Test 6 - Stream Function and Velocity Potential

Question 1 ప్రశ్న 1

Using the convention u = ∂ψ/∂y and v = -∂ψ/∂x for a two-dimensional Cartesian flow, which statement about the stream function ψ is correct?

A Curves ψ = constant are streamlines, and the velocity components are obtained from the stated cross-derivatives B Curves ψ = constant are always equipotential lines, and u = ∂ψ/∂x C ψ can be defined only for rotational flow D ψ has the dimensions of pressure head

Question 2 ప్రశ్న 2

For the standard two-dimensional incompressible-flow stream function ψ and velocity potential φ, what are their SI dimensions?

A Both have dimensions of velocity, LT^-1 B Both have dimensions L²T^-1 C ψ has dimensions L³T^-1 and φ is dimensionless D ψ has dimensions of pressure and φ has dimensions of acceleration

Question 3 ప్రశ్న 3

Why does a sufficiently smooth stream function defined by u = ∂ψ/∂y and v = -∂ψ/∂x automatically satisfy the two-dimensional incompressible continuity equation?

A Because ∂u/∂x + ∂v/∂y reduces to equal mixed partial derivatives with opposite signs B Because every stream function makes pressure uniform C Because ∂u/∂x and ∂v/∂y must each be zero separately D Because stream functions exist only when the velocity is spatially uniform

Question 4 ప్రశ్న 4

Using the convention u = ∂φ/∂x and v = ∂φ/∂y, why is any sufficiently smooth velocity field derived from a scalar velocity potential φ irrotational in a two-dimensional region?

A Because ∂u/∂x + ∂v/∂y is always zero for every φ B Because ∂v/∂x - ∂u/∂y reduces to equal mixed partial derivatives with opposite signs C Because φ must be constant throughout the flow D Because a potential function can exist only when the velocity is zero

Question 5 ప్రశ్న 5

In a simply connected two-dimensional region with a continuously differentiable velocity field, which condition permits a single-valued velocity potential φ to be defined using u = ∂φ/∂x and v = ∂φ/∂y?

A ∂u/∂x + ∂v/∂y = 0 only B ∂v/∂x - ∂u/∂y = 0 C u = v at every point D ∂u/∂t = ∂v/∂t = 0

Question 6 ప్రశ్న 6

Which statement about the existence of a conventional stream function ψ(x,y,t) is correct?

A It requires the flow to be steady as well as incompressible B It may be defined for a two-dimensional incompressible flow even when the velocity field is unsteady C It requires the flow to be irrotational but not incompressible D It can be defined only when u and v are constants

Question 7 ప్రశ్న 7

If an incompressible flow has a velocity potential φ such that V = ∇φ, which governing equation must φ satisfy?

A ∇²φ = 0 B ∇²φ = 1 C ∂φ/∂t = 0 in every case D ∇φ = 0

Question 8 ప్రశ్న 8

For a two-dimensional incompressible flow described by u = ∂ψ/∂y and v = -∂ψ/∂x, what additional condition makes the stream function ψ satisfy Laplace's equation?

A The flow must be irrotational B The pressure must be atmospheric C The velocity magnitude must be constant D The flow must be turbulent

Question 9 ప్రశ్న 9

In a two-dimensional incompressible and irrotational region where both φ and ψ exist with u = φ_x = ψ_y and v = φ_y = -ψ_x, how do lines φ = constant and ψ = constant intersect at ordinary points where velocity is nonzero?

A They are necessarily parallel B They intersect orthogonally C They coincide everywhere D The angle depends only on fluid density

Question 10 ప్రశ్న 10

Using u = φ_x = ψ_y and v = φ_y = -ψ_x, which pair of relations is correct?

A φ_x = ψ_x and φ_y = ψ_y B φ_x = ψ_y and φ_y = -ψ_x C φ_x = -ψ_y and φ_y = ψ_x D φ_x + ψ_x = 0 and φ_y + ψ_y = 0

Question 11 ప్రశ్న 11

A uniform two-dimensional flow of speed U is directed along the positive x-axis. Under the conventions u = ψ_y, v = -ψ_x and u = φ_x, v = φ_y, which pair represents this flow, apart from additive constants?

A ψ = Ux and φ = Uy B ψ = Uy and φ = Ux C ψ = -Uy and φ = -Ux D ψ = Uxy and φ = U(x² + y²)

Question 12 ప్రశ్న 12

In a two-dimensional incompressible flow, two streamlines have stream-function values ψ1 = 2 m²/s and ψ2 = 7 m²/s. What is the magnitude of volumetric flow rate per unit depth between them?

A 2 m²/s B 5 m²/s C 7 m²/s D 9 m²/s

Question 13 ప్రశ్న 13

A stream function is ψ = kxy, where k = 4.0 s^-1. Using u = ψ_y and v = -ψ_x, what are the velocity components and speed at (x,y) = (1 m, 2 m)?

A u = 4 m/s, v = -8 m/s, speed = 8.94 m/s B u = 8 m/s, v = -4 m/s, speed = 8.94 m/s C u = 4 m/s, v = 8 m/s, speed = 12.00 m/s D u = -8 m/s, v = 4 m/s, speed = 4.47 m/s

Question 14 ప్రశ్న 14

For the stream function ψ = k(x² - y²), where k is a nonzero constant with units s^-1, which velocity potential generates the same velocity field under u = φ_x and v = φ_y?

