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The finite volume method (FVM) is widely adopted in many different applications because of its built‐in conservation properties, its ability to deal with arbitrary mesh and its computational efficiency. In this work, we consider the Rhie–Chow stabilized Box method (RCBM) for the approximation of the Stokes problem. The Box method (BM) is a piecewise linear Petrov–Galerkin formulation on the Voronoi dual mesh of a Delaunay triangulation, whereas the Rhie–Chow (RC) stabilization is a well known stabilization technique for FVM. The first part of the article provides a variational formulation of the RC stabilization and discusses the validity of crucial properties relevant for the well‐posedness and convergence of RCBM. Moreover, a numerical exploration of the convergence properties of the method on 2D and 3D test cases is presented. The last part of the article considers the theoretically justification of the well‐posedness of RCBM and the experimentally observed convergence rates. This latter justification hinges upon suitable assumptions, whose validity is numerically explored. [ABSTRACT FROM AUTHOR]

This paper deals with an initial-boundary value problem in two-dimensional smoothly bounded domains for the system n t + u · ∇ n = Δ n - ∇ · (n S (n) ∇ v) , x ∈ Ω , t > 0 , v t + u · ∇ v = Δ v - v + w , x ∈ Ω , t > 0 , w t + u · ∇ w = Δ w - w + n , x ∈ Ω , t > 0 , u t + κ (u · ∇) u + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ · u = 0 , x ∈ Ω , t > 0 , (∗) which describes the mutual interaction of chemotactically moving microorganisms and their surrounding incompressible fluid, where κ ∈ R , the gravitational potential ϕ ∈ W 2 , ∞ (Ω) , and S (n) satisfies | S (n) | ≤ C S (1 + n) - α for all n ≥ 0 , C S > 0 and α > - 1. Under the boundary conditions (∇ n - n S (n) ∇ v) · ν = ∂ ν v = ∂ ν w = 0 , u = 0 , x ∈ ∂ Ω , t > 0 , it is shown in this paper that suitable regularity assumptions on the initial data entail the following: (i) If α > - 1 and κ = 0 , then the simplified chemotaxis-Stokes system possesses a unique global classical solution which is bounded. (ii) If α ≥ 0 and κ ∈ R , then the full chemotaxis-Navier–Stokes system admits a unique global classical solution. [ABSTRACT FROM AUTHOR]

Summary: Six face‐centred finite volume formulations are derived and compared for the simulation of Stokes flows with spatially varying viscosity. The main difference between the methods derived is the mixed variable used in the mixed formulation and the use of a weak or strong form in each element using integration by parts. A brief discussion about the properties of the different methods is provided, including comments on the computational cost and the symmetry of the resulting global system of equations. Finally, numerical examples in two and three dimensions are used to compare the accuracy of all the formulations presented. The examples include a problem where the methods are employed to simulate a steep variation of the viscosity, showing the ability to perform these simulations without using a mesh conforming to a material interface. The performance of different element types and different choices of the stabilisation is also discussed. [ABSTRACT FROM AUTHOR]

A spectropolarimeter based on spatial heterodyne spectroscopy and the channeled polarimetric technique (SSHSCP) is introduced. The SSHSCP uses a polarization grating for channel modulation, meaning that the Stokes parameter is no longer dependent on the wavenumber; this suppresses channel aliasing considerably and improves the polarization parameter reconstruction accuracy. The SSHSCP’s optical path structure is based on the Mach–Zehnder interferometer, ensuring efficient energy use. The spectral detection and polarization detection principles of the SSHSCP are derived, and a specific design example is presented. Simulations indicate that the SSHSCP can restore the target’s spectral and complete Stokes parameters with high accuracy for a broadband source and narrowband lines. The reconstructed Stokes parameter errors are of the order of 10−3 or less. The SSHSCP acquires spectral and polarization parameters in a single step and has no moving parts in a compact structure.

