3.1 Derivation from momentum balance

Consider a spherical bubble of radius $R(t)$ in a Newtonian liquid that is otherwise at rest and unbounded. Take the liquid to be incompressible (the bubble dynamics of interest will be slow compared to the speed of sound in the liquid — we relax this assumption later for the late stages of collapse) and Newtonian with dynamic viscosity $\mu$ and density $\rho$. The bubble contents have a uniform pressure $p_B(t)$ — composed of any combination of permanent gas and vapour — which the next lesson develops in detail. The ambient pressure infinitely far from the bubble is a prescribed function $p_\infty(t)$.

The goal of this lesson is to derive the equation of motion for $R(t)$ from Newton’s second law applied to the surrounding liquid. The result is the Rayleigh–Plesset equation

$\rho \left[ R \ddot R + \frac{3}{2} \dot R^2 \right] \;=\; p_B(t) - p_\infty(t) - \frac{2 \sigma}{R} - \frac{4 \mu \dot R}{R}.$

We derive each term in turn.

Incompressibility forces a $1/r^2$ velocity field

By spherical symmetry the liquid velocity field is purely radial: $\mathbf u = u(r, t) \, \hat r$. Incompressibility ($\nabla \cdot \mathbf u = 0$) in spherical coordinates is

$\frac{1}{r^2} \frac{\partial (r^2 u)}{\partial r} = 0,$

which implies $r^2 u(r, t)$ is independent of $r$ — i.e., the velocity satisfies $u(r, t) = f(t) / r^2$ for some function $f(t)$ that depends only on time.

The kinematic boundary condition at the bubble wall says the wall and the adjacent liquid have the same radial velocity: $u(R, t) = \dot R$. This determines $f(t) = R^2 \dot R$, so

$\boxed{u(r, t) = \frac{R^2 \dot R}{r^2}.}$

This is a remarkable result: as long as the liquid is incompressible and the bubble is spherical, the entire flow in the surrounding liquid is fully determined by the bubble’s instantaneous radius and radial velocity. We need only solve for $R(t)$ and the whole velocity field follows.

Momentum balance from $R$ to infinity

The Euler equation for an inviscid incompressible flow (we re-add viscosity below) is

$\rho \frac{\partial u}{\partial t} + \rho u \frac{\partial u}{\partial r} = -\frac{\partial p}{\partial r}.$

Substituting the velocity field $u = R^2 \dot R / r^2$:

• $\partial u / \partial t = (2 R \dot R^2 + R^2 \ddot R) / r^2$.
• $\partial u / \partial r = -2 R^2 \dot R / r^3$.
• $u \partial u / \partial r = -2 R^4 \dot R^2 / r^5$.

So the momentum equation becomes

$\rho \frac{2 R \dot R^2 + R^2 \ddot R}{r^2} - 2 \rho \frac{R^4 \dot R^2}{r^5} = -\frac{\partial p}{\partial r}.$

Integrate from $r = R$ to $r = \infty$, using $p(\infty) = p_\infty$ and $p(R)$ as the liquid-side pressure just outside the bubble wall:

The boundary condition at the bubble wall

The wall pressure $p_\text{liq, at wall}$ is not equal to the bubble’s internal pressure $p_B$. Two effects intervene:

Surface tension

The bubble-liquid interface has an interfacial tension $\sigma$ (water-vapour: $\sigma = 72$ mN/m at 20 °C). For a spherical interface of radius $R$, the Young-Laplace condition says the pressure jump across the interface is

$p_B - p_\text{liq, at wall} = \frac{2 \sigma}{R}.$

The factor of two arises from the spherical geometry — both principal curvatures of the sphere contribute. This is the same surface-tension boundary condition that appeared in the nucleation analysis of Lesson 2.1.

