Potential flow in two dimensions Guide, Meaning , Facts, Information and Description
In
fluid dynamics,
potential flow in two dimensions is simple to analyse using
complex numbers.
The basic idea is to define a holomorphic or meromorphic function . If we write
then the
Cauchy-Riemann equations show that
-
(it is conventional to regard all symbols as real numbers; and to write
and ).
The velocity field , specified by
-
then satisfies the requirements for potential flow:
-
and
Lines of constant are known as
streamlines and lines of constant are known as equipotential lines (see
equipotential surface).
The two sets of curves intersect at right angles, for
-
showing that, at any point, a vector perpendicular to the contour line has a dot product of zero with a vector perpendicular to the contour line (the two vectors thus intersecting at ). The identity may be proved by using the Cauchy-Riemann equations given above:
Thus the flow occurs along the lines of constant
ψ and at right angles to the lines of constant
φ.
It is interesting to note that is also satisfied, this relation being eqivalent to (the automatic condition gives ).
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