Efficient second-order accurate scheme for fluid–surfactant systems on curved surfaces with unconditional energy stability

Accurately simulating the interplay between fluids and surfactants is a challenge, especially when ensuring both mass conservation and guaranteed energy stability. This study proposes a highly accurate numerical scheme for the water–oil–surfactant system coupled with the Navier–Stokes equation. We use the second-order accurate discrete operators on triangular grids representing these surfaces. We use the Crank–Nicolson method to achieve the second-order temporal accuracy. To handle high-order nonlinearities, we employ the consistency-enhanced Scalar Auxiliary Variable method. Additionally, the projection method tackles the Navier–Stokes equations. These techniques guarantee the unconditional stability. Consequently, larger time steps are permissible. Our method achieves the high accuracy with lower computational cost. This efficiency stems from solely utilizing surface information, eliminating the need for calculations across the entire 3D space. The proposed scheme's accuracy, stability, and robustness can be confirmed through various numerical experiments.