A solenoidal basis is constructed to compute velocities using a certain finite element method for the Stokes problem. The method is conforming, with piecewise linear velocity and piecewise constant pressure on the Powell-Sabin split of a triangulation. Inhomogeneous Dirichlet conditions are supported by constructing an interpolating operator into the solenoidal velocity space. The solenoidal basis reduces the problem size and eliminates the pressure variable from the linear system for the velocity. A basis of the pressure space is also constructed that can be used to compute the pressure after the velocity, if it is desired to compute the pressure. All basis functions have local support and lead to sparse linear systems. The basis construction is confirmed through rigorous analysis. Velocity and pressure system matrices are both symmetric, positive definite, which can be exploited to solve their corresponding linear systems. Significant efficiency gains over the usual saddle-point formulation are demonstrated computationally. [ABSTRACT FROM AUTHOR]

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics { (n 1) t + u ⋅ ∇ n 1 = Δ n 1 − χ ∇ ⋅ (n 1 ∇ w) + n 1 (λ 1 − μ 1 n 1 + a n 2) , x ∈ Ω , t > 0 , (n 2) t + u ⋅ ∇ n 2 = Δ n 2 + ξ ∇ ⋅ (n 2 ∇ z) + n 2 (λ 2 − μ 2 n 2 − b n 1) , x ∈ Ω , t > 0 , w t + u ⋅ ∇ w = Δ w − w + n 2 , x ∈ Ω , t > 0 , u ⋅ ∇ z = Δ z − z + n 1 , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + (n 1 + n 2) ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 under no-flux boundary conditions for n 1 , n 2 , w and z in three-dimensional bounded domains and no-slip boundary conditions for u , this is ∂ n 1 ∂ ν = ∂ n 2 ∂ ν = ∂ w ∂ ν = ∂ z ∂ ν = 0 , u = 0 , x ∈ ∂ Ω , t > 0 , where χ > 0 , ξ > 0 , μ 1 ≥ 0 , μ 2 ≥ 0 , λ 1 ≥ 0 , λ 2 ≥ 0 , a ≥ 0 , b ≥ 0 and ϕ ∈ W 2 , ∞ (Ω). This system is a coupled system of the chemotaxis equations and viscous incompressible fluid equations. Under appropriate assumptions, this problem exhibits a global classical bounded solution. [ABSTRACT FROM AUTHOR]

Water has engaged in a series of activities during plant growth. Therefore, the moisture content is one of the most important indicators of plant growth. Water deficiency can lead to a decrease in the stomatal degree on the leaf surface, where the carbon dioxide can flow into the plant leaf. The resulting photosynthetic efficiency can greatly reduce the appearance, nutrition, and total biomass of forestry plants. However, it is still lacking in the rapid detection of moisture content in plants. In this study, a polarization imaging system was proposed using a split-focal plane polarization camera. Polarization images were then analyzed to establish the dependence between the linear polarization and moisture content, in order to reveal the interaction between polarized light and water in plants. The rapid and nondestructive detection was realized for the moisture content of wintergreen leaves. The specific research was as follows. 1) The holly leaves were first collected and dried in a gradient manner. The moisture content was obtained after weighing; 2) A sub-focal plane polarization system was developed to detect the polarization of the gradient-dried holly leaves; 3) A systematic investigation was carried out to clarify the influence of growth states on the moisture content of holly leaves under various months, different parts, as well as sunny or rainy weather conditions. 40 leaves were collected and then divided into 8 groups, with 5 leaves for the control in each group; 4) The holly leaves were detected for the polarization. Stokes parameters were obtained to convert into the linear polarization degrees with clear physical significance and imaging. It was found that the linear polarization degree increased gradually, as the moisture content decreased. There was a strong negative correlation between the leaf moisture content and the linear polarization degree. 5) Rapid detection of moisture content was achieved by the quantitative relationship between moisture content and linear polarization degree after fitting. The degree of linear polarization was only used to detect the moisture content of the target leaf. 6) Other holly leaves were collected and then substituted into the fitting model, in order to verify the prediction of the model. The results show that the moisture content of holly spear leaves was represented by the linear polarization degree in the three growth conditions (months, weather, and parts). Furthermore, the linear polarization degree increased gradually with the decrease in the gradient of moisture content. A prediction model was obtained using the large number of plant leaves. The correlation coefficient reached 0.85, and the MSE was only 0.45%. Verification through the leaf moisture detection system showed that the average relative error of the moisture detection system was 9.50%, and the RMSE was 4.53%. The true moisture content of the leaf was characterized by linear polarization degree. The findings can also provide optical applications in plant health detection. [ABSTRACT FROM AUTHOR]