Viscous stress

If the liquid has nonzero viscosity, the normal component of the viscous stress tensor at the wall is not zero — even in radial flow. Working out the viscous stress for the radial velocity field $u = R^2 \dot R / r^2$:

$\tau_{rr}\big|_{r=R} = 2 \mu \frac{\partial u}{\partial r}\bigg|_{r=R} = -2 \mu \cdot \frac{2 R^2 \dot R}{R^3} = -\frac{4 \mu \dot R}{R}.$

The negative sign reflects that an expanding bubble ($\dot R > 0$) feels a viscous pressure-like stress pulling inward on the wall — the liquid resists the expansion. The viscous correction to the wall pressure is

$p_\text{liq, at wall} - p_\text{liq, at wall, no-viscosity} = -\tau_{rr} = \frac{4 \mu \dot R}{R}.$

Equivalently, the liquid-side wall pressure with viscosity is higher than without by $4 \mu \dot R / R$.

Putting it together

Combining the two boundary corrections, the effective wall pressure on the right-hand side of the Rayleigh equation is

$p_\text{liq, at wall} = p_B - \frac{2 \sigma}{R} - \frac{4 \mu \dot R}{R}.$

Substituting back into the Rayleigh equation gives the Rayleigh–Plesset equation:

$\rho \left[R \ddot R + \frac{3}{2} \dot R^2 \right] = p_B(t) - p_\infty(t) - \frac{2 \sigma}{R} - \frac{4 \mu \dot R}{R}.$

Four terms on the right, each with a clear physical role:

1. $p_B(t)$ — the bubble’s internal pressure, which depends on its contents (next lesson).
2. $-p_\infty(t)$ — the ambient pressure, the drive.
3. $-2 \sigma / R$ — surface tension, which always resists expansion ($\sigma > 0$ for stable interfaces). At small $R$, this term dominates: a 1 μm bubble feels a Laplace pressure of $2 \sigma / R \approx 1.4$ atm just from surface tension.
4. $-4 \mu \dot R / R$ — viscous damping. The minus sign means the term resists motion: an expanding bubble ($\dot R > 0$) feels a retarding stress, a collapsing bubble ($\dot R < 0$) feels an opposing stress. For water at room temperature ($\mu = 10^{-3}$ Pa·s), viscous damping is negligible compared to surface tension and inertia for bubbles larger than a few μm — but it becomes important in glycerol, biological fluids, and oils.

What the equation is and is not

The Rayleigh–Plesset equation is exact under the four assumptions we made:

• The bubble is spherical (no shape modes).
• The surrounding liquid is incompressible (so the velocity field is $R^2 \dot R / r^2$ everywhere; no acoustic radiation).
• The liquid is Newtonian (linear stress-strain relation; no non-Newtonian effects).
• The liquid is unbounded (no nearby walls or other bubbles).

Each assumption fails in some physically interesting regime:

• Spherical symmetry fails near collapse, when even small perturbations are amplified by Rayleigh–Taylor-like instabilities of the contracting interface. Microjet formation near walls (Lesson 5.3) lives entirely in this regime.
• Incompressibility fails when $\dot R$ approaches the sound speed of the liquid — characteristic of the final stages of inertial collapse, where wall speeds can reach 1 km/s and sound-wave emission carries away substantial energy. The Keller–Miksis and Gilmore equations are compressibility-corrected versions of R–P used in this regime.
• Newtonian fails for blood plasma at small bubble sizes, for polymer melts, etc. Biomedical-ultrasound applications increasingly require non-Newtonian generalisations.
• The unbounded-liquid assumption is broken whenever a bubble sits near a wall (microjet physics), near another bubble (Bjerknes-force coupling), or in a finite vessel (acoustic resonance modes).

Despite these limitations the Rayleigh–Plesset equation captures the leading-order behaviour of single-bubble dynamics across a remarkable range of conditions, and it is the starting point of essentially every analysis in the book.

The next lesson works out what $p_B(t)$ — the internal pressure of the bubble — actually is.