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Integration with variables Notes_Part_7

Integration with variables Notes_Part_7 - if and only if it...

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if and only if it has vanishing divergence: ∇ · v = ∂u ∂x + ∂v ∂y = 0 . (7 . 36) Incompressibility means that the fluid volume does not change as it flows. Most liquids, including water, are, for all practical purposes, incompressible. On the other hand, the flow is irrotational if and only if it has vanishing curl: ∇ × v = ∂v ∂x ∂u ∂y = 0 . (7 . 37) Irrotational flows have no vorticity, and hence no circulation. A flow that is both incom- pressible and irrotational is known as an ideal fluid flow . In many physical regimes, liquids (and, although less often, gases) behave as ideal fluids . Observe that the two constraints (7.36–37) are almost identical to the Cauchy–Riemann equations (7.18); the only difference is the change in sign in front of the derivatives of v . But this can be easily remedied by replacing v by its negative v . As a result, we establish a profound connection between ideal planar fluid flows and complex functions. Theorem 7.14. The velocity vector field v = ( u ( x,y ) ,v ( x,y )) induces an ideal fluid flow if and only if f ( z ) = u ( x,y ) i v ( x,y ) (7 . 38) is a complex analytic function of z = x + i y . Thus, the components u ( x,y ) and v ( x,y ) of the velocity vector field for an ideal fluid flow are necessarily harmonic conjugates. The corresponding complex function (7.38) is, not surprisingly, known as the complex velocity of the fluid flow. When using this result, do not forget
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