---
title: "Recalculating the induced-charge flow around a cylinder"
description: "Read the neutral-cylinder figure in Squires and Bazant, then reconstruct a limited weak-field steady result from potential, slip and the Stokes field."
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last_modified: "2026-10-28"
---

[Electrochemistry and ion transport](https://sandboxphysics.com/stories/electrochemistry/)

L4 · Chapter 8

# Recalculating the induced-charge flow around a cylinder

Sandbox Physics · 2026-10-28

Open Squires and Bazant’s Induced-charge electro-osmosis at arXiv:physics/0304090v2. The four-lobed flow in Fig. 3(a) can be calculated from one exterior cylinder potential. Identify which charges belong to the metal and which to the liquid, check the slip direction, then use Table 1 to redraw the flow. Those steps test whether a reconstruction actually obeys the paper’s model.

## Before you begin

Use the slip relation in How surface ions drag the surrounding liquid. You should be able to read polar coordinates, partial derivatives and a first-order differential equation. The charge-sign and special-position calculations can be done before learning numerical integration.

[Review the preceding chapter](https://sandboxphysics.com/stories/electrochemistry/impedance-identifiability/)

- Locate the figures, equations and boundaries in a fixed source version.
- Check the analytic velocity at special positions before judging an image.
- Separate a published theoretical figure, your calculated reference and experimental data.

## Put the two titles and versions in order

The long JFM article, Induced-charge electro-osmosis, is fixed here to physics/0304090v2, revised 9 February 2004 and subsequently published in Journal of Fluid Mechanics 509, 217–252. Use section 3.1 and Fig. 3(a). Equation numbers below refer to that PDF; some HTML renderings renumber them sequentially, so locate the equation by its content and section if necessary.

The shorter Induced-charge Electrokinetic Phenomena: Theory and Microfluidic Applications is physics/0306100v2, revised 8 February 2004, published as Physical Review Letters 92, 066101. Its Fig. 1(c) gives the companion neutral-cylinder picture. Reading them together is useful; a calculation still needs one stated version and equation set.

The selected slice is steady DC flow around a fixed neutral cylinder in a dilute electrolyte with a thin double layer and weak applied field. AC forcing, charging transients, net cylinder charge and surface coatings have additional conditions and remain separate calculations.

Reading: Squires & Bazant · Bazant & Squires

## Read the charge signs before the arrows in Fig. 3(a)

Let the imposed-field direction define $\theta=0$. Radius $a$ runs from the axis to the metal surface; radial coordinate $r$ increases into the liquid. The cylinder is fixed and has zero total charge. Neutrality means the charge integrates to zero around its circumference, allowing positive and negative regions locally.

At the pole facing the imposed field, metal potential exceeds the nearby outer-liquid potential. The metal charge is positive and its diffuse ionic screening charge is negative. Signs reverse at the other pole. Confusing metal and liquid charge is enough to reverse a predicted slip direction.

Now follow the near-surface arrows: liquid moves from the poles towards the equator. Outer liquid approaches along the poles and leaves along the equatorial direction. There is no frequency axis, time axis or experimental error bar. This is a theoretical steady-flow structure. Quantitative comparison also checks speed, boundaries and conservation.

Reading: Squires & Bazant

## Check three signs using the same potential

PDF Eq. (3.4) gives the charged steady exterior potential. Set metal potential to zero, take the positive distant field along the polar axis and measure angle anticlockwise. Insert $r=a$ and subtract that outer-liquid potential from the metal potential to obtain Eq. (3.5), $\zeta$. Differentiate the same potential around the circumference for the tangential field in Eq. (3.6).

At $\theta=45^\circ$, zeta potential is positive, diffuse charge is negative and tangential electric field is negative. Electrical force on those negative ions is positive in the angular direction. Liquid therefore moves towards the equator. The slip relation should give a positive tangential velocity there.

$$
\phi=-E_0\left(r+\frac{a^2}{r}\right)\cos\theta,\quad\zeta=2E_0a\cos\theta,\quad E_\theta(a)=-2E_0\sin\theta
$$

All three quantities follow from one potential. The outer liquid is neutral to leading order and normal outer current vanishes; the metal itself remains equipotential.

$$
q_d=-\frac{\varepsilon\zeta}{\lambda_D},\quad u_s=-\frac{\varepsilon\zeta E_\theta}{\eta}=2U_0\sin(2\theta),\quad U_0=\frac{\varepsilon aE_0^2}{\eta}
$$

Here $q_d$ is diffuse ionic charge per surface area, $\varepsilon$ is liquid permittivity and $\eta$ is dynamic viscosity. PDF Eqs. (3.7)–(3.8) give slip and its velocity scale.

Reading: Squires & Bazant

## Calculate three values before tracing a streamline

Choose $a=10.0\,\mu\mathrm m$, $E_0=0.450\,\mathrm{V\,mm^{-1}}$, $\varepsilon=6.95\times10^{-10}\,\mathrm{F\,m^{-1}}$ and $\eta=10^{-3}\,\mathrm{Pa\,s}$. Convert field to $450\,\mathrm{V\,m^{-1}}$ and radius to metres before multiplying. The result is $U_0=1.407375\,\mu\mathrm{m\,s^{-1}}$.

At forty-five degrees on the surface, speed should be $2U_0$. Then use PDF Eq. (3.9) or Table 1 at two positions twice as far from the axis as the cylinder radius. Normalizing by $U_0$ removes the chosen water properties and gives useful hand-calculated checks.

$$
u_r=2U_0\frac{a(a^2-r^2)}{r^3}\cos(2\theta),\qquad u_\theta=2U_0\frac{a^3}{r^3}\sin(2\theta)
$$

Both components determine a streamline. Extending surface slip directly into the liquid would omit the radial flow required by incompressibility.

