Potential flow

In fluid dynamics, potential flow, also known as irrotational flow (of incompressible fluids) is steady flow defined by the equations

Examples: Power laws

If

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:

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w=Az^n

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then, writing x+iy=re^{i heta}, we have

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:

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phi=Ar^ncos n heta

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and

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:

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psi=Ar^nsin n heta

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Power law with $n1$

If w=Az^1, that is, a power law with n=1, the streamlines (ie lines of constant psi) are a system of straight lines parallel to the x-axis.

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This is easiest to see by writing in terms of real and imaginary components:

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:

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f(x+iy)=A imes(x+iy)=Ax+icdot Ay

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thus giving phi=Ax and psi=Ay.

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Power law with $n2$

If n=2, then w=Az^2 and the streamline corresponding to a particular value of psi are those points satisfying

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:

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psi=Ar^2sin 2 heta

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which is a system of rectangular hyperbolae. This may be seen by again rewriting in terms of real and imaginary components. Noting that sin 2 heta=2sin heta,cos heta and rewriting sin heta=y/r and cos heta=x/r it is seen (on simplifying) that the streamlines are given by

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:

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psi=2Axy.

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The velocity field is given by ablaphi, or

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:

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(u,v)= left( {partial phi over partial x}, {partial phi over partial y} ight) =

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left( {partial psi over partial y}, - {partial psi over partial x} ight) =

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left(2Ax,-2Ay ight)

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In fluid dynamics, the flowfield near the origin corresponds to a stagnation point. Note that the fluid at the origin is at rest (this follows on differentiation of f(z)=z^2 at z=0).

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The psi=0 streamline is particularly interesting: it has two (or four) branches, following the coordinate axes, ie x=0 and y=0.

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As no fluid flows across the x-axis, it (the x-axis) may be treated as a solid boundary (remember that the physical system to which this analysis corresponds is an inviscid (ie zero viscosity) fluid; there are thus no boundary layers to worry about). It is thus possible to ignore the flow in the lower half-plane where y

Related Topics:
Viscosity - Boundary layer

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With this interpretation, the flow is that of a vertically directed jet impinging on a horizontal flat plate.

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The flow may also be interpreted as flow into a 90 degree corner if the regions specified by (say) x

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Power law with $n3$

If n=3 the resulting flow is a sort of hexagonal version of the n=2 case considered above. Streamlines are given by x^2y-y^3=psi.

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Power law with $n-1$

if n=,-1, the streamlines are given by

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:

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psi=-rac{A}{r}sin heta.

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This is more easily interpreted in terms of real and imaginary components:

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: psi = {-A y over r^2} = {-A y over x^2 + y^2},

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: x^2 + y^2 + {A y over psi} = 0,

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:

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x^2+left(y+rac{A}{2psi} ight)^2=left(rac{A}{2psi} ight)^2.

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Thus the streamlines are circles that are tangent to the x-axis at the origin.

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The velocity field is given by

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:

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(u,v)=left( {partial psi over partial y}, - {partial psi over partial x} ight) =

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left(Arac{y^2-x^2}{(x^2+y^2)^2},-Arac{2xy}{(x^2+y^2)^2} ight)

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The circles in the upper half-plane thus flow clockwise, those in the lower half-plane flow anticlockwise. Note that speeds go as r^{-2}; and the speed at the origin is infinite.

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Power law with n equals minus 2

{this section is to be completed}

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See also: Laplacian field, conformal mapping.

Related Topics:
Laplacian field - Conformal mapping

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