FLUID KINEMATICS
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⃗ = 10𝑥 2 𝑦𝑖̂ + 15𝑦𝑧𝑗̂ − (3𝑥𝑦 − 25𝑡)𝑘̂ , 0 in each of the 3 directions.
Given a velocity field 𝑉
An incompressible fluid flows past a solid flat plate. The x and y coordinates are measured from the
⃗ and acceleration
leading edge and surface of the plate. If 𝑢 = 𝑥 2 𝑦 2 + 2𝑥𝑦 find the velocity field 𝑉
𝑎 at (2,1).
2𝑥
⃗ if v=0 at y=0. Find the
In a 2-D incompressible flow, the x velocity is given by 𝑢 = 𝑥 2 +𝑦2 . Evaluate 𝑉
direction of streamline with respect to x axis at (2,3).
The stream function in a 2 D flow is given by 𝜓 = 6𝑥 − 4𝑦 + 7𝑥𝑦. Verify whether the flow is
irrotational. Estimate the acceleration of a fluid element and the direction of the streamline at (1, -1).
⃗ = 3𝑥 2 𝑦𝑖̂ + 2𝑥𝑦𝑗̂ + (2𝑧𝑦 + 3𝑡)𝑘̂. Find the
The velocity field in a fluid medium is given by 𝑉
magnitudes and direction of (i) translational velocity, (ii) rotational velocity, (iii) acceleration, and (iv)
vorticity of a fluid element at (1, 2, 1) at t=3.
Find the acceleration of a fluid particle at r=2a, θ=π/2 for a 2D flow given by,
𝑎2
𝑣𝑟 = −𝑢 [1 − 𝑟2 ] cos 𝜃 ,
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𝑎2
𝑣𝜃 = 𝑢 [1 + 𝑟2 ] sin 𝜃
⃗ = 𝑢0 (1 + 2𝑥) 𝑖̂, where x is the distance
For a steady flow through a nozzle, the field is given by 𝑉
𝐿
along the axis of the nozzle from its inlet plane and L is the length of the nozzle. Find, (a) an expression
for acceleration of a particle in the flow field, and (b) the time taken by a fluid particle to travel from
the inlet tot the exit of the nozzle.
The x component of velocity in a 2D incompressible flow field is prescribed as 𝑢 = 𝐵𝑦 3 − 𝐴𝑥 4 , where
A and B are constant. Assume that for all values of x, v=0 at y=0. Check if the flow is irrotational.
⃗ = 𝑏𝑦𝑖̂ − 𝑎𝑥𝑗̂, where a, b are constants. Find the
The velocity field for a steady 2D flow is given by 𝑉
1
1
equation of the streamline passing through the point ( 2𝑎 ,
).
√
10.
√2𝑏
𝜕𝑢
𝜕𝑢
Viscous stress in a Newtonian fluid is related with strain rate 𝜏𝑖𝑗 = 𝜇 (𝜕𝑦 𝑖 + 𝜕𝑥𝑗) as Consider a velocity
𝑗
𝑖
⃗ = 𝑥 2 𝑦𝑖̂ + 2𝑥𝑗̂ − 2𝑧𝑦𝑘̂. Find the force due to viscous stresses at a point (x,y,z) on an
field given by 𝑉
area 𝐴 = 2𝑖̂ + 3𝑗̂ + 𝑘̂.
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Consider sphere of radius R immersed in a uniform
stream of flow with velocity U0 as shown in the figure.
The fluid velocity along AB is given as:
Find the position of maximum
acceleration along the line AB.
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A velocity field is given by
and irrotational.
. Check whether the flow is incompressible
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Consider low field V  axt i  b j with a=1/4 s-2 and b=1/3 m/s. Coordinates are measured in
meters. Find the equations for the streamline and the pathline.
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Consider a flow with velocity components
Is this a one, two- or three-dimensional flow? Is this an incompressible flow? Derive the stream
functions and comment whether they will represent a family of streamlines.
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The stream function of a flow field is 𝜓 = 𝐴𝑥 3 + 𝐵(𝑥𝑦 2 + 𝑥 2 − 𝑦 2 ), where ψ, A, B, x and y are all
dimensionless. Find the relation between A and B for this to be an irrotational flow. Find the velocity
potential.
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An incompressible two-dimensional flow is described by the velocity field:
Here a,b are non-zero constants. Find the angular velocity of a fluid particle at a point in this field.
Draw a graph of the streamlines for the case a=b=1.
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A two-dimensional flow is described by the velocity field
. Find the equation of the
streamline passing through the point (1,1).
19. Consider the velocity field, in normalized coordinates, given by: 𝒗 = (𝑥 2 − 𝑦 2 + 𝑥)𝒊 − (2𝑥𝑦 + 𝑦)𝒋.
(i) From the description of the velocity field, is it possible to conclude whether such velocity field is
physically realistic or not? Justify your answer.
(ii) What is the total derivative of the density at a point (x, y) in the above flow field?
(iii) Find the component of acceleration of flow along the line x = y.
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A flow field is defined by the following velocity components, where a and b are dimensional constants:
u  0 , v  a  x  z  , w  by .
(ii) Obtain an expression for the stream function, if it exists.
(iii) Obtain an expression for the velocity potential if it exists, for (i) a  b , (ii) a  b .
(iv) Sketch the deformed configuration of a rectangular fluid element that was having its edges
originally parallel to the y and z axes, respectively, for the cases (i) a  b , (ii) a  b .