Temporal discretization of viscous stress terms of incompressible Navier–Stokes equations with surface tension effect

In this study, four temporal discretization methods for viscous stress terms were examined through the finite element method for incompressible Navier–Stokes equations with surface tension effect. The temporal stability and numerical accuracy of the four methods were evaluated in static and rising bubble benchmark problems with severe temporal restrictions caused by viscous stress terms. First, conservative and non-conservative discretization methods were compared in terms of stability and accuracy. The stability and accuracy of the numerical solutions were further investigated through three temporal discretization methods (i.e., fully implicit, semi-implicit, and fully explicit) for viscous stress terms. Nonconservative discretization may yield an incorrect solution although it provides a compact element matrix that reduces CPU time for matrix–vector multiplication. Among the three temporal discretization methods for the conservative treatment of viscous stress terms, the fully implicit method is recommended for cases with strong viscous stress. This method requires an element matrix that is larger than those of the other methods. However, the time-step limit caused by the explicit or semiimplicit treatment of viscous stress terms can be avoided.