VIDEO: introductory comments on potential flow solutions.

Potential flow with plane symmetry has a number of applications, including airfoil theory and prediction of flow in
shallow microfluidic devices. In both cases, the variations in one dimension (e.g., the depth of a microchannel) are
assumed minor as compared to the variations in the other direction. When this is the case, the flow may be
approximated as two-dimensional. For 2D irrotational flows, the velocity potential and stream function are
orthogonal harmonic functions—isopotential contours are orthogonal to streamlines and these two functions can be
combined and manipulated using complex algebra.

In this section, we create flow solutions from basic elements, such as uniform flows, vortexes, sources, sinks, and
doublets. Sources, sinks, and doublets are components of a multipolar expansion of the flow solution, as discussed in
Appendix F. First, we describe how complex algebra is used for bookkeeping, then we describe basic elements,
then we describe how these elements can be superposed. This is a Green’s function approach to the solution of the
governing equation.

7.3.1 Complex algebra and its use in plane-symmetric potential flow

We use complex algebra in this section to simplify the mathematics for plane-symmetric potential flows. The
algebra of complex variables is convenient for describing velocity potential and stream function, for
reasons analogous to those for circuit analysis with sinusoidal functions, as summarized in Table 7.1.

Topic

Sinusoidal functions

Plane-symmetric Potential Flow

Key functions

sint, cost

ϕ_{v}(x,y), ψ(x,y)

Relation between functions

sint =cos(t = π∕2); functions
are 90 degrees out of phase

isocontours of ϕ_{v} and
ψ are spatially orthogonal to each
other—rotated 90 degrees in the
xy-plane.

Equations the functions satisfy

f′′+ f = 0; 2nd order
homogeneous ODE

∇^{2}f = 0; 2nd order
homogeneous PDE

Role of complex algebra in
simplifying the functions

expjωt =cost+jsint; not really
simpler

= ϕ_{v}+ jψ; two functions
combined into one

Role of complex algebra is
simplifying derivatives

(expjωt) = jωexpjωt

derivatives not really easier

Role of complex algebra in
simplifying solutions

solutions to the equations are
always of the form Aexpjωt +α

solution to the
equations are always of the form
= f(only)

Table 7.1: Analogies between the role of complex algebra in manipulating sinusoidal functions and the role
of complex algebra in treating plane-symmetric potential flow.

Complex distance
Given two points in the xy-plane separated by an x-distance Δx = x_{2}-x_{1} and a y-distance Δy = y_{2}-y_{1}, we can
define a complex distance :

where j = . This complex distance is a complex number that contains both x- and y-distance
information. The undertilde is used to denote a complex representation of two real, physical quantities. This
distance can also be written in terms of one length and one angle, in a manner analogous to cylindrical
coordinates:

where Δr= and Δθ =tan^{-1}. Here, Δr is the distance between the points, and Δθ is the angle the
line from point 1 to point 2 makes with the x-axis. The value of tan^{-1} is understood to fall between 0
and 2π, cosΔθ is equal to , and sinΔθ is equal to . Also, recall Euler’s formula
(expjα =cosα+jsinα).

Rotation
An immediate example of the utility of complex descriptions of distances in the xy-plane is the ease with
with a distance can be rotated. A complex distance can be rotated by an angle α by multiplying
it by exp[jα]. To rotate a flow by the angle α, we replace with exp[-jα] in the formulas for .

Complex velocity potential
We define also a complex velocity potential :

where ϕ_{v} is the velocity potential and ψ is the stream function. This combines the velocity potential and the stream
function into one complex function. Just as the x-direction and y-direction are orthogonal and conveniently
represented with a complex variable , ϕ_{v} and ψ are orthogonal and conveniently represented with a complex value
.

Complex velocity
Define also a complex velocity :

where u and v are the velocity components in the x- and y-directions.