The aim of this work is to propose a new nodal treatment of the pressure for tetrahedral or triangular meshes devoted to the simulation of incompressible and nearly incompressible material flows. The approach proposed has the interest of fulfilling numerically the LBB condition for P1‐type discretizations over a wide range of element sizes. Thus, the existence of an error estimate is ensured and there is no need for a stabilization technique. For more convenience, the new P1/RNP0 formulation is first detailed for the Stokes problem. It is based on a RNP0 (Random Nodal P0) approximation of the pressure with constant values on nodal subcells whose size is defined by means of a pseudo‐random number generator. For nearly incompressible material flows, the numerical approach is extended to problems involving von Mises elasto‐plasticity. Examples are presented to show the relevance of the new approach for Eulerian and Lagrangian formalisms. [ABSTRACT FROM AUTHOR]

We derive an ultraweak variational formulation of the quad-curl problem in two and three dimensions. We present a discontinuous Petrov–Galerkin (DPG) method for its approximation and prove its quasi-optimal convergence. We illustrate how this method can be applied to the Stokes problem in two dimensions, after an application of the curl operator to eliminate the pressure variable. In this way, DPG techniques known from Kirchhoff–Love plates can be used. We present an a priori error estimate that improves a previous approximation result for effective shear forces by using a less restrictive regularity assumption. Numerical experiments illustrate our findings. [ABSTRACT FROM AUTHOR]

The chemotaxis‐Stokes system nt+u·∇n=Δnϕ(c),ct+u·∇c=Δc−c+n,ut=Δu+∇P+n∇Φ,∇·u=0$$ \left\{\begin{array}{rcl}{n}_t+u\cdotp \nabla n& =& \Delta \left(n\phi (c)\right),\\ {}{c}_t+u\cdotp \nabla c& =& \Delta c-c+n,\\ {}{u}_t& =& \Delta u+\nabla P+n\nabla \Phi, \kern2em \nabla \cdotp u=0\end{array}\right. $$is considered along with homogeneous boundary conditions of no‐flux type for n$$ n $$ and c$$ c $$, and of Dirichlet type for u$$ u $$, in a smoothly bounded domain Ω⊂ℝ2$$ \Omega \subset {\mathrm{\mathbb{R}}}^2 $$. Under the assumption that Φ∈W2,∞(Ω)$$ \Phi \in {W}^{2,\infty}\left(\Omega \right) $$, that ϕ∈C0((0,∞))$$ \phi \in {C}^0\left(\left(0,\infty \right)\right) $$ is bounded on each of the intervals (ξ0,∞)$$ \left({\xi}_0,\infty \right) $$ with arbitrary ξ0>0$$ {\xi}_0>0 $$, and that with some kϕ>0$$ {k}_{\phi }>0 $$ and α>0$$ \alpha >0 $$, we have ϕ(ξ)≥kϕ·(ξ+1)−αfor allξ>0.$$ \phi \left(\xi \right)\ge {k}_{\phi}\cdotp {\left(\xi +1\right)}^{-\alpha}\kern2em \mathrm{for}\ \mathrm{all}\kern0.4em \xi >0. $$It is shown that for any suitably regular initial data, an associated initial‐boundary value problem admits a global very weak solution. [ABSTRACT FROM AUTHOR]

A new face‐centred finite volume method (FCFV) for Stokes problems involving sharp interfaces is proposed. Two formulations, based on two strong forms of the Stokes problem, and using different mixed variables, are presented. Particular attention is paid to the symmetry of the resulting system of global equations, and a simple rewriting of the interface boundary condition is proposed to ensure that one of the formulations preserves the symmetry of the linear system that is usually lost when considering material interfaces. Four numerical examples are considered to test the implementation numerically by performing mesh convergence studies, in two and three dimensions. The examples account for discontinuous viscosity as well as the effect of surface tension. The results show that one of the formulations is less sensitive to the numerical stabilisation used in FCFV methods but does not preserve the symmetry of the global system, whereas the other formulation is more sensitive to the stabilisation, but preserves the symmetry of the resulting system of equations. The FCFV method appears as a promising alternative for the simulation of viscous flow involving internal boundaries on conformal meshes. The potential application of the FCFV method for the purpose of geodynamic modelling is discussed. [ABSTRACT FROM AUTHOR]

