. ঘ
      u = 4, v = 2
    ব্যাখ্যা

    ৭.
    Stream function: Ψ = x2 + y2
    The streamlines represent-
    1. ক
      Circles around the origin
    2. খ
      Straight lines along x-axis
    3. গ
      Straight lines along y-axis
    4. ঘ
      It is impossible to define
    ব্যাখ্যা

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

    ৮.
    Stream function:
    Ψ = rsin⁡θ Find radial velocity at r = 2,θ = π/2.
    1. ক
      0
    2. খ
      1
    3. গ
      2
    4. ঘ
      - 1
    ব্যাখ্যা

    ৯.
    For 2D flow: Ψ = xy 
    Compute the velocity magnitude at (x = 2, y = 1).
    1. ক
      2
    2. খ
      3
    3. গ
      √5
    4. ঘ
      0
    ব্যাখ্যা

    ১০.
    Why does the use of stream function automatically satisfy the continuity equation in 2D incompressible flow?
    1. ক
      Because velocity is always constant
    2. খ
      Because ∂u/∂x​ + ∂v/∂y ​= 0 holds identically
    3. গ
      Because flow is always steady
    4. ঘ
      Because pressure gradient is zero
    ব্যাখ্যা

    ১১.
    If a velocity field has a stream function Ψ\psiΨ, does it guarantee irrotational flow?
    1. ক
      Yes, always
    2. খ
      No, stream function only ensures continuity
    3. গ
      Yes, if pressure is constant
    4. ঘ
      Only in potential flow
    ব্যাখ্যা

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

    ১২.
    For Ψ = 2, what does the curve represent?
    1. ক
      A pathline
    2. খ
      A streamline
    3. গ
      A velocity potential line
    4. ঘ
      None of these
    ব্যাখ্যা

    Ψ = const is always a streamline.

    ১৩.
    If both a stream function Ψ and a velocity potential φ exist for a 2D flow, the flow must be:
    1. ক
      Only incompressible
    2. খ
      Only irrotational
    3. গ
      Incompressible and irrotational
    4. ঘ
      Steady but rotational
    ব্যাখ্যা

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

    ১৪.
    If Ψ and φ exist simultaneously, which condition holds?
    1. ক
      ∂Ψ/∂x​ = ∂φ/∂y​, ∂Ψ/∂y = - ∂φ​/∂x
    2. খ
      ∂Ψ/∂x ​= ∂φ/∂y​, ∂Ψ​/∂y = ∂φ/∂x
    3. গ
      ∂Ψ/∂y​ = ∂φ/∂x​, ∂Ψ​/∂y = - ∂φ​/∂x
    4. ঘ
      ∂Ψ/∂y​ = ∂φ/∂x​, ∂Ψ​/∂y = ∂φ​/∂x
    ব্যাখ্যা

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

    ১৫.
    Given Ψ = x2 - y2, does a velocity potential φ exist?
    1. ক
      Yes, because continuity holds
    2. খ
      Yes, because Cauchy–Riemann equations are satisfied
    3. গ
      No, because ∇ × q ≠ 0
    4. ঘ
      No, because flow is compressible
    ব্যাখ্যা

    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 ×

    ১৬.
    Given φ =x2 - y2, find the corresponding stream function Ψ. 
    1. ক
      Ψ = 2xy
    2. খ
      Ψ = - 2xy
    3. গ
      Ψ = x2 + y2
    4. ঘ
      Ψ = ln(x2 + y2)
    ব্যাখ্যা





    ১৭.
    If φ and Ψ exist, the function, w(z) = Φ + iΨ  represents-
    1. ক
      Pressure distribution
    2. খ
      Velocity distribution
    3. গ
      Complex potential of flow
    4. ঘ
      Lamb’s equation
    ব্যাখ্যা

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

    ১৮.
    Velocity potential in polar coordinates: φ = rcosθ Find qr​ and qθ​.
    1. ক
      qr ​= cosθ, qθ ​= - sinθ
    2. খ
      qr​ = sinθ, qθ​ = cosθ
    3. গ
       qr​ = -cosθ, qθ ​= sinθ
    4. ঘ
      qr ​= r, qθ​ = 0
    ব্যাখ্যা

    ১৯.
    For the same flow, φ = rcosθ, the stream function Ψ is
    1. ক
      - rsinθ
    2. খ
      - r2sinθ
    3. গ
      lnr
    4. ঘ
      -sinr
    ব্যাখ্যা

    qr​ = -cosθ, qθ ​= sinθ.

    Check with Ψ  =-rsinθ:

    qr ​= (1/r) ∂Ψ/∂θ = (1/r)(-rcos⁡θ) = -cos⁡θ, 
    qθ = -∂Ψ /∂r = sin⁡θ
    Matches the velocity field.

    ২০.
    Stream function: Ψ = r2cos⁡θ
    Find qr at r = 2,  θ = π/3.
    1. ক
      0
    2. খ
      1
    3. গ
      - √3
    4. ঘ
      - 1
    ব্যাখ্যা

    ২১.
    Given stream function, Ψ = x2y - y3
    Find (u,v) and check whether the flow is rotational or irrotational.
    1. ক
      u = x2−3y2, v = −2xy, rotational
    2. খ
      u = x2−3y2, v = −2xy, irrotational
    3. গ
      u = 2xy, v = 3y2−x2, rotational
    4. ঘ
      u = 2xy, v = 3y2−x2, irrotational
    ব্যাখ্যা

    ২২.
    Stream function, Ψ = r3sin⁡θ​ 
    Find transverse velocity at r = 1,  θ = π/4.
    1. ক
      3/√2
    2. খ
      -3/√2
    3. গ
      1/√2
    4. ঘ
      -1/√2
    ব্যাখ্যা