A φ = 2kxy + C B φ = -2kxy + C C φ = k(x² + y²) + C D φ = k(x² - y²) + C

Question 15 ప్రశ్న 15

A velocity potential is φ = k(x² - y²), where k = 1.0 s^-1. What are the velocity components at (x,y) = (1 m, 2 m), and what is the speed?

A u = 2 m/s, v = -4 m/s, speed = 4.47 m/s B u = -2 m/s, v = 4 m/s, speed = 4.47 m/s C u = 2 m/s, v = 4 m/s, speed = 6.00 m/s D u = 4 m/s, v = -2 m/s, speed = 2.24 m/s

Question 16 ప్రశ్న 16

The velocity potential is φ = kxy, where k = 2.0 s^-1. Under the convention u = φ_x, v = φ_y and u = ψ_y, v = -ψ_x, which stream function represents the same flow?

A ψ = x² + y² + C B ψ = y² - x² + C C ψ = x² - y² + C D ψ = 2xy + C

Question 17 ప్రశ్న 17

A stream function is ψ = k(xy² - x³/3), where k = 1.0 m^-1 s^-1. Which velocity potential corresponds to the same two-dimensional flow under the conventions u = ψ_y, v = -ψ_x and u = φ_x, v = φ_y?

A φ = k(x²y - y³/3) + C B φ = k(x²y + y³/3) + C C φ = k(xy² - x³/3) + C D φ = k(x³/3 - xy²) + C

Question 18 ప్రశ్న 18

A two-dimensional velocity field is u = ax - by and v = bx + ay, where a = 2 s^-1 and b = 3 s^-1. Which statement is correct in a simply connected region?

A It is incompressible and irrotational, so both ψ and φ exist in the standard sense B It is incompressible but rotational, so only φ exists C It is irrotational but has nonzero divergence, so only the conventional incompressible stream function exists D It has both nonzero divergence and nonzero vorticity, so it satisfies neither the standard incompressible stream-function condition nor the velocity-potential condition

Question 19 ప్రశ్న 19

For the velocity field u = ax and v = -ay, where a = 2 s^-1, which pair of scalar functions represents the field under the conventions used in this test?

A φ = (a/2)(x² - y²) and ψ = axy B φ = (a/2)(x² + y²) and ψ = -axy C φ = axy and ψ = (a/2)(x² - y²) D φ = -(a/2)x² + (a/2)y² and ψ = (a/2)(x² + y²)

Question 20 ప్రశ్న 20

A stream function is ψ = k(x² + y²), where k = 1.0 s^-1. Under u = ψ_y and v = -ψ_x, which statement about the resulting two-dimensional flow is correct?

A It violates incompressible continuity B It is incompressible but rotational, so a velocity potential does not exist over an ordinary simply connected region C It is incompressible and irrotational, so ψ must be harmonic D It is irrotational but compressible

Question 21 ప్రశ్న 21

A velocity potential is φ = k(x² + y²), where k = 1.0 s^-1. Under u = φ_x and v = φ_y, which statement is correct?

A The field is irrotational but has divergence 4k, so it does not satisfy the constant-density incompressible continuity equation B The field is rotational and incompressible C The field has zero velocity everywhere D The field is both incompressible and irrotational because every potential automatically satisfies Laplace's equation

Question 22 ప్రశ్న 22

At an ordinary point in a two-dimensional potential flow, which geometrical statement about an equipotential line φ = constant and the local velocity vector is correct?

A The velocity is tangent to the equipotential line B The velocity is normal to the equipotential line C The velocity must vanish on every equipotential line D The velocity direction is unrelated to the potential gradient

Question 23 ప్రశ్న 23

Why is a curve ψ = constant a streamline under the convention u = ψ_y and v = -ψ_x?

A Because dψ = 0 gives -v dx + u dy = 0, hence dy/dx = v/u B Because dψ = 0 requires dx = dy = 0 C Because ψ = constant implies u = v = 0 D Because every constant scalar field is automatically a velocity potential

Question 24 ప్రశ్న 24

For φ = k(x²y - y³/3), where k = 1.0 m^-1 s^-1, which statement is correct for the two-dimensional field defined by u = φ_x and v = φ_y?

A The field has divergence zero and vorticity zero B The field has divergence 2ky and vorticity zero C The field has divergence zero and vorticity 4kx D The field has divergence 4ky and vorticity 2kx

Question 25 ప్రశ్న 25

Consider the following statements for a sufficiently smooth two-dimensional flow using u = ψ_y, v = -ψ_x and u = φ_x, v = φ_y: I. A stream-function representation automatically satisfies incompressible continuity. II. A velocity-potential representation automatically gives zero vorticity. III. If an incompressible potential flow exists, φ satisfies Laplace's equation. IV. If the same flow is both incompressible and irrotational, constant-φ and constant-ψ lines are orthogonal at regular points. Which option is correct?

A I and II only B I, II and III only C II, III and IV only D I, II, III and IV

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