Test: Potential Function - Civil Engineering (CE) MCQ

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/∂x² + ∂²ϕ/∂y² = 0, it is a case of steady incompressible irrotational flow

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.

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 
= 

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/∂x² + ∂²ϕ/∂y² = 0, it is a case of steady incompressible irrotational flow

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

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.

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.

Concept:
Valid potential function satisfies the place equation,

For option 1:
ϕ = Axy

Hence the option 1 is correct

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.

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.