Cornell University - Visit www.cornell.edu Kirby Research Group at Cornell: Microfluidics and Nanofluidics : - Home College of Engineering - visit www.engr.cornell.edu Cornell University - Visit www.cornell.edu
Cornell University, College of Engineering Search Cornell
News Contact Info Login

Donations keep this resource free! Give here:

Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

[Return to Table of Contents]


Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

7.3 Potential flows with plane symmetry [potential flow top]

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

ϕv(x,y), ψ(x,y)

Relation between functions

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

isocontours of ϕv 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

2f = 0; 2nd order homogeneous PDE

Role of complex algebra in simplifying the functions

expjωt = cost+jsint; not really simpler

microfluidics textbook nanofluidics textbook Brian Kirby Cornell = ϕv + ; two functions combined into one

Role of complex algebra is simplifying derivatives

microfluidics textbook nanofluidics textbook Brian Kirby Cornell(expjωt) = expjωt

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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell = f(microfluidics textbook nanofluidics textbook Brian Kirby Cornellonly)





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 = x2 -x1 and a y-distance Δy = y2 -y1, we can define a complex distance microfluidics textbook nanofluidics textbook Brian Kirby Cornell:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where j = microfluidics textbook nanofluidics textbook Brian Kirby Cornell. 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:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where Δr = microfluidics textbook nanofluidics textbook Brian Kirby Cornell and Δθ = tan-1microfluidics textbook nanofluidics textbook Brian Kirby Cornell. 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell, and sinΔθ is equal to microfluidics textbook nanofluidics textbook Brian Kirby Cornell. Also, recall Euler’s formula (exp= 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell can be rotated by an angle α by multiplying it by exp[]. To rotate a flow by the angle α, we replace microfluidics textbook nanofluidics textbook Brian Kirby Cornell with microfluidics textbook nanofluidics textbook Brian Kirby Cornellexp[-] in the formulas for microfluidics textbook nanofluidics textbook Brian Kirby Cornell.


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


Complex velocity potential We define also a complex velocity potential microfluidics textbook nanofluidics textbook Brian Kirby Cornell:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where ϕv 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell, ϕv and ψ are orthogonal and conveniently represented with a complex value microfluidics textbook nanofluidics textbook Brian Kirby Cornell.

Complex velocity Define also a complex velocity microfluidics textbook nanofluidics textbook Brian Kirby Cornell:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 (ur) and counterclockwisecircumferential (uθ) velocities. In this case, we can equivalently define the complex velocity as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Note that ur and uθ 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell = ϕv:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where the star denotes the complex conjugate. The derivative with respect to microfluidics textbook nanofluidics textbook Brian Kirby Cornell can also be written as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell alone (microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornell(microfluidics textbook nanofluidics textbook Brian Kirby Cornell)) automatically satisfies the Laplace Equation and is 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell(microfluidics textbook nanofluidics textbook Brian Kirby Cornell) satisfies the Laplace equation, so microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornell, microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornell3, microfluidics textbook nanofluidics textbook Brian Kirby Cornell = lnmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell, and microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornell all satisfy the Laplace equation. This is not true if we specify any ϕv(x,y)—for example, ϕv = x2, ϕv = y-1, ϕv = x3 +y3, and ϕv = 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell(microfluidics textbook nanofluidics textbook Brian Kirby Cornell) 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 ϕv = microfluidics textbook nanofluidics textbook Brian Kirby Cornelllnr and a stream function of ψ = microfluidics textbook nanofluidics textbook Brian Kirby Cornellθ, and leads to an exclusively radial velocity of ur = microfluidics textbook nanofluidics textbook Brian Kirby Cornell. 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

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

corresponds to the flow induced by a point source of fluid, where Λ is the source strength or volume outflow rate per unit depth [m2s] and microfluidics textbook nanofluidics textbook Brian Kirby Cornell is the complex distance measured from the location of the source.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 7.2: Isopotentials, streamlines, velocity magnitudes, and net volumetric outflow for a 2D point source with strength Λ.


After complex differentiation of microfluidics textbook nanofluidics textbook Brian Kirby Cornell, we examine the real part to find

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and examine the imaginary part to find

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.24)

and the stream function is
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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).


