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Question 1
ప్రశ్న 1
In an Eulerian description of fluid motion, what does the velocity field V(x,y,z,t) specify?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The Eulerian description specifies flow variables as functions of position and time. Thus V(x,y,z,t) gives the velocity associated with a spatial location at a specified instant. Different fluid particles may pass through that same location at different times. Following the complete history of an individually identified particle instead corresponds to the Lagrangian description. Therefore Option B correctly describes the meaning of an Eulerian velocity field.
వివరణ:
Question 2
ప్రశ్న 2
For a velocity field V(x,y,z,t), which expression gives the acceleration of a fluid particle in Eulerian form?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Acceleration of a particular fluid particle is obtained by taking the material derivative of its velocity. In Eulerian form, the material derivative contains both temporal and spatial contributions. The term ∂V/∂t is the local or temporal acceleration. The term (V·∇)V is the convective acceleration produced as the particle moves through spatial velocity gradients. Hence the total acceleration is a = ∂V/∂t + (V·∇)V, making Option C correct.
వివరణ:
Question 3
ప్రశ్న 3
Which statement correctly distinguishes local acceleration from convective acceleration?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Local acceleration is the partial time derivative of velocity evaluated at a fixed spatial location. Convective acceleration arises when a moving fluid particle enters locations where the velocity field has different values. A steady flow has zero local acceleration but may have nonzero convective acceleration. Turbulence is not required for convective acceleration because a perfectly steady nozzle flow can possess it. Therefore Option B correctly distinguishes the two contributions.
వివరణ:
Question 4
ప్రశ్న 4
A fluid moves steadily through a converging passage and its speed increases in the direction of motion. Which statement about acceleration is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Steady flow means that velocity at each fixed spatial point does not change with time. Therefore the local acceleration ∂V/∂t is zero. The velocity nevertheless changes from one spatial position to another in the converging passage. A particle moving through these different velocity values experiences convective acceleration. Hence Option B correctly describes a steady but spatially non-uniform flow.
వివరణ:
Question 5
ప్రశ్న 5
A spatially uniform velocity field is V = (5 + 2t)i m/s, with t in seconds and the coefficient 2 having units of m/s². What is the acceleration of a fluid particle?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The velocity changes with time but has no spatial dependence. Therefore the convective acceleration is zero because all spatial derivatives of velocity vanish. The local acceleration is ∂V/∂t = 2i m/s². The material acceleration equals the local acceleration in this spatially uniform unsteady flow. Hence Option B is correct.
వివరణ:
Question 6
ప్రశ్న 6
A steady one-dimensional velocity field is u = ax, where a = 4.0 s^-1. What is the fluid-particle acceleration at x = 2.0 m?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For one-dimensional flow, material acceleration is a_p = ∂u/∂t + u(∂u/∂x). The field is steady, so the local term ∂u/∂t is zero. At x = 2.0 m, the velocity is u = ax = 4 × 2 = 8 m/s. Also, ∂u/∂x = a = 4 s^-1, giving a_p = 8 × 4 = 32 m/s². Therefore Option C is correct.
వివరణ:
Question 7
ప్రశ్న 7
A one-dimensional velocity field is u = ax + bt, where a = 2.0 s^-1 and b = 1.0 m/s². At x = 1.0 m and t = 2.0 s, what is the material acceleration?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For one-dimensional motion, material acceleration is a_p = ∂u/∂t + u(∂u/∂x). At the stated position and time, u = 2(1) + 1(2) = 4 m/s. The local acceleration is ∂u/∂t = b = 1 m/s². The convective acceleration is u(∂u/∂x) = 4 × 2 = 8 m/s². Thus the total acceleration is 1 + 8 = 9 m/s², making Option C correct.
వివరణ:
Question 8
ప్రశ్న 8
A two-dimensional velocity field is u = ax + bt and v = -cy + dt, where a = 2 s^-1, b = 1 m/s², c = 1 s^-1 and d = 2 m/s². What is the acceleration vector at x = 1 m, y = 2 m and t = 1 s?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
At the specified point, u = 2(1) + 1(1) = 3 m/s and v = -1(2) + 2(1) = 0. For the x-component, ax = ∂u/∂t + u∂u/∂x + v∂u/∂y = 1 + 3(2) + 0 = 7 m/s². For the y-component, ay = ∂v/∂t + u∂v/∂x + v∂v/∂y = 2 + 0 + 0 = 2 m/s². Hence the acceleration vector is 7i + 2j m/s². Therefore Option B is correct.
