Given a velocity field with u = 2y and v = x^2 ,find:

(a) Equations for streamline passing through points (0,0) and (0,1)
(b) Plot several streamlines in all 4 quadrants along with directions
(c) What is the relationship between these Streamlines to the Stream Function (written)?
(d) What is the velocity potential for this flow
(e) Does the flow satisfy conservation of mass everywhere?

a) y² = (1/3) x³  (Pic 1)

y² = (1/3) x³ + 1  (Pic 2)

b) The streamlines are shown in the pics.

c) The stream function is used to plot the streamlines of the flow (which represent the trajectories of particles in a steady flow) and find the velocity.

d) V = 2y i + x² j

e) This flow does not satisfy conservation of mass.

Explanation:

Given

u = 2y

v = x²

a) So the streamlines are given by

dy/dx = v/u = x²/2y

Thus

∫2ydy = ∫x²dx

⇒ y² = (1/3) x³ + C

where C is a constant.

For the streamline that goes through x = y = 0, C = 0

Hence

y² = (1/3) x³

For the streamline that goes through x = 0 and y = 1,

(1)² = (1/3) (0)³ + C = 0

⇒ C = 1

Hence

y² = (1/3) x³ + 1

b) The streamlines are plotted in the pic shown.

c) Stream functions are defined for two-dimensional flow and for three-dimensional axial symmetric flow. The stream function is used to plot the streamlines of the flow (which represent the trajectories of particles in a steady flow) and find the velocity.

d) V = 2y i + x² j

e) We apply ∇V = 0

then

∇V = ∂V/∂x + ∂V/∂y + ∂V/∂z

⇒  ∇V = (0, 2x, 0) + (2, 0 , 0) + (0, 0, 0) = (2, 2x, 0) ≠ 0

This flow does not satisfy conservation of mass.

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explanation: