fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...

, the Hadamard–Rybczynski equation gives the terminal velocity of slowly moving spherical

bubble Bubble, Bubbles or The Bubble may refer to: Common uses * Bubble (physics), a globule of one substance in another, usually gas in a liquid ** Soap bubble * Economic bubble, a situation where asset prices are much higher than underlying fundame ...

through an ambient

fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...

. It is named after Jacques Hadamard and

Witold Rybczynski Witold Rybczynski (born 1 March 1943) is a Canadian American architect, professor and writer. He is currently the Martin and Margy Meyerson Professor Emeritus of Urbanism at the University of Pennsylvania. Early life Rybczynski was born in Ed ...

: :

W_\mathrm = \frac \frac 
 \frac

where *

R

is the radius of the bubble. *

g

the gravitational acceleration. *

\rho_\mathrm

the density of the bubble. *

\rho_0

the density of the ambient fluid. *

\mu_\mathrm

the viscosity of the bubble. *

\mu_0

the viscosity of the ambient fluid. *

W_\mathrm

the resultant velocity of the bubble. The Hadamard–Rybczynski equation can be derived from the Navier–Stokes equations by considering only the

buoyancy force Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the ...

and

drag force In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...

acting on the moving bubble. The surface tension force and inertia force of the bubble are neglected.

References

Further reading