NationMaster - Encyclopedia: Stream function

The stream function is defined for two-dimensional flows of various kinds. The stream function can be used to plot stream lines, which are perpendicular to equipotential lines. In most cases, the stream function is the imaginary part of the complex potential, while the potential function is the real part. In Mathematics and Physics (especially Electronics), a region is called equipotential if every point in it is at the same potential. ...

Considering the particular case of fluid dynamics, the difference between the stream function values at any two points gives the volumetric flow rate (or flux) through a line connecting the two points. Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... flux in science and mathematics. ...

Note that since streamlines are tangent to the flow, the value of the stream function must be the same along a streamline. If there were a flux across a line, it would necessarily not be tangent to the flow, hence would not be a streamline. Look up streamline in Wiktionary, the free dictionary. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...

The usefulness of the stream function lies in the fact that the velocity components in the x- and y- directions at a given point are given by the partial derivatives of the stream function at that point. A stream function may be defined for any flow of dimensions greater than two, however the two dimensional case is generally the easiest to visualize and derive. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...

Taken together with the velocity potential, the stream function may be used to derive a complex potential for a fluid flow. A velocity potential is used in fluid dynamics, when a fluid is irrotational. ... A potential flow is characterized by an irrotational velocity field. ...

1 Two dimensional stream function
2 See also
3 References

Two dimensional stream function

The stream function $ψ$ for a two dimensional flow is defined such that the flow velocity can be expressed as:

$mathbf{u}= nabla times boldsymbol{psi}$

Where $boldsymbol{psi} = (0,0,psi)$ if the velocity vector $mathbf{u} = (u,v,0)$ .

In Cartesian coordinate system this is equivalent to Fig. ...

$u= frac{partialpsi}{partial y},qquad v= -frac{partialpsi}{partial x}$

Where $u$ and $v$ are the velocities in the $hat{mathbf{x}}$ and $hat{mathbf{y}}$ directions, respectively. In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

This formulation of the stream function satisfies the two dimensional continuity equation: All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

$frac{partial u}{partial x} + frac{partial v}{partial y} = 0$

Derivation of the two dimensional stream function

Consider two points A and B in two dimensional plane flow. If the distance between these two points is very small: δn, and a stream of flow passes between these points with an average velocity, q perpendicular to the line AB, the volume flow rate per unit thickness, δΨ is given by:

$delta psi = q delta n,$

As δn → 0, rearranging this expression, we get:

$q = frac{partial psi}{partial n},$

Now consider two dimensional plane flow with reference to a coordinate system. Suppose an observer looks along an arbitrary axis in the direction of increase and sees flow crossing the axis from right to left. A sign convention is adopted such that the velocity of the flow is positive. However, this sign convention is not universal and thus ought to be taken with caution. A couple of examples ought to clarify this point:

Flow in Cartesian coordinates

By observing the flow into an elemental square in an x-y Cartesian coordinate system, we have: Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

$delta psi = -u delta y,$

$delta psi = v delta x,$

where u is the velocity parallel to and in the direction of the x-axis, and v is the velocity parallel to and in the direction of the y-axis. In this case, we have a negative velocity of -u since the flow crosses the y-axis from left to right. Thus as δn → 0 and by rearranging, we have:

$u = - frac{partial psi}{partial y},$

$v = frac{partial psi}{partial x},$

Flow in Polar coordinates

By observing the flow into an elemental square in an r-θ Polar coordinate system, we have: This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$delta psi = -v_r ( r delta theta ),$

$delta psi = v_theta delta r,$

where v_r is the velocity parallel to and in the direction of the r-axis, and v_θ is the velocity parallel to and in the direction of the θ-axis. In this case, we have a negative velocity of -v_r since the flow crosses the θ-axis from left to right. Thus as δn → 0 and by rearranging, we have:

$v_r = - frac{1}{r} frac{partial psi}{partial theta},$

$v_theta = frac{partial psi}{partial r},$

Continuity: The Derivation

Consider two dimensional plane flow within a Cartesian coordinate system. Continuity states that if we consider flow into an elemental square, the flow into that small element must equal the flow out of that element. All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

The total flow into the element is given by:

$delta psi_{in} = u delta y + v delta x,$

The total flow out of the element is given by:

$delta psi_{out} = left( u + frac{partial u}{partial x}delta x right) delta y + left( v + frac{partial v}{partial y}delta y right) delta x,$

Thus we have:

$delta psi_{in} = delta psi_{out},$

$u delta y + v delta x = left( u + frac{partial u}{partial x}delta x right) delta y + left( v + frac{partial v}{partial y}delta y right) delta x,$

and simplifying to:

$frac{partial u}{partial x} + frac{partial v}{partial y} = 0$

(NB: Whilst we have ignored the earlier sign convention for Ψ, the mathematics ought work out fine but this proof is perhaps slightly tidier and easier to follow). Substituting the expressions of the stream function into this equation, we have:

$frac{partial^2 psi}{partial x partial y} - frac{partial^2 psi}{partial y partial x} = 0$

Vorticity

In Cartesian coordinates, the stream function can be found from vorticity using Poisson's equation: Vorticity is a mathematical concept used in fluid dynamics. ... In mathematics, Poissons equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. ...

$vec nabla ^2 psi = -omega$

where $vec omega = ( 0, 0, omega )$ and $vec omega = vec nabla times vec v .$

References

B. Massey, "Mechanics of Fluids (Seventh Edition)"
T. W. Gamelin, "Complex Analysis", Springer, NY, (2001). ISBN 0-387-95093-1.

Categories: Engineering | Fluid dynamics | Fluid mechanics

Results from FactBites:

Stream function - Wikipedia, the free encyclopedia (674 words)
	The usefulness of the stream function lies in the fact that the velocity components in the x- and y- directions at a given point are given by the partial derivatives of the stream function at that point.
	A stream function may be defined for any flow of dimensions greater than two, however the two dimensional case is generally the easiest to visualize and derive.
	Taken together with the velocity potential, the stream function may be used to derive a complex potential for a fluid flow.

ANSI and GNU Common Lisp Document: Streams (3537 words)
	The action of closing a stream marks the end of its use as a source or sink of data, permitting the implementation to reclaim its internal data structures, and to free any external resources which might have been locked by the stream when it was opened.
	A concatenated stream is an input stream which is a composite stream of zero or more other input streams, such that the sequence of data which can be read from the concatenated stream is the same as the concatenation of the sequences of data which could be read from each of the constituent streams.
	A stream that is an alias for another stream, which is the value of a dynamic variable whose name is the synonym stream symbol of the synonym stream.

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