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell2 = r2 and microfluidics textbook nanofluidics textbook Brian Kirby Cornell= rexp. Given that
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.32)

we can write
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.33)

And from microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornellexp, we find

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 ϕv = microfluidics textbook nanofluidics textbook Brian Kirby Cornellθ and a stream function of ψ = -microfluidics textbook nanofluidics textbook Brian Kirby Cornelllnr, and leads to an exclusively circumferential velocity of uθ = microfluidics textbook nanofluidics textbook Brian Kirby Cornell.

More generally, the complex velocity potential

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

corresponds to the flow induced by a plane-symmetric point vortex, where Γ is the vortex strength or circulation per unit depth [m2s] and microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell followed by an examination of the real and imaginary parts of the result, leading to

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.39)

and the stream function is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell2 = r2 and microfluidics textbook nanofluidics textbook Brian Kirby Cornell= rexp. Given that
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.41)

we can write
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.42)

And from microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornellexp,

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and

microfluidics textbook nanofluidics textbook Brian Kirby Cornell


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


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 ϕv = microfluidics textbook nanofluidics textbook Brian Kirby Cornell and a stream function of ψ = -microfluidics textbook nanofluidics textbook Brian Kirby Cornell. It leads to a radial velocity of ur = -microfluidics textbook nanofluidics textbook Brian Kirby Cornell cosθ and a circumferential velocity of uθ = -microfluidics textbook nanofluidics textbook Brian Kirby Cornell sinθ.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 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

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where κ is the dipole moment per unit depth [m3s]. Here microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell. The real component of the result is

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

while the imaginary part is

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.55)

while the stream function is
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell2 = r2 and microfluidics textbook nanofluidics textbook Brian Kirby Cornell= rexp. Given that
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.57)

we set α = 0, Δr = r, and Δθ = θ:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.58)

Since this flow is centered on the origin, cylindrical coordinates are convenient, and we look for velocity in the form of microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornellexp:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.59)

from which we see that

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

__________________________________________________________________________________________________________________________________________________________
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

Solution: See Figure 7.5.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 7.5: Streamlines for a doublet flow.


__________________________________________________________________________________________________________________________________________________________

7.3.5 Uniform flow with speed U

A uniform flow moving with speed U in the x-direction is given simply by ϕv = Ux and ψ = Uy. Uniform flow is the degenerate case of the Legendre polynomial solution to Laplace’s equation, where A0 = U.

More generally, the complex velocity potential

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 U cosα and U sinα. This flow is by definition uniform in space, so here microfluidics textbook nanofluidics textbook Brian Kirby Cornell can be measured relative to any arbitrary point. Usually, microfluidics textbook nanofluidics textbook Brian Kirby Cornell is measured from the origin.

Uniform Flow: velocity by complex differentiation. The uniform flow is easy to characterize by differentiation of microfluidics textbook nanofluidics textbook Brian Kirby Cornell, from which we can see that

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The velocity potential is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.65)

and the stream function is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and for the velocities we find

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

7.3.6 Flow around a corner [potential flow top]

Consider a geometry consisting of a wall starting at the origin and proceeding to r→∞ along a line at some angleθ = θ0, 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 = θ0 wall at speed U, runs along the surface, and exits along the x-axis, again at speed U (see Figure 7.6).


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 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 θ0 = π. In the domain for which 0 < Δθ < θ0, the solution for this flow is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(7.71)

The corner in this flow can be located arbitrarily (by measuring microfluidics textbook nanofluidics textbook Brian Kirby Cornell with respect to any point); further, this flow can also be rotated so that it applies to flow entering along a wall at Δθ = θ0 +α and exiting along a wall at Δθ = α. In this general case, the complex velocity potential is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Here microfluidics textbook nanofluidics textbook Brian Kirby Cornell is measured relative to the location of the corner.

The velocity potential for this flow is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where n = π∕θ0, and the stream function for this flow is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure 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 = microfluidics textbook nanofluidics textbook Brian Kirby Cornell:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell be a function of microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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.

[Return to Table of Contents]



Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


Ad revenue from these pages is used to support student research. The presence of an advertisement on these pages does not constitute an endorsement by the Kirby Research Group or Cornell University.

Donations keep this resource free! Give here: