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*In this survey article, we present results from the last several decades on several systems of PDE that model water waves.*

- Numerical Problems on Surface Tension
- Surface tension problems solved with the Particle Finite Element Method using large time-steps
- On the Existence of Solutions to the Muskat Problem with Surface Tension

## Numerical Problems on Surface Tension

Google Scholar. Antontsev, C. Meirmanov, Local existence of classical solutions to the well-posed Hele-Shaw problem,, Port. Differential Equations , 8 , Begehr and R. Cheng, D. Coutand and S.

Nigro , S. Idelsohn , E. In previous works [1,2], the authors have presented a highly efficient extension of the Particle Finite Element Method, called PFEM-2, to solve two-phase flows. Although that is the target range of problems to solve with PFEM-2, most of real problems that fall in these categories also includes other flow regimes in certain regions of the domain. Maybe the most common secondary regime is when the surface tension dominates, as an example when drops or bubbles are released from the main flow, and this feature must be taken into account in any complete numerical strategy. Attending to that, in this work the treatment of the surface tension to PFEM-2 is included.

## Surface tension problems solved with the Particle Finite Element Method using large time-steps

Example — A needle 5 cm long can just rest on the surface of the water without wetting. What is its weight? A light square wireframe each side of which is 10 cm long hangs vertically in the water with one side just touching the water surface. A rectangular film of liquid is formed in a frame of wire and a movable rod of length 4 cm. A thin and light ring of the material of radius 3 cm is rested flat on the liquid surface.

## On the Existence of Solutions to the Muskat Problem with Surface Tension

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Given: Length of needle = l = 5 cm = 5 × m, Surface tension = T = N/m. To Find: Weight of needle = F =? Solution: In case of a needle.

Surface Tension. A.H. Shapiro and A.A. Sonin Preamble: If we make a diagram of the leg, we realize that this is a highly idealized problem; the position of.