Given the x - component of the velocity u = 6xy - 2x2, the y-component of the flow velocity is given by:a)6y2 - 4xyb)-6xy + 2x2c)6x2 - 2xyd)4xy - 3y2Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Given: x-component of the velocity u = 6xy - 2x^2

To find: y-component of the flow velocity

Solution:

The velocity vector can be represented as (u, v), where u is the x-component and v is the y-component.

We know that the velocity vector is the gradient of the velocity potential function φ.

i.e., (u, v) = (∂φ/∂x, ∂φ/∂y)

We are given the x-component of the velocity u. So, we can find the velocity potential function φ as follows:

φ = ∫(6xy - 2x^2)dx = 3x^2y - (2/3)x^3 + g(y)

where g(y) is the constant of integration with respect to x.

Taking the partial derivative of φ with respect to y, we get:

∂φ/∂y = 3x^2 + dg(y)/dy

But, we know that ∂φ/∂y = v (y-component of the velocity)

Therefore, v = 3x^2 + dg(y)/dy

To find g(y), we can use the fact that the velocity is irrotational, i.e., the curl of the velocity vector is zero.

i.e., ∂v/∂x - ∂u/∂y = 0

Substituting u and v in the above equation, we get:

∂(3x^2)/∂x - ∂(3x^2 + dg(y)/dy)/∂y = 0

Simplifying, we get:

dg(y)/dy = -4xy

Integrating with respect to y, we get:

g(y) = -2xy^2 + C

where C is the constant of integration with respect to y.

Substituting this value of g(y) in the expression for v, we get:

v = 3x^2 - 4xy

Therefore, the y-component of the flow velocity is given by option 'D': 4xy - 3y^2.

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