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For a two dimensional flow, the stream function is given by 1/2 = 2xy. The velocity at a point (3, 4) is equal to :-
- a)6m/sec
- b)8m/ sec
- c)10 m/sec
- d)12m/ sec
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
For a two dimensional flow, the stream function is given by 1/2= 2xy. ...
u = dy/dx = 2y = 2*4 = 8.
v = dy/dy = 2x = 2*3 = 6.
|v| = (8^2+6^2)^1/2 = 10.
Most Upvoted Answer
For a two dimensional flow, the stream function is given by 1/2= 2xy. ...
Stream Function and Velocity in Two Dimensional Flow
Stream Function:
- Stream function is a mathematical function used to describe the flow of a two-dimensional fluid.
- It is defined as the function whose gradient gives the velocity components of the fluid flow.
- For a two-dimensional flow, the stream function is a scalar function of two coordinates (x,y) and is denoted by the symbol ψ.
Velocity Components from Stream Function:
- The velocity components in x and y directions can be obtained by taking the partial derivatives of the stream function with respect to y and x, respectively.
- The velocity components are given by:
- Vx= - ∂ψ/∂y
- Vy= ∂ψ/∂x
Given Stream Function:
- The given stream function is 1/2= 2xy.
- Taking the partial derivatives with respect to x and y, we get:
- Vx= - ∂ψ/∂y = -2x
- Vy= ∂ψ/∂x = 2y
- Substituting the coordinates (3,4), we get:
- Vx= -2(3) = -6
- Vy= 2(4) = 8
Velocity at (3,4):
- The velocity at point (3,4) can be obtained by finding the magnitude of the velocity vector using the velocity components obtained above.
- The velocity vector is given by:
- V = sqrt(Vx^2 + Vy^2)
- Substituting the calculated values, we get:
- V = sqrt((-6)^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 m/sec
Answer:
- Therefore, the velocity at point (3,4) is 10 m/sec, which is option (c).
Stream Function:
- Stream function is a mathematical function used to describe the flow of a two-dimensional fluid.
- It is defined as the function whose gradient gives the velocity components of the fluid flow.
- For a two-dimensional flow, the stream function is a scalar function of two coordinates (x,y) and is denoted by the symbol ψ.
Velocity Components from Stream Function:
- The velocity components in x and y directions can be obtained by taking the partial derivatives of the stream function with respect to y and x, respectively.
- The velocity components are given by:
- Vx= - ∂ψ/∂y
- Vy= ∂ψ/∂x
Given Stream Function:
- The given stream function is 1/2= 2xy.
- Taking the partial derivatives with respect to x and y, we get:
- Vx= - ∂ψ/∂y = -2x
- Vy= ∂ψ/∂x = 2y
- Substituting the coordinates (3,4), we get:
- Vx= -2(3) = -6
- Vy= 2(4) = 8
Velocity at (3,4):
- The velocity at point (3,4) can be obtained by finding the magnitude of the velocity vector using the velocity components obtained above.
- The velocity vector is given by:
- V = sqrt(Vx^2 + Vy^2)
- Substituting the calculated values, we get:
- V = sqrt((-6)^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 m/sec
Answer:
- Therefore, the velocity at point (3,4) is 10 m/sec, which is option (c).
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For a two dimensional flow, the stream function is given by 1/2= 2xy. The velocity at a point (3, 4) is equal to :-a)6m/secb)8m/ secc)10 m/secd)12m/ secCorrect answer is option 'C'. Can you explain this answer?
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