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Query: "keywords" (boundary value problems) .

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1.
Formulation of the method of fundamental solutions for two-phase Stokes flow
Zlatko Rek, Božidar Šarler, 2024, original scientific article

Abstract: The method of fundamental solutions with a subdomain technique is used for the solution of the free boundary problem associated with a two-phase Stokes flow in a 2D geometry. The solution procedure is based on the collocation of the boundary conditions with the Stokeslets. It is formulated for the flow of unmixing fluids in contact, where the velocity, pressure field, and position of the free boundary between the fluids must be determined. The standard formulation of the method of fundamental solutions is, for the first time, upgraded for the case with mixed velocity and pressure boundary conditions and verified on a T-splitter single-phase flow with unsymmetric pressure boundary conditions. The standard control volume method is used for the reference solution. The accurate evaluation of the velocity derivatives, which are required to calculate the balance of forces at the free boundary between the fluids, is achieved in a closed form in contrast to previous numerical attempts. An algorithm for iteratively calculating the position of the free boundary that involves displacement, smoothing and repositioning of the nodes is elaborated. The procedure is verified for a concurrent flow of two fluids in a channel. The velocity and velocity derivatives show fast convergence to the analytical solution. The developed boundary meshless method is easy to code, accurate and computationally efficient since only collocation at the fixed and free boundaries is needed.
Keywords: Stokes equations, two-phase flow, free boundary problems, method of fundamental solutions, subdomain technique
Published in DiRROS: 28.02.2024; Views: 116; Downloads: 64
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2.
Mixed Riemann-Hilbert boundary value problem with simply connected fibers
Miran Černe, 2024, original scientific article

Abstract: We study the existence of solutions of mixed Riemann-Hilbert or Cherepanov boundary value problem with simply connected fibers on the unit disk $\Delta$. Let ▫$L$▫ be a closed arc on $\partial\Delta$ with the end points $\omega_{-1}, \omega_1$ and let $a$ be a smooth function on $L$ with no zeros. Let $\{\gamma_{\xi}\}_{\xi\in\partial\Delta\setminus\mathring{L}}$ be a smooth family of smooth Jordan curves in $\mathbb C$ which all contain point $0$ in their interiors and such that $\gamma_{\omega_{-1}}$, $\gamma_{\omega_{1}}$ are strongly starshaped with respect to $0$. Then under condition that for each $w\in \gamma_{\omega_{\pm 1}}$ the angle between $w$ and the normal to $\gamma_{\omega_{\pm 1}}$ at $w$ is less than $\frac{\pi}{10}$, there exists a Hölder continuous function $f$ on $\overline{\Delta}$, holomorphic on $\Delta$, such that ${\rm Re}(\overline{a(\xi)} f(\xi)) = 0$ on $L$ and $f(\xi)\in\gamma_{\xi}$ on $\partial\Delta\setminus\mathring{L}$.
Keywords: boundary value problems, mixed Riemann-Hilbert problem, Cherepanov problem
Published in DiRROS: 15.02.2024; Views: 126; Downloads: 62
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