When a flow is centered on the origin, it is useful to describe flow in terms ofradial (u_{r}) and
counterclockwisecircumferential (u_{θ}) velocities. In this case, we can equivalently define the complex velocity
as

Note that u_{r} and u_{θ} are defined with respect to the origin, as are the distances r and θ. The distances Δr and Δθ, in
contrast, are measured with respect to a point of interest.

Relation between complex velocity, distance, and velocity potential
The relation between the complex velocity, complex potential, and complex distance is analogous to the relation
= ∇ϕ_{v}:

where the star denotes the complex conjugate. The derivative with respect to can also be written
as

(7.20)

See Appendix G for more detailed information about complex differentiation.

Solutions to the Laplace Equation
The use of complex variables has an even more elegant result—any well-behaved (i.e., suitably differentiable)complex velocity potential that is a function of alone (= ()) automatically satisfies the Laplace Equation andis thus a solution of the fluid mechanical equations. This mathematical concept is discussed in Section G.1.2, but is
stated here as well. Any () satisfies the Laplace equation, so = , = ^{3}, =ln, and = all satisfy the
Laplace equation. This is not true if we specify any ϕ_{v}(x,y)—for example, ϕ_{v}= x^{2}, ϕ_{v}= y^{-1}, ϕ_{v}= x^{3}+y^{3}, and
ϕ_{v}= yxlnx all fail to satisfy the Laplace equation. The reason the complex variable technique works is because the
use of a complex variable enforces a specific spatial relation between the derivatives with respect to x and
y.

Thus, a Green’s function-type approach (i.e., first finding solutions to the governing equations, then combining
them to solve boundary conditions) becomes simple–any complex velocity potential () is already a solution, so
all that is required is to combine functions to match the intended boundary conditions, rather than actually solving
the Laplace equation directly. Since the Laplace equation is linear and homogeneous, we can superpose
solutions at will to match boundary conditions. Some of these solutions are described in the following
sections.

7.3.2 Monopolar flow: plane-symmetric (line) source with volume outflow per unit depth Λ

A plane-symmetric source with volume outflow per unit depth Λ located at the origin is denoted by a velocity
potential of ϕ_{v}=lnr and a stream function of ψ =θ, and leads to an exclusively radial velocity of
u_{r}= . A source is a monopole, i.e., a point singularity that induces a flow in the field around it (see
Section F.1.2).

More generally, the complex velocity potential

corresponds to the flow induced by a point source of fluid, where Λ is the source strength or volume outflow
rate per unit depth [m^{2}∕s] and is the complex distance measured from the location of the source.

Figure 7.2: Isopotentials, streamlines, velocity magnitudes, and net volumetric outflow for a 2D point
source with strength Λ.

After complex differentiation of , we examine the real part to find

and examine the imaginary part to find

This corresponds to a flow radiating outward from the location of the source. The velocity magnitude scales
inversely with the distance from the source (as is the case for all monopolar flows) and is equivalent in all directions.
The velocity potential is

(7.24)

and the stream function is

(7.25)

Thus, the the isopotentials are concentric circles around the source, and the streamlines are lines radiating out from
the source (Figure 7.2).

Plane-symmetric source located at the origin.
For the special case of a source at the origin, Δx = x, Δy = y, Δr= r, and Δθ = θ. Thus ^{2}= r^{2} and
= rexpjθ. Given that

(7.32)

we can write

(7.33)

And from = expjθ, we find

and

7.3.3 Plane-symmetric vortex with counterclockwise circulation per unit depth Γ

A plane-symmetric vortex with counterclockwise circulation per unit depth Γ located at the origin is denoted by a
velocity potential of ϕ_{v}=θ and a stream function of ψ = -lnr, and leads to an exclusively circumferential
velocity of u_{θ}= .