This paper expands the isoparametric framework to construct a stable, H 1 -conforming, and divergence-free method for the Stokes problem in two dimensions based on the Scott-Vogelius pair on Clough-Tocher splits. The pressure space is defined through composition, whereas the velocity space is constructed via a new divergence-preserving mapping that imposes full continuity across shared edges in the isoparametric mesh. Our construction is motivated by operators and spaces found in isoparametric C 1 finite element methods. We prove the method is stable, pressure-robust, and has optimal order convergence. Numerical experiments are provided which confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The latest geoid model "HELLAS GEOID 2023" (HG2023) derived by the Hellenic Military Geographical Service is the most comprehensive model for the entire Hellenic area. Long-term gravity data, orthometric and geometric heights, seabed topography and a high-resolution digital terrain model were implemented in the calculations. Data evaluation and their accuracy estimation were of major importance to ascertain compatibility among data sources. In this study, data from neighboring countries were used as they were essential for the completeness of the model developed. The technique remove–compute–restore was adopted for the separation of the gravity signal in the heterogeneous data utilizing EIGEN 6C4 (full degree and order 2190), since the global geodynamic model was found to fit best in the Greek region. Mean gravity residual contribution to the local geoid model was calculated using Stokes' theorem in the frequency domain and implementing the Fourier transform using the Wang and Core modification. The final surface of the resulting gravimetric geoid was adapted to the existing height system of the State utilizing the Least Squares Collocation method by fitting points of known orthometric and geometric heights, distributed throughout Greece. The external accuracy of the estimated geoid model was estimated at the level of 6 cm. [ABSTRACT FROM AUTHOR]

Standard liquid simulators apply operator splitting to independently solve for pressure and viscous stresses, a decoupling that induces incorrect free surface boundary conditions. Such methods are unable to simulate fluid phenomena reliant on the balance of pressure and viscous stresses, such as the liquid rope coil instability exhibited by honey. By contrast, unsteady Stokes solvers retain coupling between pressure and viscosity, thus resolving these phenomena, albeit using a much larger and thus more computationally expensive linear system compared to the decoupled approach. To accelerate solving the unsteady Stokes problem, we propose a reduced fluid model wherein interior regions are represented with incompressible polynomial vector fields. Sets of standard grid cells are consolidated into super-cells, each of which are modelled using a quadratic field of 26 degrees of freedom. We demonstrate that the reduced field must necessarily be at least quadratic, with the affine model being unable to correctly capture viscous forces. We reproduce the liquid rope coiling instability, as well as other simulated examples, to show that our reduced model is able to reproduce the same fluid phenomena at a smaller computational cost. Futhermore, we performed a crowdsourced user survey to verify that our method produces imperceptible differences compared to the full unsteady Stokes method. [ABSTRACT FROM AUTHOR]

This paper uses the principles of polarisation to calculate the Stokes component of the polarisation image. It designs the spatial weighted fusion algorithm, focusing on the research and evaluation of the accuracy and time complexity of the spatial weighted fusion algorithm in the polarisation image fusion processing. The spatial weighted image fusion algorithm is used to fuse the polarisation degree image, the polarisation angle image, and the light intensity image. Finally, the algorithm function is realised through software programming to verify the effect and accuracy of the algorithm. Experimental results show that the spatial weighted fusion algorithm can keep the pixel value of the polarisation source image unchanged, avoid image distortion, artefacts and ringing effects algorithm is stable and efficient, with low time complexity. [ABSTRACT FROM AUTHOR]

In this paper we employ a Banach spaces-based framework to introduce and analyze new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson–Nernst–Planck equations, which is a nonlinear model describing the dynamics of electrically charged incompressible fluids. The pressure of the fluid is eliminated from the system (though computed afterwards via a postprocessing formula) thanks to the incompressibility condition and the incorporation of the fluid pseudostress as an auxiliary unknown. In turn, besides the electrostatic potential and the concentration of ionized particles, we use the electric field (rescaled gradient of the potential) and total ionic fluxes as new unknowns. The resulting fully mixed variational formulation in Banach spaces can be written as a coupled system consisting of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddle-point problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuška–Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach–Nečas–Babuška theorem. For this we also employ smallness assumptions on the data. An analogous approach, but using now both the Brouwer and Banach theorems, and invoking suitable stability conditions on arbitrary finite element subspaces, is employed to conclude the existence and uniqueness of solution for the associated Galerkin scheme. A priori error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., Raviart–Thomas elements of order k along with piecewise polynomials of degree ≤k. In addition, the latter yield approximate local conservation of momentum for all three equations involved. Finally, rates of convergence are specified and several numerical experiments confirm the theoretical error bounds. These tests also illustrate the aforementioned balance-preserving properties and the applicability of the proposed family of methods. [ABSTRACT FROM AUTHOR]

