Answer to Question #149106 in Vector Calculus for Usman

Question #149106
If for a two dimensional potential flow, the velocity potential is given by : φ = 4x(3y - 4), determine the velocity at the point (2, 3). Determine also the value of stream function ψ at the point (2, 3).
1
Expert's answer
2020-12-11T09:56:30-0500

In velocity we can write 


"u=\\dfrac{\\partial\\varphi}{\\partial x}, v=\\frac{\\partial\\varphi}{\\partial y}"


Given "\\varphi=4x(3y-4)"

"u=\\dfrac{\\partial\\varphi}{\\partial x}=12y-16, v=\\dfrac{\\partial\\varphi}{\\partial y}=12x"

Calculate the velocity at the point (2, 3)


"u=20, v=24"

"|velocity|=\\sqrt{u^2+v^2 }=\\sqrt{(20)^2+(24)^2}=4\\sqrt{61}"

"u=\\dfrac{\\partial\\psi}{\\partial y}, v=-\\dfrac{\\partial\\psi}{\\partial x}"

"u=\\dfrac{\\partial\\psi}{\\partial y}=12y-16"

Integrate with respect to "y"


"\\psi=6y^2-16y+g(x)"

"-\\frac{\\partial\\psi}{\\partial x}=-\\frac{dg}{d x}=12x"

Integrate with respect to "x"


"g(x)=-6x^2+C"

Then


"\\psi(x,y)=-6x^2+6y^2-16y+C"


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