•A linearized energy diminishing ALE-FEM for two-phase Navier-Stokes flow is proposed.•The energy diminishing property is proved theoretically and illustrated numerically.•Benchmark experiments of 2D ...and 3D rising bubbles in fluids are presented.
A linearized fully discrete arbitrary Lagrangian–Eulerian finite element method is proposed for solving the two-phase Navier–Stokes flow system and to preserve the energy-diminishing structure of the system at the discrete level, by taking account of the kinetic, potential and surface energy. Two benchmark problems of rising bubbles in fluids in both two and three dimensions are presented to illustrate the convergence and performance of the proposed method.
We consider the spatial–temporal behavior of the Navier–Stokes flow past a rigid body in R3$\mathbb {R}^3$. This paper develops analysis in Lebesgue spaces with anisotropic weights ...(1+|x|)α(1+|x|−x1)β${(1+|x|)}^\alpha {(1+|x|-x_1)}^\beta$, which naturally arise in the asymptotic structure of fluid when the translational velocity of the body is parallel to the x1‐direction. We derive anisotropically weighted Lq$L^q$‐Lr$L^r$ estimates for the Oseen semigroup in exterior domains. As applications of those estimates, we study the stability/attainability of the Navier–Stokes flow in anisotropically weighted Lq$L^q$ spaces to get the spatial–temporal behavior of nonstationary solutions.
Direct numerical simulation (DNS) of fluid dynamics in digital images of porous materials is challenging due to the cut-off length issue where interstitial voids below the resolution of the imaging ...instrument cannot be resolved. Such subresolution microporosity can be critical for flow and transport because they could provide important flow pathways. A micro-continuum framework can be used to address this problem, which applies to the entire domain a single momentum equation, i.e., Darcy-Brinkman-Stokes (DBS) equation, that recovers Stokes equation in the resolved void space (i.e., macropores) and Darcy equation in the microporous regions. However, the DBS-based micro-continuum framework is computationally demanding. Here, we develop an efficient multiscale method for the compressible Darcy-Stokes flow arising from the micro-continuum approach. The method decomposes the domain into subdomains that either belong to the macropores or the microporous regions, on which Stokes or Darcy problems are solved locally, only once, to build basis functions. The nonlinearity from compressible flow is accounted for in a local correction problem on each subdomain. A global interface problem is solved to couple the local bases and correction functions to obtain an approximate global multiscale solution, which is in excellent agreement with the reference single-scale solution. The multiscale solution can be improved through an iterative strategy that guarantees convergence to the single-scale solution. The method is computationally efficient and well-suited for parallelization to simulate fluid dynamics in large high-resolution digital images of porous materials.
•Develop a multiscale method for compressible Darcy-Stokes flow at pore scale.•Multiscale solutions are in excellent agreement with direct numerical simulation (DNS).•Multiscale method is much more computational efficient than DNS.
Inexpensive numerical methods are key to enabling simulations of systems of a large number of particles of different shapes in Stokes flow and several approximate methods have been introduced for ...this purpose. We study the accuracy of the multiblob method for solving the Stokes mobility problem in free space, where the 3D geometry of a particle surface is discretised with spherical blobs and the pair-wise interaction between blobs is described by the RPY-tensor. The paper aims to investigate and improve on the magnitude of the error in the solution velocities of the Stokes mobility problem using a combination of two different techniques: an optimally chosen grid of blobs and a pair-correction inspired by Stokesian dynamics. Different optimisation strategies to determine a grid with a given number of blobs are presented with the aim of matching the hydrodynamic response of a single accurately described ideal particle, alone in the fluid. It is essential to obtain small errors in this self-interaction, as they determine the basic error level in a system of well-separated particles. With an optimised grid, reasonable accuracy can be obtained even with coarse blob-resolutions of the particle surfaces. The error in the self-interaction is however sensitive to the exact choice of grid parameters and simply hand-picking a suitable geometry of blobs can lead to errors several orders of magnitude larger in size. The pair-correction is local and cheap to apply, and reduces the error for moderately separated particles and particles in close proximity. Two different types of geometries are considered: spheres and axisymmetric rods with smooth caps. The error in solutions to mobility problems is quantified for particles of varying inter-particle distances for systems containing a few particles, comparing to an accurate solution based on a second kind BIE-formulation where the quadrature error is controlled by employing quadrature by expansion (QBX).
