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The stream function-vorticity formulation is a useful alternative to solving the Navier-Stokes equations in two dimensional domain. One of its major difficulties is the lack of boundary conditions for vorticity, and several schemes are proposed to satisfy such a condition. In this paper we propose a finite element scheme for solving the couple problem of the stream functionvorticity formulation using linear triangle elements. A possible application of the method is also demonstrated using the calculated velocity field from the formulation in the heat transport equation to study the temperature distribution in a incompressible single-phase fluid medium. The results obtained were satisfactory when using low to moderate Reynolds number.

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The breakup of liquid threads into smaller droplets is a fundamental problem in fluid dynamics. In this study, we estimate the characteristic wavelength of the breakup process by means of many-body dissipative particle dynamics. This wavelength shows a power-law dependence on the Ohnesorge number in line with results from stability analysis. We also discover that the number of satellite droplets exhibits a power-law decay with exponent 0.72 ± 0.04 in the product of the Ohnesorge and thermal capillary numbers, while the overall size of main droplets is larger than that based on the characteristic wavelength thanks to the asynchronous breakup of the thread. Finally, we show that the formation of satellite droplets is the result of the advection of pinching points toward the main droplets in a remaining thinning neck, when the velocity gradient of the fluid exhibits two symmetric maxima.