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What is gradient operator tangent to the surface of bubble, when a gas bubble moving in fluid?. How can be solved if we using finite element method.?

Hi all. I am solving Naveir stokes equations using FEM. After imposing the Neumann boundary condition, I have surface integral in my equation. Can please someone help me how to solve it. I know to solve such as line integral for the gas bubble, but there is test function which makes me confuse. Please, how to solve it?. The attached file will help you. Thanks.

where the hand written is the detail of gradient operator.

gradient operator tangent to surface=grad(v)-n(n.grad(v)) where v is the test function and n is unit normal vector to the surface of bubble. We need to solve it.

  • $\begingroup$ Hmm .. This sounds more like a research question, probably involving several years of hard work (by yourself I suppose). $\endgroup$ – Han de Bruijn Jul 3 '18 at 15:45

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