Magnetic resonance imaging allows the measurement of the three-dimensional velocity field in blood flows. Therefore, several methods have been proposed to reconstruct the pressure field from such measurements using the incompressible Navier–Stokes equations, thereby avoiding the use of invasive technologies. However, those measurements are obtained at limited spatial resolution given by the voxel sizes in the image. In this paper, we propose a strategy for the convergence analysis of state-of-the-art pressure reconstruction methods. The methods analyzed are the so-called Pressure Poisson Estimator (PPE) and Stokes Estimator (STE). In both methods, the right-hand side corresponds to the terms that involving the field velocity in the Navier–Stokes equations, with a piecewise linear interpolation of the exact velocity. In the theoretical error analysis, we show that many terms of different order of convergence appear. These are certainly dominated by the lowest-order term, which in most cases stems from the interpolation of the velocity field. However, the numerical results in academic examples indicate that only the PPE may profit of increasing the polynomial order, and that the STE presents a higher accuracy than the PPE, but the interpolation order of the velocity field always prevails. Furthermore, we compare the pressure estimation methods on real MRI data, assessing the impact of different spatial resolutions and polynomial degrees on each method. Here, the results are consistent with the academic test cases in terms of sensitivity to polynomial order as well as the STE showing to be potentially more accurate when compared to reference pressure measurements. [ABSTRACT FROM AUTHOR]

The rising velocity of an air bubble in a non‐Newtonian shear‐thinning fluid at low Reynolds numbers is generally similar to the Newtonian case given by Stokes' law. However, when the shear‐thinning fluid is subject to mechanical oscillations, the rising velocity could significantly increase. Here, we present a series of experiments quantifying the rising velocity of single bubbles during shaking in very high‐viscosity (2,000–30,000 Pa·s) shear‐thinning silicone oils. Air bubbles (18–30 mm diameter) were injected in a tank mounted on a shaking table. The tank was horizontally oscillated, at accelerations between 0.4 and 2 g. We observed a small increase in the rising velocity of the shaking cases at our experimental conditions. The increase was larger when bubbles were large and accelerations were high. Larger accelerations experienced the largest observational errors and we emphasize the exploratory nature of our results. We also measured the change in bubble diameter during the oscillations and computed the shear rate at the bubble surface. Maximum shear rates were in the range of 0.04–0.08 s−1. At these shear rates, our analysis indicates that shear thinning behavior of our analog fluids is expected to be small and compete with elastic behavior. This transitional viscous/elastic regime helps explain the small and variable results of our experiments. Our results are relevant to the study of earthquake‐volcano interactions. Most crystal‐free silicate melts would exhibit a purely viscous, shear‐thinning behavior in a natural scenario. Seismically enhanced bubble rise could offer an explanation for the observed increased degassing and unrest following large earthquakes. Plain Language Summary: The speed at which air bubbles rise in a liquid is generally well‐known. However, in non‐Newtonian fluids (liquids with non‐constant viscosity), previous experiments have shown that this speed could be significantly increased by shaking the fluid container. In this study, we present experiments where we measure the rising speed of an air bubble in a viscous non‐Newtonian fluid while shaking it. We found that bubbles rose slightly faster during shaking, especially for strong shaking and/or large bubbles. However, the shaking of the table reduced the precision of our measurements, and our results need to be further validated. We were also able to measure how quickly the bubbles deformed during shaking. When the bubbles deform quickly, our fluids may behave like solids rather than liquids. This helps explain why our results are less striking than previous experiments. Our results can be applied to the study of volcanoes. Magma inside a volcano contains vapor bubbles. When an earthquake occurs close to a volcano, these bubbles may accelerate toward the surface and lead to more gas coming out of the volcano. Key Points: We measured the rising rate of bubbles in shear‐thinning fluids subject to mechanical shakingThe bubble rising velocity increases slightly during shaking versus non‐shaking experimentsNatural magmas are likely to exhibit shear‐thinning behavior and increased bubble rise during large earthquakes [ABSTRACT FROM AUTHOR]