వివరణ:
Question 9
ప్రశ్న 9
A steady two-dimensional velocity field is u = ax² and v = -2axy, where a = 1.0 m^-1 s^-1. What is the acceleration vector at x = 1 m and y = 1 m?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Because the field is steady, only convective acceleration terms remain. At x = y = 1 m, the velocity components are u = 1 m/s and v = -2 m/s. For the x-component, ∂u/∂x = 2ax = 2 s^-1 and ∂u/∂y = 0, giving ax = 1(2) + (-2)(0) = 2 m/s². For the y-component, ∂v/∂x = -2ay = -2 s^-1 and ∂v/∂y = -2ax = -2 s^-1, giving ay = 1(-2) + (-2)(-2) = 2 m/s². Thus the acceleration vector is 2i + 2j m/s² and Option A is correct.
వివరణ:
Question 10
ప్రశ్న 10
In a steady flow along a straight streamline, the speed varies with distance s according to V = V0 + ks, where V0 = 1 m/s and k = 2 s^-1. What is the tangential acceleration at s = 2 m?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For steady motion along a streamline, the tangential acceleration is at = V(dV/ds). At s = 2 m, the speed is V = 1 + 2(2) = 5 m/s. Since V = V0 + ks, its spatial derivative is dV/ds = k = 2 s^-1. Therefore at = 5 × 2 = 10 m/s². The streamline is stated to be straight, so no normal curvature acceleration is involved, making Option C correct.
వివరణ:
Question 11
ప్రశ్న 11
Which statement correctly defines a streamline?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A streamline is an instantaneous geometrical description of the direction of a velocity field. At every regular point on a streamline, its tangent is parallel to the local instantaneous velocity vector. It does not generally represent the complete history of one particular particle in an unsteady flow. The trajectory of an individually identified fluid particle is instead called a pathline. Therefore Option C is the correct definition.
వివరణ:
Question 12
ప్రశ్న 12
Which statement correctly defines a pathline?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A pathline follows one specific fluid particle through time. It is therefore a Lagrangian description of particle motion. Determining a pathline requires integrating the particle-position equations dx/dt = u, dy/dt = v and dz/dt = w with appropriate initial conditions. In an unsteady flow, a pathline generally need not coincide with the instantaneous streamline through the particle. Therefore Option B is correct.
వివరణ:
Question 13
ప్రశ్న 13
Which statement correctly defines a streakline?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A streakline is associated with continuous release or identification of particles at a fixed spatial location. At an observation instant, all particles that have previously passed through that location are considered together. Continuous dye injection from a fixed point in water provides a common physical realization of a streakline. A pathline instead follows only one identified particle, while a streamline is determined from the instantaneous velocity-direction field. Hence Option C correctly defines a streakline.
వివరణ:
Question 14
ప్రశ్న 14
Under what condition do streamlines, pathlines and streaklines coincide in the usual fluid-kinematics description?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
In steady flow, the velocity field at every spatial location is independent of time. Consequently, the streamline pattern does not change while particles travel through the field. An individual particle therefore follows the same curve that defines the corresponding streamline. Particles successively released from a fixed point also follow that same curve, causing the streakline to coincide with it. Thus Option C is correct, and neither spatial uniformity nor irrotationality is required.
వివరణ:
Question 15
ప్రశ్న 15
For a three-dimensional velocity field with components u, v and w, which differential relation describes a streamline at a specified instant?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The tangent to a streamline must be parallel to the instantaneous velocity vector. Hence a differential displacement along the streamline must have component ratios proportional to u, v and w. This gives dx/u = dy/v = dz/w wherever the relevant component ratios are meaningful. For a two-dimensional x-y flow, the corresponding equation reduces to dy/dx = v/u. Therefore Option B gives the correct streamline relation.
వివరణ:
Question 16
ప్రశ్న 16
For the two-dimensional velocity field u = 2x and v = -2y, which equation represents the family of streamlines in a region where x and y are nonzero?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
For a two-dimensional streamline, dy/dx = v/u. Substituting the given components gives dy/dx = (-2y)/(2x) = -y/x. Separating variables gives dy/y = -dx/x. Integration produces ln|y| = -ln|x| + constant, which can be rearranged to xy = C. Hence Option C correctly represents the streamline family.