Special positions in the neutral-cylinder field; these are recalculated analytic references.

| Position | Radial velocity $u_r/U_0$ | Tangential velocity $u_\theta/U_0$ |
| --- | --- | --- |
| $r=a,\ \theta=45^\circ$ | $0$ | $2$ |
| $r=2a,\ \theta=0$ | $-3/4$ | $0$ |
| $r=2a,\ \theta=45^\circ$ | $0$ | $1/4$ |

Reading: Squires & Bazant

## Save the calculation and check boundaries and conservation

The reference file methods-iceo-reference.json contains $181$ surface angles and $1681$ velocity-grid points. Surface columns include angle, zeta potential, tangential field, diffuse charge and slip. Grid positions $x,y$ are in micrometres; velocities $u,v$ are in micrometres per second. Points marked insideMetal belong to the solid and are excluded from exterior-flow integration.

Keep the parameter, unit and source-version fields with the samples. These are our recalculated analytic values. The paper’s theoretical figure is the reading target. Neither contains the measurement noise of a metal-wire experiment; an experimental comparison needs separate apparatus dimensions, surface characterization and acquisition conditions.

First test zero normal velocity on the cylinder and equality of tangential velocity with electrically derived slip. Integrate around a circle to check zero net volume flux and zero total diffuse charge. Finite differences first test divergence, the net outflow from a small box. They then give vorticity, a measure of local rotation. The sum of its second spatial derivatives tests the pressure-eliminated steady Stokes equation. Trace particles and redraw the four-lobed pattern only after these checks.

At the default $\lambda_D=0.100\,\mu\mathrm m$, layer-to-radius ratio is $0.0100$ and maximum induced potential is $9.00\,\mathrm{mV}$. These scales support the limited weak-field, thin-layer demonstration. Increasing field requires rechecking induced potential relative to thermal voltage before extrapolating the square law.

Reading: Squires & Bazant · Sandbox Physics analytic ICEO reference

## Use the same reading order on the next paper

Fix the version and figure, then label each curve as theory, simulation or measurement. Extract geometry and coordinates from the caption, and boundaries and approximations from the text. Choose two or three limits you can calculate by hand. Save the parameters and intermediate quantities before plotting the full result.

If the next paper adds reactions, walls or concentration changes, identify the altered boundary or conservation equation before reusing a formula. A small reconstruction is most useful when it tells you exactly which step must be checked again.

## Try the prediction

### [Test three special positions in the four-lobed flow](https://sandboxphysics.com/labs/induced-charge-electroosmosis/)

1. Choose Four lobes of flow and download the calculated profiles. Read surface angles and velocity-grid units.
2. Check the surface slip at forty-five degrees, then the radial and tangential velocities at twice the cylinder radius.
3. Reverse the field and compare charge signs, velocity values and streamlines. Explain which factors change sign and which remain equal.

## Exercises and explanations

1. A reconstruction looks four-lobed, but fluid crosses inward through the cylinder surface. Is the visual similarity enough?

   Read the explanation

   No. A fixed impermeable cylinder requires zero normal surface velocity. Check polar components, radius normalization and the Cartesian transformation before comparing streamline appearance.

2. Double field strength and halve radius. What happens to maximum induced potential and velocity scale? Does this guarantee the model remains applicable?

   Read the explanation

   The potential depends on field times radius, so it stays fixed. Velocity depends on radius times field squared, so it doubles. At fixed Debye length, relative layer thickness also doubles. The potential condition is unchanged, but thin-layer separation becomes less accurate.

3. The calculated reference has no error bars. Does this mean the paper’s experimental uncertainty is zero?

   Read the explanation

   No. The file contains analytic-model evaluations and the selected figure is a theoretical flow picture. Neither estimates experimental random error. Numerical precision is checked through boundaries, conservation and finite-difference error; experimental uncertainty requires measured data.

[Previous: What can one impedance arc tell us?](https://sandboxphysics.com/stories/electrochemistry/impedance-identifiability/) [Next: When a depleted layer starts to move](https://sandboxphysics.com/stories/electrochemistry/electroconvection-frontier/)

## Sources and calculation scope

1. [Squires & Bazant, Induced-charge electro-osmosis, arXiv:physics/0304090v2 (9 February 2004)](https://arxiv.org/abs/physics/0304090v2)

   Frozen preprint of Journal of Fluid Mechanics 509, 217–252: section 3.1, PDF equations (3.4)–(3.9), Figure 3(a) and Table 1. The limited reproduction calculates the steady field, charge, slip and outer Stokes flow for a fixed neutral cylinder. The HTML conversion renumbers equations consecutively.

2. [Bazant & Squires, Induced-charge Electrokinetic Phenomena: Theory and Microfluidic Applications, arXiv:physics/0306100v2 (2004)](https://arxiv.org/abs/physics/0306100v2)

   Physical Review Letters 92, 066101, especially equation (1) and Figure 1(c). This short paper supplies the physical picture; it is distinct from the longer JFM paper. Its original figures are linked, not redistributed.

3. [Sandbox Physics analytic ICEO reference, version 1 (original calculated values)](https://sandboxphysics.com/data/frontier/fr17/methods-iceo-reference.json)

   Original computed surface and velocity samples from the stated cylinder solution, with units and parameters. These are theoretical values, not experimental measurements or digitized points from a published figure.