Friction of Bubbles
Date: May 2, 2011
What does the drag on an air bubble rising through a liquid mostly comprises of, skin friction or pressure drag? Why?
There are two commonly used models for drag force and they depend on
the conditions. If you have high speed flow where viscous forces
are not dominating, the drag can usually be expressed as a function
of air density, cross-sectional area and velocity squared. But in
situations where you have slow speed through a viscous medium such
that viscous forces are dominating, the drag there is usually
expressed via Stokes drag. Sir George Gabriel Stokes found that in
such situations the viscous shear over a sphere's surface (aka "skin
friction") was the dominant retarding force and this force was
linearly proportional to viscosity, velocity and the radius of the sphere.
John C Strong
When bubbles are very small (10 micrometers?), viscous (linear-law) drag dominates.
I think that's what you meant by skin friction. (Sometimes people use it a different way.)
When they are fairly large (1 cm), square-law mass-dynamic drag dominates,
as you called it: pressure drag.
When bubbles are larger still, they distort and become non-spherical.
It is still pressure-drag, but the frontal area for a given volume is larger,
and the non-spherical shape also has a higher coefficient of drag.
So, a plot of experimental data may not fit very clearly to any one power law.
I think you should look at Reynold's Number in Wikipedia or elsewhere.
The formula combines size, speed, fluid viscosity, and fluid density
into one number ("Re"), which in turn determines which type of drag dominates.
Try to make a table of Reynolds number vs. the size of possible bubbles.
I think Re < 1 means that viscous drag will dominate.
Vortices and turbulence appear somewhere around R3=~100-300.
So for Re > 1000, pressure drag will clearly dominate.
in between 1 and 100 will probably be a gradual transition from viscous to mass-dynamic.
If I wanted to fit measured data to a "theoretical" curve, I would use equations like:
F_drag = (F_visc) + (F_md)
F_visc = Cd1 * frontal area * liquid_viscosity * speed (always spherical )
F_md = Cd2(speed, size) * frontal_area(size) * liquid_density * speed^2
The small-bubble, low-speed, low-Re part of the curve can fit well,
because viscous drag is very simple and the bubble is always a sphere.
The exact Cd1 for a sphere is available.
Unfortunately Cd2, the shape-coefficient of drag for pressure-drag,
is a somewhat arbitrary function of speed. You might have to get a couple of likely values,
draw a reasonable curve between them, and implement it as a look-up table.
Your main controlling bubble-size variable would have to be volume
(or diameter of sphere with equivalent volume).
function of size:
Cd2(sphere) = ~ 0.2-0.3 (bubble diameter~2mm) ,
but a 3cm bubble will flatten out so much it's Cd will be almost 1.0,
in addition to the actual frontal area being 2-4 times larger than you'd expect from a sphere.
function of speed:
Cd(Re~500) is one thing,
then Cd(Re~2000) may take a step change higher or lower,
as flow detachment changes its position between the equator and 45 degrees to the rear of the sphere.
Usually those changes will be less than a factor of 2.
I like to call pressure-drag "mass-dynamic" drag because the density of the liquid matters,
much more so than the viscosity.
I interpret it as: the moving object imparts its speed to some percentage of the liquid volume it passes through,
then that imparted kinetic energy is stored in vortices and slowly lost to viscosity.
So naturally the kinetic energy lost is proportional to the liquid's density and the square of the speed.
Notice at which size the bubble starts wobbling as it ascends.
That should be the onset of vortex-shedding at Re=~200.
Pressure drag should dominate for that size or larger.
At bubble diameter smaller than that one by roughly a factor of 10, viscous-drag should dominate.
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Update: June 2012