More generally, the complex velocity potential

corresponds to the flow induced by a plane-symmetric point vortex, where Γ is the vortex strength or circulation per
unit depth [m^{2}∕s] and is measured from the location of the vortex. The two-dimensional vortex (Figure 7.3) is
given by an expression analogous to that of the complex potential for a two-dimensional source, but multiplied by
-j. In this sense, a vortex is a monopole with an imaginary strength.

Figure 7.3: Isopotentials, streamlines, velocity magnitudes, and net solid body rotation for a 2D point
vortex with strength Γ.

The flow induced by a plane-symmetric vortex can be determined by differentiation of followed by an
examination of the real and imaginary parts of the result, leading to

and

This corresponds to a flow rotating counterclockwise (for positive Γ) around the location of the vortex. The velocity
magnitude scales inversely with the distance from the vortex and is equivalent in all directions.

The velocity potential is given by

(7.39)

and the stream function is given by

(7.40)

From this, we see the structure of isopotentials (lines radiating from the vortex) and streamlines (concentric circles
around the vortex).

Plane-symmetric vortex located at the origin.
For the case of a vortex at the origin, Δx = x, Δy = y, Δr= r, and Δθ = θ. Thus ^{2}= r^{2} and = rexpjθ.
Given that

(7.41)

we can write

(7.42)

And from = expjθ,

and

7.3.4 Dipolar flow: plane-symmetric doublet with dipole moment κ

A plane-symmetric doublet is mathematically equivalent to a source and a sink (both of equal and infinite
strength) separated by an infinitesimal distance. We define the vector dipole moment of the doublet in the
direction from the source to the sink. The doublet is a dipole, which is the mathematical limit of a
combination of two oppositely-signed, infinitely strong monopoles separated by an infinitesimal distance (see
Section F.1.2).

A plane-symmetric doublet with doublet strength or dipole moment κ located at the origin and aligned along the
x-axis is denoted by a velocity potential of ϕ_{v}= and a stream function of ψ = -. It leads to a radial
velocity of u_{r}= -cosθ and a circumferential velocity of u_{θ}= -sinθ.

Figure 7.4: Isopotentials, streamlines, and velocity magnitudes for a 2D doublet with strength κ.

The general complex velocity potential for a plane-symmetric doublet is given by

where κ is the dipole moment per unit depth [m^{3}∕s]. Here is measured from the location of the source,
and α indicates the angle of the vector dipole moment measured counterclockwise with respect to the
positive x-axis. The dipole moment thus has components of κcosα and κsinα. We characterize the
flow induced by a plane-symmetric doublet by differentiation of . The real component of the result
is

while the imaginary part is

This corresponds (for positive κ) to a flow emanating from a point just left of the doublet, circling
around, and returning to a point just right of the doublet. The velocity magnitude scales inversely with
the squared distance from the doublet (this is the case for all dipolar flows) and is equivalent in all
directions.

The velocity potential for this flow is

(7.55)

while the stream function is

(7.56)

The streamlines of the doublet flow are circles tangent to the axis of the doublet, and the isopotentials are circles
tangent to the normal to the axis of the doublet (Figure 7.4).

Plane-symmetric doublet located at the origin with dipole moment aligned along positive x-axis.
For the case of a doublet at the origin with dipole moment aligned with positive x-axis, Δx = x, Δy = y, Δr= r,
Δθ = θ, and α = 0. Thus ^{2}= r^{2} and = rexpjθ. Given that

(7.57)

we set α = 0, Δr= r, and Δθ = θ:

(7.58)

Since this flow is centered on the origin, cylindrical coordinates are convenient, and we look for velocity in the form
of = expjθ:

(7.59)

from which we see that

and

__________________________________________________________________________________________________________________________________________________________ EXAMPLE PROBLEM 7.4:

Consider 2D potential flow. In the domain -20 < x < 20, -20 < y < 20, plot the streamlines for a doublet
located at the origin with strength 50π and dipole moment aligned in the x-direction

A uniform flow moving with speed U in the x-direction is given simply by ϕ_{v}= Ux and ψ = Uy. Uniform flow is the
degenerate case of the Legendre polynomial solution to Laplace’s equation, where A_{0}= U.