This work investigates the creeping flow problem via a swarm of porous circular cylindrical particles using the cell model technique. Brinkman equation inside the porous cylindrical region is used, and Stokes equation for the clear fluid would be satisfied. The stress-jump condition, together with the continuity of the velocity components and the continuity of normal stress, are employed at the porous-liquid interfaces. In contrast, no-couple stress condition and no-spin condition are used on the outer boundary of the cell. In addition, we are applying no-slip boundary conditions on the surface of the solid core. Our problem is solved analytically, and stream function expressions are derived for the inside and outside flow fields. The influence of the drag force on each porous cylindrical particle in the cell is calculated. The graph depicts the variation of hydrodynamic permeability with various parameters and especially the impact of the jump coefficient. The results of this model may be used to examine the membrane filtration process.

We construct and analyze a CutFEM discretization for the Stokes problem based on the Scott–Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. We show stability of the scheme as well as a divergence–free property of the discrete velocity outside an O(h) neighborhood of the boundary. To mitigate the error caused by the violation of the divergence–free condition, we introduce local grad–div stabilization. The error analysis shows that the grad–div parameter can scale like O(h−1), allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates. [ABSTRACT FROM AUTHOR]

The Hadamard–Rybczynski equation describes the steady‐state buoyant rise velocity of an unconfined spherical bubble in a viscous liquid. This solution has been experimentally validated for the case where the liquid viscosity is held constant. Here, we extend this result for non‐isothermal conditions, by developing a solution for bubble position in which we account for the time‐dependent liquid viscosity, liquid and gas densities, and bubble radius. We validate this solution using experiments in which spherical bubbles are created in a molten silicate liquid by cutting gas cavities into glass sheets, which are stacked, then heated through the glass transition interval. The bubble‐bearing liquid, which has a strongly temperature‐dependent viscosity, is subjected to various heating and cooling programs such that the bubble rise velocity varies through the experiment. We find that our predictions match the final observed position of the bubble measured in blocks of cooled glass to within the experimental uncertainty, even after the application of a complex temperature–time pathway. We explore applications of this solution for industrial, artistic, and natural volcanological applied problems. [ABSTRACT FROM AUTHOR]

We discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs. [ABSTRACT FROM AUTHOR]

A spectropolarimeter based on spatial heterodyne spectroscopy and the channeled polarimetric technique (SSHSCP) is introduced. The SSHSCP uses a polarization grating for channel modulation, meaning that the Stokes parameter is no longer dependent on the wavenumber; this suppresses channel aliasing considerably and improves the polarization parameter reconstruction accuracy. The SSHSCP's optical path structure is based on the Mach–Zehnder interferometer, ensuring efficient energy use. The spectral detection and polarization detection principles of the SSHSCP are derived, and a specific design example is presented. Simulations indicate that the SSHSCP can restore the target's spectral and complete Stokes parameters with high accuracy for a broadband source and narrowband lines. The reconstructed Stokes parameter errors are of the order of 10−3 or less. The SSHSCP acquires spectral and polarization parameters in a single step and has no moving parts in a compact structure. [ABSTRACT FROM AUTHOR]

We present a new enriched Galerkin (EG) scheme for the Stokes equations based on piecewise linear elements for the velocity unknowns and piecewise constant elements for the pressure. The proposed EG method augments the conforming piecewise linear space for velocity by adding an additional degree of freedom which corresponds to one discontinuous linear basis function per element. Thus, the total number of degrees of freedom is significantly reduced in comparison with standard conforming, non-conforming, and discontinuous Galerkin schemes for the Stokes equation. We show the well-posedness of the new EG approach and prove that the scheme converges optimally. For the solution of the resulting large-scale indefinite linear systems we propose robust block preconditioners, yielding scalable results independent of the discretization and physical parameters. Numerical results confirm the convergence rates of the discretization and also the robustness of the linear solvers for a variety of test problems. [ABSTRACT FROM AUTHOR]

