Potential flow

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

Analysis

Potential flow in two dimensions is simple to analyse using complex numbers, viewed for convenience on the Argand diagram.

Related Topics:
Complex number - Argand diagram

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The basic idea is to define a holomorphic function f. If we write

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:

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f(x+iy)=phi+ipsi

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then the Cauchy-Riemann equations show that

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:

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rac{partialphi}{partial x}=rac{partialpsi}{partial y},

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qquad

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rac{partialphi}{partial y}=-rac{partialpsi}{partial x}.

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(it is conventional to regard all symbols as real numbers; and to write

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z=x+iy and w=phi+ipsi).

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The velocity field underline{u}=(u,v), specified by

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:

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u=rac{partialphi}{partial x},qquad

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v=rac{partialphi}{partial y}

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then satisfies the requirements for potential flow:

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:

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ablacdotunderline{u}=

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abla^2phi=

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rac{partial^2phi}{partial x^2}+rac{partial^2phi}{partial y^2}=

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ight) =

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0

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and

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:

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left| abla imesunderline{u} ight|=

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rac{partial v}{partial x}-rac{partial u}{partial y}=

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rac{partial^2phi}{partial xpartial y}-

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rac{partial^2phi}{partial ypartial x}=0.

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psi is defined as the stream function. Lines of constant psi are known as streamlines and lines of constant phi are known as equipotential lines (see equipotential surface).

Related Topics:
Stream function - Streamline - Equipotential surface

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The two sets of curves intersect at right angles, for

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:

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abla phi cdot abla psi =

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rac{partialphi}{partial x}rac{partialpsi}{partial x}+

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rac{partialphi}{partial y}rac{partialpsi}{partial y}=

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