    ২৩.
    For the stream function, Ψ = y3
    Determine the nature of the flow
    1. ক
      Rotational, velocity potential exists
    2. খ
      Rotational, no velocity potential exists
    3. গ
      Uniform flow
    4. ঘ
      Irrotational, velocity potential exists
    ব্যাখ্যা

    ২৪.
    Given stream function, Ψ = x2y2
    Check if velocity potential φ\phiφ exists.
    1. ক
      Yes, φ = x2y2
    2. খ
      Yes, φ = x3−y3
    3. গ
      Yes, φ = 2xy
    4. ঘ
      No velocity potential exists
    ব্যাখ্যা

    ২৫.
    For the velocity field, u = 2x, v = −2y
    Identify the type of flow.
    1. ক
      Uniform flow
    2. খ
      Rotational flow
    3. গ
      Pure strain (extensional flow)
    4. ঘ
      Doublet flow
    ব্যাখ্যা



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

    ২৬.
    If φ = x3 - 3xy2 then velocity components are-
    1. ক
      u = 3x2 - 3y2, v = − 6xy
    2. খ
      u = 3x2 + 3y2, v = - 6xy
    3. গ
      u = 2x, v = - 2y
    4. ঘ
      u = - 3x2 + 3y2,  v = 6xy
    ব্যাখ্যা

    ২৭.
    For Ψ = r2sin(2θ) then find qθ
    1. ক
      −2rsin(2θ)
    2. খ
      2rsin(2θ)
    3. গ
      −2r2cos(2θ)
    4. ঘ
      2r2cos(2θ)
    ব্যাখ্যা

    ২৮.
    Stream function Ψ = sin(x)sin(y) then find (u, v).
    1. ক
      u = sin(x)sin(y), v = −cos(x)cos(y)
    2. খ
      u = cos(x)sin(y), v = −sin(x)cos(y)
    3. গ
      u = cos(x)cos(y), v = −sin(x)sin(y)
    4. ঘ
      u = sin(x)cos(y), v = −cos(x)sin(y)
    ব্যাখ্যা

    ২৯.
    φ = ln(r), Find qr​ in polar coordinates.
    1. ক
      1/r
    2. খ
      r
    3. গ
      -1/r
    4. ঘ
      0
    ব্যাখ্যা

    ৩০.
    A flow has velocity components u = y, v = x.
    Which of the following statements is correct?
    1. ক
      Flow is rotational and velocity potential exists
    2. খ
      Flow is irrotational and velocity potential exists
    3. গ
      Flow is rotational and velocity potential does not exist
    4. ঘ
      Flow is irrotational but velocity potential does not exist
    ব্যাখ্যা



    Flow is irrotational → velocity potential exists.

    ৩১.
    The difference in stream function values between two streamlines in 2D incompressible flow represents:
    1. ক
      Pressure difference
    2. খ
      Velocity magnitude difference
    3. গ
      Volumetric flow rate per unit depth between the streamlines
    4. ঘ
      Vorticity
    ব্যাখ্যা

    In 2D incompressible flow, Δψ=flow rate per unit depthΔΨ = flow rate per unit depth between two streamlines.

    ৩২.
    For Ψ = x2 + y2, the flow is
    1. ক
      Rotational with constant vorticity
    2. খ
      Irrotational
    3. গ
      Uniform flow
    4. ঘ
      Potential flow only in polar coordinates
    ব্যাখ্যা

    ৩৩.
    If Ψ = y2, the flow rate per unit depth between y = 1 and y = 3 is:
    1. ক
      6
    2. খ
      8
    3. গ
      9
    4. ঘ
      4
    ব্যাখ্যা

    ৩৪.
    The condition for existence of a 2D stream function is:
    1. ক
      (∂u/∂x) + (∂v/∂y) = 0
    2. খ
      (∂u/∂y) - (∂v/∂x) = 0
    3. গ
      ∂u/∂x = ∂v/∂y
    4. ঘ
      (∂u/∂y) + (∂v/∂x) = 0
    ব্যাখ্যা

    Continuity condition for incompressible flow.

    ৩৫.
    In irrotational 2D flow, streamlines and equipotential lines are:
    1. ক
      Always perpendicular
    2. খ
      Always parallel
    3. গ
      Identical
    4. ঘ
      Never related
    ব্যাখ্যা

    ৩৬.
    The velocity potential is:
    Φ(r, θ) = r2cos2θ
    Find qr​ and qθ​ at r=1,θ = π/4.
    1. ক
      -1 & 2 respectively
    2. খ
      0 & 2
    3. গ
      -2 & 2
    4. ঘ
      -1 & 2
    ব্যাখ্যা


    ৩৭.
    The velocity potential for a 2D flow is Φ(x, y) = 3xy
    Find the velocity components u and v at the point (x, y) = (2,1)
    1. ক
      u=−6, v=−3 
    2. খ
      u=6, v=3 
    3. গ
      u=−3, v=−6 
    4. ঘ
      u=3, v=6 
    ব্যাখ্যা



    At (x,y)=(2,1): u=−3, v=−6 

    ৩৮.
    For Φ = ln(x2+y2), the velocity components are:
    1. ঘ
      None of these
    ব্যাখ্যা

    ৩৯.
    Stream function:
    Ψ = r2cos⁡θ Find qθ at r = 2,  θ = π/3.
    1. ক
      0
    2. খ
      - 1
    3. গ
      1
    4. ঘ
      - 2
    ব্যাখ্যা