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The velocity components of an incompressible, two-dimensional velocity field are given by the equations u= y2 – x(1 + x) v = y(2x + 1) Show that the flow is irrotational and satisfies conservation of mass.



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The velocity flow field is irrotational and the conservation of mass is preserved.

Irrotational flow for a two-dimensional velocity field happens when

[tex]\frac{dv}{dx} \:-\: \frac{du}{dy} \:=\: 0[/tex]

Conservation of mass for an incompressible happens when

[tex]\frac{du}{dx} \:+\: \frac{dv}{dy} \:=\: 0[/tex]

Perimeter

  • u = y² - x (1+x)
    u = y² - x - x²
  • v = y (2x + 1)
    v = 2xy + y

Calculate the derivate in each function.

  • [tex]\frac{du}{dx} = 0 - 1 - 2x = - 2x - 1[/tex]
  • [tex]\frac{du}{dy} = 2y - 0 - 0 = 2y[/tex]
  • [tex]\frac{dv}{dx} = 2y + 0 = 2y[/tex]
  • [tex]\frac{du}{dy} = 2x + 1[/tex]

Checking the irrotational flow

[tex]\frac{dv}{dx} \:-\: \frac{du}{dy} \:=\: 2y - 2y = 0[/tex]

Checking conservation of mass for an incompressible

[tex]\frac{du}{dx} \:+\: \frac{dv}{dy} \:=\: - 2x - 1 + 2x + 1 = 2x - 2x + 1 - 1 = 0[/tex]

So the velocity flow field is irrotational and the conservation of mass is preserved.

Learn more about incompressible flow here https://brainly.com/question/12950504

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