Buoyancy Explained: Complex Pressure and Line Integral

[skook]

I have worked through potential/streamline function theory and conformal mapping during a final-year pre-graduate course. But was wondering whether there was a simple explanation for buoyancy in the same way as by integration around a surface in a vector pressure field (say hydrostatic). If it is possible to do this using a line integral around a closed contour, then there must be a singularity somewhere inside the contour.I can't quite see how this works..... perhaps it's wrong. Anybody out there that can explain this in fairly simple terms?

cheers

 

[LightHero]

! Buoyancy is a force that acts on an object submerged in a fluid. It is the result of the pressure difference between the bottom and top surfaces of the object due to the weight of the fluid above it. This pressure difference is caused by gravity, which produces a net upward force on the object, called the buoyant force. The buoyant force can be calculated using Archimedes' Principle, which states that the buoyant force is equal to the weight of the displaced fluid. In other words, the buoyant force is equal to the weight of the fluid that is displaced by the object.The buoyant force can also be calculated using a line integral around a closed contour. The line integral is taken along the boundary of the submerged object, and the integral is evaluated using the pressure field of the surrounding fluid. The integral can be written as:\int_{C} \mathbf{F}.\mathrm{d}\mathbf{r} = \int_{C} (P_2 - P_1)\mathrm{d}Awhere C is the closed contour, F is the buoyant force, P_2 is the pressure at the top surface of the object, P_1 is the pressure at the bottom surface of the object, and dA is the area element.The line integral measures the net force acting on the submerged object, which is the buoyant force. The integral is evaluated by breaking the contour into small elements and summing up the pressure differences over each element. The result is the buoyant force acting on the object.