---
title: "How surface ions drag the surrounding liquid"
description: "Derive electro-osmotic slip from viscous force balance, then use one cylinder potential to calculate induced charge and quadrupolar flow."
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language: "en"
page_type: "story"
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last_modified: "2026-10-28"
---

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

L3 · Chapter 5

# How surface ions drag the surrounding liquid

Sandbox Physics · 2026-10-28

An electric field pulls excess ions near a surface, and those ions drag liquid with them. A very thin charge layer can move liquid far beyond it. If the field also induces the charge, reversing the field reverses both factors, leaving a flow proportional to field strength squared.

## Before you begin

Use chapter three’s thin double layer. Viscosity resists relative sliding of neighbouring liquid layers. At low Reynolds number, inertia is weak and instantaneous force balance determines velocity. The cylinder uses plane polar coordinates.

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

- Derive the direction of electro-osmotic slip from electrical and viscous forces.
- Calculate induced charge and tangential field from a single cylinder potential.
- Explain the quadrupolar flow and field-squared scaling around a fixed neutral cylinder.

## Begin with a stationary flat wall

Take wall-normal coordinate $y$ into the liquid, and tangential coordinate $s$ along the positive tangential field $E_s$. The wall is stationary with no slip. Let $\psi(y)$ be potential relative to the outer liquid. Its surface value $\zeta$ is the zeta potential, measured in volts.

Electrical force per volume is $\rho_eE_s$. With negligible inertia and no imposed pressure gradient, this balances the gradient of viscous stress. Replace charge density by the second derivative of potential through Poisson’s equation. Treat the tangential field as constant across the thin layer and integrate twice.

$$
\eta\partial_y^2u_s+\rho_eE_s=0,\quad\rho_e=-\varepsilon\partial_y^2\psi,\quad u_s(\infty)=-\frac{\varepsilon\zeta}{\eta}E_s
$$

Use zero wall velocity and vanishing shear outside the layer. Viscosity $\eta$ is in $\mathrm{Pa\,s}$. The outer effective slip is the Helmholtz–Smoluchowski relation.

Reading: MIT 10.626 · Squires & Bazant

## Check direction and magnitude

A negative zeta potential has excess cations beside the surface. A rightward field pulls them and the liquid rightward. With permittivity $6.95\times10^{-10}\;\mathrm{F\,m^{-1}}$, viscosity $10^{-3}\;\mathrm{Pa\,s}$, $\zeta=-10.0\;\mathrm{mV}$ and $E_s=10^3\;\mathrm{V\,m^{-1}}$, the outer speed is $6.95\;\mathrm{\mu m\,s^{-1}}$ to the right.

This velocity is defined outside the charge layer. The solid surface still obeys no slip. Effective slip compresses the action of an unresolved thin layer into a boundary condition. It requires separation of layer and outer length scales, with corrections such as surface conduction small.

## Let a cylinder induce its own charge layer

Place a fixed, initially neutral metal cylinder of radius $a$ in an infinite planar electrolyte. Set metal potential to zero and distant field strength to $E_0$. Polar angle $\theta$ increases anticlockwise from the applied-field direction; $r$ measures distance from the axis. No Faradaic reaction occurs. After charging, normal outer current vanishes at the cylinder.

The neutral outer solution obeys Laplace’s equation. A uniform distant field, cylinder symmetry and zero surface normal field determine its dipole correction. Subtract outer surface potential from metal potential to obtain zeta potential. Differentiate that same potential around the circumference to get tangential field.

$$
\Phi=-E_0\left(r+\frac{a^2}{r}\right)\cos\theta,\qquad\zeta=2E_0a\cos\theta
$$

For a weak-potential thin layer, metal charge is $q_m=\varepsilon\zeta/\lambda_D$ and diffuse charge is $q_d=-q_m$. They have opposite signs.

$$
E_\theta(a)=-2E_0\sin\theta,\qquad u_\theta(a)=2U_0\sin2\theta,\quad U_0=\frac{\varepsilon aE_0^2}{\eta}
$$

Use zeta potential and tangential field from the same solution in the slip law. One power of field establishes charge and the other drives it.

Reading: Squires & Bazant · Bazant & Squires

## Follow liquid from poles to equator

Liquid approaches the cylinder near its poles, moves along the surface toward the two equators, then moves outward. The four quadrants form a quadrupolar flow. Reversing the applied field reverses both induced charge and tangential field, so their product drives the same flow.

The symmetric fixed neutral cylinder selects no net pumping direction. Electrophoresis would require a moving object and overall force and torque conditions. Pumping requires broken symmetry in boundaries, charge or shape. Here the question is electro-osmosis around a stationary surface.

$$
u_r=2U_0\frac{a(a^2-r^2)}{r^3}\cos2\theta,\qquad u_\theta=2U_0\frac{a^3}{r^3}\sin2\theta
$$

This analytic outer Stokes field has zero radial surface velocity, the required tangential slip and vanishing distant velocity. Together the two components satisfy incompressibility.

Reading: Squires & Bazant

## Calculate the default experimental scales

Take $a=10.0\;\mathrm{\mu m}$, $E_0=0.450\;\mathrm{V\,mm^{-1}}$ and $\lambda_D=0.100\;\mathrm{\mu m}$ with the water properties above. Maximum induced zeta potential is $9.00\;\mathrm{mV}$, or $0.350$ thermal voltages. The layer-to-radius ratio is $0.0100$.

The velocity scale is $U_0=1.41\;\mathrm{\mu m\,s^{-1}}$. At forty-five degrees on the surface, slip reaches $2.81\;\mathrm{\mu m\,s^{-1}}$. With $D=10^{-9}\;\mathrm{m^2\,s^{-1}}$, charging scale $\lambda_Da/D$ is about $1.00\;\mathrm{ms}$. The flow is the charged steady solution; tracer playback may be accelerated, so physical speed and animation time must be read separately.

Doubling field quadruples this ideal velocity and doubles the induced potential. The second change progressively challenges weak-potential screening. Check the charge model’s assumptions along with the field-squared law.

## Try the prediction

### [Reverse the charge and test the flow](https://sandboxphysics.com/labs/induced-charge-electroosmosis/)

1. Predict what a half-turn of the field will do to surface charge and flow.
2. Rotate the field and compare pole colours with tracer streamlines. Then change only field magnitude.
3. Record maximum slip and test field-squared scaling. Use induced potential to judge whether extrapolation remains appropriate.

## Exercises and explanations

1. Induced charge is largest at the cylinder poles. Why is surface slip zero there?

   Read the explanation

   Tangential field vanishes at the poles. Electro-osmotic slip needs that field to act on charge. Maximum slip lies near forty-five degrees, where both induced charge and tangential field are nonzero.

2. Double radius at fixed field. How do speed, maximum zeta potential and charging time change? Which assumption improves and which worsens?

   Read the explanation

   All three double. At fixed Debye length, the relative layer becomes thinner, improving thin-layer separation. Induced potential grows, making weak-potential screening less accurate. The useful speed increase ends as induced potential leaves the weak-field limit.

[Previous: How ion depletion creates a transport limit](https://sandboxphysics.com/stories/electrochemistry/concentration-polarization/) [Next: When ions exchange electrons at an electrode](https://sandboxphysics.com/stories/electrochemistry/butler-volmer/)

## Sources and calculation scope

1. [MIT 10.626, Electrokinetics, 2011 Lecture 29](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/resources/mit10_626s14_s11lec29/)

   Thin-double-layer electro-osmosis and viscous force balance. The chapter derives slip with the wall normal, zeta-potential sign and tangential field defined before use.

2. [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.

3. [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.