More generally, the complex velocity potential

corresponds to a uniform flow at speed U in a direction rotated α counterclockwise from the positive x-axis,
i.e., a flow with x- and y-velocity components Ucosα and Usinα. This flow is by definition uniform
in space, so here can be measured relative to any arbitrary point. Usually, is measured from the
origin.

Uniform Flow: velocity by complex differentiation.
The uniform flow is easy to characterize by differentiation of , from which we can see that

and

The velocity potential is given by

(7.65)

and the stream function is given by

(7.66)

The streamlines for this flow are aligned at an angle α with respect to the x-axis, and isopotentials are aligned at an
angle α with respect to the y-axis.

Uniform Flow: α = 0.
For the α = 0 case (flow from left to right), we substitute in zero for α and find

Consider a geometry consisting of a wall starting at the origin and proceeding to r→∞ along a line at some
angleθ = θ_{0}, as well as a second wall starting at the origin and proceeding to r→∞ along the x-axis (θ = 0).
Consider flow that enters along the θ = θ_{0} wall at speed U, runs along the surface, and exits along the x-axis, again
at speed U (see Figure 7.6).

Figure 7.6: Flow through a corner

This flow is closely related to the uniform flow from the previous section; in fact, the previous section gives the
solution for the case where θ_{0}= π. In the domain for which 0 < Δθ < θ_{0}, the solution for this flow is given
by

(7.71)

The corner in this flow can be located arbitrarily (by measuring with respect to any point); further, this flow
can also be rotated so that it applies to flow entering along a wall at Δθ = θ_{0}+α and exiting along a wall at Δθ = α.
In this general case, the complex velocity potential is given by

Here is measured relative to the location of the corner.

The velocity potential for this flow is given by

where n = π∕θ_{0}, and the stream function for this flow is given by

The corner flow is commonly observed in any microdevice with a sharp corner. The Laplace equation solutions
predict stagnation (zero velocity) at any sharp concave corner and infinite velocity at any sharp convex corner, as
shown in Figure 7.7. In any real system, of course, corners have finite curvature. Thus, the velocities at
concave corners of finite radius of curvature are low, but finite; and the velocities of convex corners of
finite radius of curvature are high, but finite. If these corners had radii of curvature smaller than the
Debye length, the use of the Laplace equation to describe the flow outside the boundary layer would be
invalid.

Figure 7.7: Infinite velocities at sharp convex corners (left) and stagnation at sharp concave corners (right).

7.3.7 Flow over a circular cylinder

A doublet with strength κ and a uniform flow with speed U, both aligned in the same direction, can be superposed to
describe the potential flow over a circular cylinder with radiusa = :

(7.75)

This flow is observed in shallow microdevices with circular obstacles.

7.3.8 Conformal mapping

For plane symmetric potential flow, certain mapping functions make it easy to transform problems spatially in a
manner that still satisfies the kinematic and mass conservation relations. Since all that is required for a
plane-symmetric potential flow to satisfy the Laplace equations is that be a function of only, any of a variety of
transforms can be useful. Rotation in the plane (Section 7.3.1) is the simplest transform, and orients flows with
respect to the coordinate axes. Perhaps the most well-known of these mapping functions is theJoukowski transform
(see Section G.5 in the appendix), which maps a family of conic sections. It is particularly useful when mapping
results for circular objects to predict flow over elliptical or linear obstacles. Finally, the Schwarz-Christoffel
transform maps the upper half of the complex plane onto arbitrary polygonal shapes and thus facilitates calculation
of plane-symmetric potential flow through arbitrary polygonal channels. Given the relative ease of numerical
solutions of the Laplace equation, the use of these more advanced transforms by practising engineers has
become relatively uncommon; when the transforms are applicable, though,they can greatly simplify
analysis.