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Test: Potential Function - Civil Engineering (CE) MCQ


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10 Questions MCQ Test Fluid Mechanics for Civil Engineering - Test: Potential Function

Test: Potential Function for Civil Engineering (CE) 2024 is part of Fluid Mechanics for Civil Engineering preparation. The Test: Potential Function questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Potential Function MCQs are made for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Potential Function below.
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Test: Potential Function - Question 1

Velocity potential function exist only when

Detailed Solution for Test: Potential Function - Question 1

Velocity Potential Function:
It is defined as the scalar function of space and time, such that its negative derivative with respect to any direction gives the velocity in that direction.
It is denoted by ϕ and defined for two-dimensional as well as three-dimensional flow.
u = −∂ϕ / ∂x; v = −∂ϕ / ∂y; w = −∂ϕ / ∂z

Properties of Stream function:

  • If velocity potential function exists, the flow should be irrotational
  • If the velocity potential function satisfies the Laplace equation i.e. ∂2ϕ/∂x+ ∂2ϕ/∂y2 = 0, it is a case of steady incompressible irrotational flow
Test: Potential Function - Question 2

Verify whether the following functions are valid potential functions.
(i) ϕ = A(X− Y2)
(ii) ϕ = Acos x

Detailed Solution for Test: Potential Function - Question 2

Properties of Potential functions:
1. If velocity potential (ϕ) exists, the flow should be irrotational.
2. If velocity potential (ϕ) satisfies the Laplace equation, it represents the possible steady incompressible irrotational flow.
Calculation:
Given:

(i) ϕ = A(X2 − Y2), (ii) ϕ = Acos x.
(i) ϕ = A(X2 − Y2),

(ii) ϕ = Acos x

Thus (i) is a valid potential function as it satisfies the Laplace equation, whereas (ii) is not a valid potential function as it does not satisfy Laplace equation.

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Test: Potential Function - Question 3

The velocity potential function in a two-dimensional flow field is given by ϕ = x2 – y2
The magnitude of velocity at point P (1, 1) is

Detailed Solution for Test: Potential Function - Question 3

The velocity potential function, stream function, and velocity components are related as shown below

where ϕ - velocity potential function, ψ - stream function.
Calculation:
Given:

Velocity potential function is ϕ = x2 – y2;
Now the velocity components will be 

At P(1,1), U = -2 m/s, V = 2 m/s;
Now the magnitude of velocity at point P (1, 1) will be 


 

Test: Potential Function - Question 4

A velocity potential function exists when the flow is:

Detailed Solution for Test: Potential Function - Question 4

Velocity Potential Function:
It is defined as the scalar function of space and time, such that its negative derivative with respect to any direction gives the velocity in that direction.
It is denoted by ϕ and defined for two-dimensional as well as three-dimensional flow.
u = −∂ϕ / ∂x; v = −∂ϕ / ∂y; w = −∂ϕ / ∂z

Properties of Stream function:

  • If velocity potential function exists, the flow should be irrotational
  • If the velocity potential function satisfies the Laplace equation i.e. ∂2ϕ/∂x+ ∂2ϕ/∂y2 = 0, it is a case of steady incompressible irrotational flow
Test: Potential Function - Question 5

If the fluid velocity for a potential flow is given by V(x, y) = u(x, y)i + v(x, y)ĵ  with usual notations, then the slope of the potential line at (x,y) is

Detailed Solution for Test: Potential Function - Question 5

From potential function definition,
u = −∂ϕ / ∂x and v = −∂ϕ / ∂y
when, ϕ = ϕ(x, y) 
dϕ = ∂ϕ / ∂xdx + ∂ϕ / ∂ydy
dϕ = - u dx - v dy
Since along potential line, ϕ is constant, so dϕ = 0
0 = - udx - vdy
slope = ∂y/dx = -u/v

Test: Potential Function - Question 6

The line where the velocity potential is constant is called as______.

Detailed Solution for Test: Potential Function - Question 6

Velocity potential:
The velocity potential is defined as a scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction.
The line where the velocity potential is constant is called an equipotential line.
​Flowline:

The flow pattern of any fluid flow system may be described by means of streamlines, pathlines, streak lines.

Velocity gradient:
The difference in velocity between adjacent layers of the fluid is known as a velocity gradient and is given by dv/dx, where dv is the velocity difference and dx is the distance between the layers.

Test: Potential Function - Question 7

In a flow field, the streamlines and equipotential lines

Detailed Solution for Test: Potential Function - Question 7

Concept:
Stream line is an imaginary curve drawn through a flowing fluid in such a way so that the tangents to it at any point gives the direction of the instantaneous velocity of flow at that point.
Equipotential lines are formed by joining the different points having the same value of velocity potential function.
Properties:
(dy/dx)ϕ × (dy/dx)ψ = −1
Slope of equipotential Line × Slope of stream function = -1
i.e. streamlines and equipotential lines always meet orthogonally.
Meshes formed by them may be in square, rectangular or curvilinear.

Test: Potential Function - Question 8

Which of the following functions are valid velocity potential functions(ϕ)?

Detailed Solution for Test: Potential Function - Question 8

Concept:
Valid potential function satisfies the place equation,

For option 1:
ϕ = Axy

Hence the option 1 is correct

Test: Potential Function - Question 9

Which of the following satisfies the Laplace Equation?

Detailed Solution for Test: Potential Function - Question 9

Velocity potential (ϕ) satisfied the Laplace equation only if it represents the flow field. This can be proved in the following way:
By definition of ϕ:
u = - δϕ/δx and v = - δϕ/δy
If ϕ represents the flow field, then it must satisfy the continuity equation i.e.
δu/δx + δv/δy = 0
or
Substitute the values of u and v in the above continuity equation, we get the Laplace Equation.

 

Test: Potential Function - Question 10

If the velocity potential function ϕ =5 (x2 – y2), the velocity components at the points (4, 5) will be

Detailed Solution for Test: Potential Function - Question 10

Velocity potential function (ϕ):

  • If ϕ exists → Irrotational flow
  • Equation of equipotential lines → dy/dx = −uv
  • It is a scalar function of space and time such that


Here,
u, v, and w - components of velocity vector along x, y, and z directions respectively.
Calculation:
Given,
ϕ = 5(x2 - y2)
From (1)

Hence,
At (4, 5)
u = -4 × 10 = -40
v = 5 × 10 = 50.

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