వివరణ:
Question 17
ప్రశ్న 17
A steady two-dimensional velocity field has u = U and v = kx, where U is a positive constant velocity and k is a positive constant with units of s^-1. Which equation represents the streamline passing through the origin?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The streamline equation is dy/dx = v/u. For the given velocity field, dy/dx = kx/U. Integrating with respect to x gives y = (k/2U)x² + C. Because the required streamline passes through the origin, the integration constant is C = 0. Therefore the streamline is y = (k/2U)x² and Option B is correct.
వివరణ:
Question 18
ప్రశ్న 18
A one-dimensional unsteady field has velocity u = αt, where α = 1 m/s². A particle is at x = 1 m when t = 0. Where is the particle at t = 2 s?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A pathline is obtained from the particle equation dx/dt = u. Here dx/dt = αt, so integration gives x - x0 = αt²/2. With x0 = 1 m, α = 1 m/s² and t = 2 s, the displacement is 1 × 2²/2 = 2 m. The final particle position is therefore x = 1 + 2 = 3 m. Hence Option B is correct.
వివరణ:
Question 19
ప్రశ్న 19
In a steady one-dimensional velocity field u = ax, where a = 0.5 s^-1, a particle is at x = 1.0 m when t = 0. What is its position at t = 2.0 s?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
The particle path satisfies dx/dt = ax. Separating the variables gives dx/x = a dt. Integration with the initial condition gives x = x0 exp(at). Substituting x0 = 1.0 m, a = 0.5 s^-1 and t = 2.0 s gives x = exp(1) m. Since exp(1) is approximately 2.718, the position is about 2.72 m and Option C is correct.
వివరణ:
Question 20
ప్రశ్న 20
Consider the unsteady two-dimensional field u = U and v = αt, where U = 1 m/s and α = 1 m/s². At t = 2 s, which statement correctly compares the instantaneous streamline through the origin with the position of a particle that started from the origin at t = 0?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
At t = 2 s, the instantaneous velocity components are u = 1 m/s and v = 2 m/s. The streamline slope at that instant is therefore dy/dx = v/u = 2 throughout the field. For the particle released from the origin at t = 0, integrating dx/dt = 1 gives x = 2 m at t = 2 s. Integrating dy/dt = t gives y = t²/2 = 2 m. Thus the particle is at (2 m, 2 m), while the instantaneous streamline through the origin is y = 2x, so Option A is correct.
వివరణ:
Question 21
ప్రశ్న 21
For the unsteady field u = U and v = αt, particles are continuously released from the origin. At observation time T, which equation describes the streakline, where x and y are measured from the release point?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Consider a particle released from the origin at an earlier time τ and observed at time T. Its horizontal position is x = U(T - τ), while integration of dy/dt = αt gives y = (α/2)(T² - τ²). From the horizontal-position relation, τ = T - x/U. Substituting this expression into y and simplifying gives y = (αT/U)x - [α/(2U²)]x². Therefore Option A correctly describes the streakline and explicitly demonstrates why it need not equal the instantaneous streamline in this unsteady flow.
వివరణ:
Question 22
ప్రశ్న 22
Which statement correctly describes a stream tube?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A stream tube is an imaginary tubular region whose side boundary is formed by a family of streamlines. Because the velocity vector is tangent to every streamline, it has no normal component through the side surface of the stream tube at the instant considered. Fluid therefore does not cross that side boundary in the instantaneous streamline description. The cross-sectional area of a stream tube need not remain constant, and a stream tube need not correspond to a physical pipe. Hence Option A is correct.
వివరణ:
Question 23
ప్రశ్న 23
Why can two distinct streamlines not cross at an ordinary point where the instantaneous velocity is nonzero and uniquely defined?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A streamline is tangent to the local instantaneous velocity vector. If two distinct streamlines crossed at an ordinary point with different tangents, the velocity at that single point would have to possess two different directions simultaneously. A regular single-valued velocity field cannot satisfy that requirement. The restriction is kinematic and does not depend on whether the fluid is incompressible. Therefore Option A is correct, while stagnation or singular points require separate care because the velocity direction may not be uniquely defined there.
