Buoyant Force Equations
Date: Summer 2013
I would like to know how to calculate the buoyant upward acceleration of a container of air in sea water. I found an equation on Wikipedia and one on a site called "Beyond Archimedes" but they do not say what the units are for the terms and some terms are ill-defined. I realize that the shape of the container has much to do with the drag and viscous forces. I am assuming a flexible container of 1 cubic meter of air at 10 m depth (2 atm ambient and air pressure) rising to the surface (but not breaking it). What would be the acceleration from bottom and what would be the terminal velocity and at what depth would it reach this if the container were assumed to be two opposed cones (i.e. 2 cones facing apart with a maximum diameter of 0.5m)? If you could also provide me an algebraic equation to get a close estimate I would appreciate it so I can solve with other variables.
My way of thinking about buoyancy is this “simple” principle, but do not read it too quickly, because there is a lot of information embedded in the statement. An object (without regard to shape, size, …) experiences an upward force against the force of GRAVITY that equals [the WEIGHT of the VOLUME] of the liquid displaced. Note that buoyant force depends on the “force of gravity”. This volume and weight depends on the shape, size, … etc. of the object, but also does not necessarily mean the Earth’s gravitational force. So, for example, if there is no gravitational force, there is no buoyancy. Similarly, if the fluid is on the Moon, the WEIGHT of the displaced VOLUME is much smaller, because it is the weight of the totally displaced fluid that counts. This criterion makes the principle clearer than trying to provide a numerical calculation. It also makes no difference whether or not the “object” is barely submerged or totally submerged. Keeping the qualitative principle in mind does not confuse the issue with a “bunch” of numbers.
It is impossible to give you an "algebraic equation" that will give you
the answer you want! Your question is vastly more complex than that,
and will likely require use of Computational Fluidflow Dynamics (CFD)
software (and a powerful computer) to get even close to an accurate
While it is easy to calculate the instantaneous acceleration the instant
that the container is released (when there is no fluid drag yet), as soon
as your container begins to move upwards, fluid drag comes into the
picture. Movement of an object through a fluid, and the resulting drag
is complex and highly nonlinear, and emphatically cannot be solved by
simple algebraic equations. Nor can any simple formula even
determine if the container will begin tumbling or oscillating when rising,
either of which would massively affect drag.
Hope this is not too discouraging!
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