I am graduate student of chemical engineering. I got this problem and solution in a paper but cannot find how they solved it. Consider I have a sphere that is equally divided into two different patch(P, S). The sphere can rotate and translate(up-down) at the air-water interface. The angle alpha defines the position of the sphere (alpha increase means the sphere goes towards the air). The angle beta defines the rotation of the sphere. So the surface area of any patch into any phase changes with the position and orientation(example: area of P inside air phase will change with alpha and beta angle). Now the calculation of the surface area of P part of the sphere in air is given by the following equation. I understand the limit for θ direction but did not find out how the limit comes for φ direction.
Below is visual represntation of the problem. Please keep in mind that the figure is only 2D presentation of a 3D sphere where the coordinate system refers as (r, θ, φ) .
Below is visual represntation of the problem. Please keep in mind that the figure is only 2D presentation of a 3D sphere where the coordinate system refers as (r, θ, φ) .
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