If both liquid and air/gas are modeled, then the specific difficulty is modeling the interface between the two fluids/phases. If not treated correctly it may lead to instabilities and/or excessive dispersion between the phases (mixing). Some surface capturing algorithms are available to counter this, making the phase boundaries sharp.

Dear Saqib Zia.

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There are several types of problems depending on the type of disturbance that has the flow (roll waves, aerated flow, translatory waves of , shock waves and other).

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Generally in the case of flow of rivers and canals, (one-dimensional case) depends on the type of wave that is present in the flow, in the case for shallow water equations (Saint Venand equations ), no big problems as the modeling, the only problem is the boundary conditions.

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If there are disturbances described in the first paragraph models begin to complicate.

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The same as in the unidimensional case scenario is typically have a two dimensional configuration occurs.

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If you have to resort to three-dimensional simulations (hydraulic jump and other singularity), the answer is that covered by Halldór Pálsson wrote.

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It would be better to specify the physical situation to become more punctual response.

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In some of the free surface flow problems, e.g. Marangoni effect, there is a coupling between the flow in bulk phases, and effects at interfaces, like surface tension gradients introduced by surfactants, or temperature. A prominent example are the tears of wine, see http://en.wikipedia.org/wiki/Tears_of_wine

But there are also advantages. Some of the free surface flow problems can be used with a simplified set of equations reducing the dimensionality of the PDEs by 1. If the aspect ratio is very large, e.g. for a very thin film, then the so-called lubrication approximation can be used, see http://en.wikipedia.org/wiki/Lubrication_theory

The boundaries of the domain of Free surface flows are not known a priori, so modelling the evolution of the boundary is a severe challenge. Usually, the equations of fluid dynamics used in these flows are only applicable in the regions where there is fluid, so in the regions where there is no fluid, you have to worry how to handle such areas. This a kind of wetting and drying problem. So, correctly computing interfaces, propagating shocks and other discontinuities at the correct speeds, and satisfying other physical conditions such as steady state conditions are some eal challenges in this area.