I am launching a high-altitude balloon as a part of a physics project I am working on. I know that the amount of helium I need corresponds to about 40 newtons of lift for the launch, which is all I need to know for practical purposes. But if I wanted to account for the ever-changing buoyant force on a weather balloon, I would need a calculus-based model, either an integral or some differential equation.
From the derivation on Wikipedia, I have:
$$ F_b = \oint \sigma dA $$
where $\sigma$ is the Cauchy stress tensor and $dA$ is the surface area in contact with the fluid, or of the latex balloon, in my case. I am an undergrad student and have virtually no knowledge of tensor calculus.
Are there any other calculus-based models for buoyancy? If not, is numeric integration an appropriate strategy for this problem?