వివరణ:
Question 24
ప్రశ్న 24
Which experimental observation is correctly matched with the corresponding flow line?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
A tagged particle followed through time traces its actual trajectory, which is a pathline. Continuous dye introduced from one fixed spatial point marks particles that have successively passed through that location and therefore generates a streakline. A streamline is instead constructed from the instantaneous velocity-direction field at a specified time. In steady flow these different flow-line descriptions coincide, but in unsteady flow they need not. Therefore Option C is the only correctly matched pair.
వివరణ:
Question 25
ప్రశ్న 25
Consider the following statements: I. Local acceleration vanishes in steady flow. II. Convective acceleration vanishes in a spatially uniform velocity field. III. Streamlines, pathlines and streaklines coincide in steady flow. IV. The side surface of a stream tube is formed by streamlines and has no normal flow through it in the instantaneous streamline description. Which option is correct?
Select an option first. || ముందుగా ఒక సమాధానాన్ని ఎంచుకోండి.
Explanation:
Statement I is correct because steady flow has no explicit time variation of velocity at a fixed point, so local acceleration is zero. Statement II is correct because a spatially uniform velocity field has zero spatial velocity gradients and therefore zero convective acceleration. Statement III is correct because a time-independent velocity field causes streamline, pathline and streakline descriptions to coincide. Statement IV is correct because a stream-tube side boundary is composed of streamlines and the local velocity is tangent to that boundary. Consequently all four statements are correct and Option D is the valid answer.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: B. The velocity that would be observed at each spatial location and time in the flow field సమాధానం: B.
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Question 2 ప్రశ్న 2Answer: C. a = ∂V/∂t + (V·∇)V సమాధానం: C.
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Question 3 ప్రశ్న 3Answer: B. Local acceleration measures time variation at a fixed point, whereas convective acceleration arises because a particle moves through a spatially varying velocity field సమాధానం: B.
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Question 4 ప్రశ్న 4Answer: B. Local acceleration is zero, but convective acceleration can be nonzero సమాధానం: B.
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Question 5 ప్రశ్న 5Answer: B. 2i m/s² సమాధానం: B.
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Question 6 ప్రశ్న 6Answer: C. 32 m/s² సమాధానం: C.
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Question 7 ప్రశ్న 7Answer: C. 9 m/s² సమాధానం: C.
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Question 8 ప్రశ్న 8Answer: B. 7i + 2j m/s² సమాధానం: B.
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Question 9 ప్రశ్న 9Answer: A. 2i + 2j m/s² సమాధానం: A.
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Question 10 ప్రశ్న 10Answer: C. 10 m/s² సమాధానం: C.
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Question 11 ప్రశ్న 11Answer: C. It is a curve whose tangent at every point gives the instantaneous direction of the local velocity vector సమాధానం: C.
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Question 12 ప్రశ్న 12Answer: B. The actual trajectory traced through space by a particular identified fluid particle సమాధానం: B.
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Question 13 ప్రశ్న 13Answer: C. The locus, at a specified instant, of particles that have previously passed through a particular fixed point సమాధానం: C.
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Question 14 ప్రశ్న 14Answer: C. For steady flow సమాధానం: C.
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Question 15 ప్రశ్న 15Answer: B. dx/u = dy/v = dz/w సమాధానం: B.
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Question 16 ప్రశ్న 16Answer: C. xy = C సమాధానం: C.
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Question 17 ప్రశ్న 17Answer: B. y = (k/2U)x² సమాధానం: B.
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Question 18 ప్రశ్న 18Answer: B. 3 m సమాధానం: B.
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Question 19 ప్రశ్న 19Answer: C. 2.72 m సమాధానం: C.
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Question 20 ప్రశ్న 20Answer: A. The streamline through the origin has slope 2, and the particle is at (2 m, 2 m) సమాధానం: A.
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Question 21 ప్రశ్న 21Answer: A. y = (αT/U)x - [α/(2U²)]x² సమాధానం: A.
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Question 22 ప్రశ్న 22Answer: A. It is a tube whose side surface is formed by streamlines, so the instantaneous velocity is tangent to the side surface and there is no flow across that surface సమాధానం: A.
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Question 23 ప్రశ్న 23Answer: A. Because streamline intersection would require two different velocity directions at the same point and instant సమాధానం: A.
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Question 24 ప్రశ్న 24Answer: C. Long-exposure tracking of one tagged particle – pathline సమాధానం: C.
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Question 25 ప్రశ్న 25Answer: D. I, II, III and IV సమాధానం: D.