Rising Bubble in a Viscous Fluid

Buoyancy versus drag

A gas bubble released in a viscous liquid rises because it is less dense than the liquid around it - the same buoyancy that floats a boat - but it does not accelerate forever. As it speeds up, viscous drag from the surrounding fluid grows with velocity until it exactly balances the buoyant force, and the bubble settles into a constant terminal velocity. Everything interesting about how a bubble rises - whether it stays a smooth sphere, flattens into a cap, or wobbles chaotically - comes down to how a handful of dimensionless numbers weigh buoyancy, viscosity and surface tension against each other.

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Stokes drag and the small-bubble limit

For a small bubble moving slowly enough that inertia is negligible compared with viscosity (low Reynolds number), the drag force follows Stokes law: F_drag = 6*pi*mu*r*v, where mu is the liquid's dynamic viscosity and r is the bubble radius. Setting Stokes drag equal to the net buoyant force (weight of displaced liquid minus the bubble's own negligible weight) gives the classic terminal velocity result:

F_buoyancy = (4/3)*pi*r^3 * delta_rho * g
F_drag     = 6*pi*mu*r*v_terminal            (Stokes drag, valid for Re < ~1)
v_terminal = (2/9) * delta_rho * g * r^2 / mu   // solving F_buoyancy = F_drag

This is the same formula, sign flipped, that describes a dense sphere sinking slowly through a viscous fluid, and it holds well for very small bubbles - a few tenths of a millimetre or less in water - where surface tension is strong enough to keep the bubble almost perfectly spherical throughout its rise, so the drag really is the simple Stokes drag on a rigid-looking sphere. Larger bubbles depart from this picture quickly, both because inertial effects that Stokes drag ignores become significant and because the bubble's shape stops being spherical.

Two numbers that set the shape

The Eotvos number, Bo = delta_rho * g * d^2 / sigma, compares the buoyant force trying to deform the bubble against the surface tension sigma trying to hold it spherical. Small Bo (surface tension dominant) keeps the bubble round; large Bo (buoyancy dominant, typically bigger bubbles) lets it flatten into an oblate ellipsoid or, at the extreme, a wobbling "spherical cap" shape with a flat or even concave base.

The Morton number, Mo = g*mu^4*delta_rho / (rho^2*sigma^3), is unusual among dimensionless groups because it depends only on the fluid properties themselves, not on the bubble's size or velocity at all - it is a fixed number for a given liquid-gas pair (water-air, glycerin-air, and so on) that measures how viscous, dense fluids are relative to their surface tension. Together, Eotvos and Morton numbers (or equivalently a Reynolds-Eotvos-Morton triple) fix a bubble's shape and rise regime almost completely, and researchers plot them on a standard reference chart - the Grace diagram - that partitions this parameter space into spherical, ellipsoidal, wobbling, and spherical-cap regions, letting an engineer read off the expected bubble shape and terminal velocity from fluid properties and bubble size alone, without running a simulation.

Path instability: why bubbles wobble and zigzag

Beyond a critical bubble size (and correspondingly Reynolds number, typically several hundred), a rising bubble's path stops being a straight vertical line. Small asymmetric perturbations in the wake behind the bubble - vortex shedding, much like the vortex street behind a cylinder in cross-flow - couple back to the bubble's motion and shape, producing a rectilinear-to-zigzag or rectilinear-to-spiral transition. This path instability is a genuinely three-dimensional, unsteady phenomenon: the bubble's wake sheds vorticity asymmetrically, pushing the bubble sideways, which changes the wake, which pushes it again, in a self-sustaining oscillation that also correlates with shape oscillations (wobbling) for bubbles that are not perfectly rigid.

Why bubble rise matters beyond the lab

Bubble dynamics governs gas-liquid mass transfer in bubble-column chemical reactors, aeration in wastewater treatment, boiling heat transfer in power plants, CO2 outgassing from carbonated drinks, and the degassing of magma in volcanic conduits, where large, rapidly rising gas bubbles are implicated in explosive eruption dynamics. In nearly all of these, the practical quantity engineers care about - the interfacial area available for mass or heat transfer, or the residence time a bubble spends in a reactor - depends directly on the terminal velocity and shape regime the Eotvos-Morton-Reynolds framework predicts.

Frequently asked questions

Why doesn't Stokes drag work for large bubbles?

Stokes drag assumes viscous forces dominate over inertial ones (low Reynolds number) and that the bubble stays essentially spherical. Larger bubbles rise fast enough that fluid inertia in the wake becomes significant, and buoyancy becomes strong enough relative to surface tension (high Eotvos number) to flatten the bubble into an ellipsoid or cap shape, both of which invalidate the assumptions Stokes drag relies on.

What is the practical difference between the Eotvos and Morton numbers?

The Eotvos number compares buoyancy to surface tension for a specific bubble of a given size, so it changes as the bubble size changes. The Morton number depends only on the fixed properties of the liquid and gas (viscosity, density difference, surface tension) and is the same number for every bubble size in that fluid pair - it characterises the fluid, not the bubble, and is what makes the Grace diagram a two-parameter chart rather than needing a separate curve for every fluid.

Why do rising bubbles zigzag instead of going straight up?

Above a critical size, the wake behind a rising bubble sheds vorticity asymmetrically rather than staying perfectly symmetric, similar to vortex shedding behind a cylinder. That asymmetric wake pushes the bubble sideways, which alters the wake further, producing a self-sustaining zigzag or spiral path instability rather than a straight vertical rise.

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Rising Bubble in a Viscous Fluid: Shape, Speed and Path Instability