•Rigid rods and spheres are studied in Stokes flow in 3D free space.•The accuracy of the multiblob method is improved at no extra cost.•An optimal grid of blobs matches the hydrodynamic interaction of a model particle.•A self-interaction error dominant in the far-field is reduced with the optimal grid.•Pair-corrections of Stokesian dynamics type reduce errors in near-field.
In this work we propose and analyze an HDG method for the Stokes equation whose domain is discretized by two independent polygonal with different meshsizes. This causes a non-conformity at the ...intersection of the subdomains or leaves a gap (unmeshed region) between them. In order to appropriately couple these two different discretizations, we propose suitable transmission conditions to preserve the high order convergence of the scheme. Furthermore, stability estimates are established in order to show the well-posedness of the method and the error estimates. In particular, for smooth enough solutions, the L2 norm of the errors associated to the approximations of the velocity gradient, the velocity and the pressure are of order order hk+1, where h is the meshsize and k is the polynomial degree of the local approximation spaces. Moreover, the method presents superconvergence of the velocity trace and the divergence-free postprocessed velocity. Finally, we show numerical experiments that validate our theory and the capacities of the method.
In this article, we give a comprehensive characterization of $L^1$-summability for the Navier-Stokes flows in the half space, which is a long-standing problem. The main difficulties are that ...$L^q-L^r$ estimates for the Stokes flow don't work in this end-point case: $q=r=1$; the projection operator $P: L^1\longrightarrow L^1_\sigma$ is not bounded any more; useful information on the pressure function is missing, which arises in the net force exerted by the fluid on the noncompact boundary. In order to achieve our aims, by making full use of the special structure of the half space, we decompose the pressure function into two parts. Then the knotty problem of handling the pressure term can be transformed into establishing a crucial and new weighted $L^1$-estimate, which plays a fundamental role. In addition, we overcome the unboundedness of the projection $P$ by solving an elliptic problem with homogeneous Neumann boundary condition.
•Hydrodynamic cavitation of nematic liquid crystal 5CB in the Stokes flow regime.•The critical Reynolds number of cavitation inception from 2695 to 6596.•Using the Strouhal number to describe the ...unsteady oscillating cavitation flow.•Dynamics characteristics of cavitation are analyzed.
The impact of hydrodynamic cavitation is one of the key factors in the fabrication and application of metal materials. The hydrodynamic cavitation in anisotropic fluids can be much different from common fluids due to the unique fluid dynamics characteristics related to the director field. In this paper, the hydrodynamic cavitation of nematic liquid crystal 5CB flowing around a cylindrical pillar in a microchannel in the Stokes flow regime is studied at various Reynolds numbers and blockage ratios by experiments. Once the Reynolds number rises over a threshold value, the hydrodynamic cavitation of nematic liquid crystal is generated in the stokes flow in the microchannel, which is the unique phenomenon for anisotropic fluids. The critical Reynolds number of cavitation inception decreases with the increase of the blockage ratio. The Strouhal number is induced to describe this unsteady oscillating cavitation flow, while the Strouhal number increases and decreases with the increase of the blockage ratio and Reynolds number, respectively. Dynamics characteristics of the cavitation including cavity volume, oscillation frequency, and pressure difference are analyzed at different Reynolds numbers and blockage ratios by image binarization treatment and Fast Fourier Transformation analysis. The results of this study are useful for the application of microfluidic chips based on nematic liquid crystals and protection of devices in the environment of similar anisotropic fluids.
Display omitted
Summary
Flow calculations in an unbounded domain have limitations and challenges due to its infiniteness. A common approach is to impose a far‐field asymptotic condition to determine a unique flow. ...The leading behavior of the flow is identified at the far field, and then an unknown coefficient is assumed for the second behavior. This allows us to propose an efficient numerical method to solve two‐dimensional steady Stokes and potential flows in a truncated domain along with the coefficients. The second term provides crucial hydrodynamic information for the flow and is referred to as the informative boundary condition. The truncation creates artificial boundaries requiring boundary conditions for the approximate solution. The axial Green function method (AGM), combined with a specific one‐dimensional Green function over a semi‐infinite axis‐parallel line extended to infinity, allows us to implement the informative boundary condition in the truncated domain. AGMs, designed for complicated domains, are now applied to infinite domain cases because AGMs' versatility enables implementing the informative boundary condition by changing only the axial Green function. This approach's efficiency, accuracy, and consistency are investigated through several appealing Stokes flow problems including potential ones in infinite domains.
PDF
