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The velocity potential which follow the equation of continuity is ________
- a)x2y
- b)x2- y2
- c)cos x
- d)x2 + y2
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
The velocity potential which follow the equation of continuity is ____...
Velocity Potential function (ϕ) is given as = x2 - y2
u = - 2x
V = 2y
For continuity to be satisfied
-2 + 2 = 0
(Hence satisfied)
Most Upvoted Answer
The velocity potential which follow the equation of continuity is ____...
Explanation:
The continuity equation in fluid dynamics is given by:
∇.V = 0
where V is the velocity vector. Taking the curl of both sides of the equation, we get:
∇x(∇.V) = ∇x0 = 0
Now, using the vector identity ∇x(∇xA) = ∇(∇.A) - ∇2A, we can write:
∇(∇.V) - ∇2V = 0
Since the flow is assumed to be irrotational (i.e., ∇xV = 0), we have:
∇2Φ = 0
where Φ is the velocity potential, defined as V = ∇Φ. This is the Laplace equation, which is a second-order homogeneous partial differential equation.
The general solution to the Laplace equation in two dimensions is given by:
Φ(x,y) = A(x^2 - y^2) + B
where A and B are constants of integration.
Comparing this with the given options, we can see that option B, Φ(x,y) = x^2 - y^2, satisfies the Laplace equation and hence the continuity equation.
The continuity equation in fluid dynamics is given by:
∇.V = 0
where V is the velocity vector. Taking the curl of both sides of the equation, we get:
∇x(∇.V) = ∇x0 = 0
Now, using the vector identity ∇x(∇xA) = ∇(∇.A) - ∇2A, we can write:
∇(∇.V) - ∇2V = 0
Since the flow is assumed to be irrotational (i.e., ∇xV = 0), we have:
∇2Φ = 0
where Φ is the velocity potential, defined as V = ∇Φ. This is the Laplace equation, which is a second-order homogeneous partial differential equation.
The general solution to the Laplace equation in two dimensions is given by:
Φ(x,y) = A(x^2 - y^2) + B
where A and B are constants of integration.
Comparing this with the given options, we can see that option B, Φ(x,y) = x^2 - y^2, satisfies the Laplace equation and hence the continuity equation.
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Community Answer
The velocity potential which follow the equation of continuity is ____...
Option b
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