In this paper, we aim to determine the effective permeability of biporous solids containing fractures in a computational homogenization framework. Precisely, at the local scale, we use the Brinkman law for the porous solids and the Stokes equations within fractures. The resolution of the Brinkman/Stokes coupled equations is performed with the fast Fourier transform iterative scheme on periodic unit cells. The role of the dimensions and orientation of the cracks is investigated. The simulation results are also compared with an analytical formulation based on Mori–Tanaka estimates. The findings indicate that, depending on the considered flow direction, the dimensions and orientation of cracks strongly affect the effective permeability of fractured porous media. Besides, we also determine the macroscopic permeability for a population of fractures with regular and random orientations, dimensions and shapes. The results are then given and discussed. [ABSTRACT FROM AUTHOR]

Physical light transport (PLT) algorithms can represent the wave nature of light globally in a scene, and are consistent with Maxwell's theory of electromagnetism. As such, they are able to reproduce the wave-interference and diffraction effects of real physical optics. However, the recent works that have proposed PLT are too expensive to apply to real-world scenes with complex geometry and materials. To address this problem, we propose a novel framework for physical light transport based on several key ideas that actually makes PLT practical for complex scenes. First, we restrict the spatial coherence shape of light to an anisotropic Gaussian and justify this restriction with general arguments based on entropy. This restriction serves to simplify the rest of the derivations, without practical loss of generality. To describe partially-coherent light, we present new rendering primitives that generalize the radiometric radiance and irradiance, and are based on the well-known Stokes parameters. We are able to represent light of arbitrary spectral content and states of polarization, and with any coherence volume and anisotropy. We also present the wave BSDF to accurately render diffractions and wave-interference effects. Furthermore, we present an approach to importance sample this wave BSDF to facilitate bi-directional path tracing, which has been previously impossible. We show good agreement with state-of-the-art methods, but unlike them we are able to render complex scenes where all the materials are new, coherence-aware physical optics materials, and with performance approaching that of "classical" rendering methods. [ABSTRACT FROM AUTHOR]

The relaxation of a rigid particle suspended in a one-dimensional oscillating flow is calculated according to different drag models and the results are compared. Conditions are derived under which relaxation can be neglected or drag models can be substituted by simpler ones. This investigation is conducted analytically and graphically via the plane defined by the Reynolds number and amplitude parameter. This work matches various, mostly analytic drag models together to consider simple particle relaxation with a few, broad range input parameters and cover large parts of the plane spanned by Reynolds number and amplitude parameter. [ABSTRACT FROM AUTHOR]

The aim of this paper is to simulate the two-dimensional stationary Stokes problem. In vorticity-Stream function formulation, the Stokes problem is reduced to a biharmonic one; this approach leads to a formulation only based on the stream functions and therefore can only be applied to two-dimensional problems. The idea developed in this paper is to use the discretization of the Laplace operator by the nonconforming P 1 finite element. For the solutions which admit a regularity greater than H 0 2 (Ω) , in the general case, the convergence of the method is shown with the techniques of compactness. For solutions in C 4 (Ω ̄) , an error estimate is proved, and numerical experiments are performed for the steady-driven cavity problem. [ABSTRACT FROM AUTHOR]

This work presents a method to obtain RGB pulsed lasers (683/532/436 nm) with pulse energies above 100 mJ for green and red laser beams. The maximum pulse energy of the blue laser beam is 16.5 mJ, resulting in a conversion efficiency of 5.9 %, a pulse width of 5.6 ns, and a maximum peak power of 2.9 MW. The pump source is a 532 nm pulsed laser, and the Raman medium is low-pressure hydrogen. The authors achieved a 683 nm Stokes Raman laser through stimulated Raman scattering and a 436 nm anti-Stokes Raman laser (AS1) through a four-wave mixing process. The conversion efficiency of the blue laser beam was optimized by adjusting the H 2 pressure and focus configuration. Additionally, a theoretical model was developed to explore methods for improving the blue laser conversion efficiency. By implementing a Herriot multiple-pass cavity with a special coating, the conversion efficiency of the AS1 laser reached 17.5 %. In conclusion, this work provides a practical technique for efficiently obtaining RGB lasers and demonstrates the potential for generating lasers with shorter wavelengths than the pump lasers. [ABSTRACT FROM AUTHOR]

Most numerical approaches for coupling fluid flow with porous media flow rely either on Stokes equations in the fluid part of the domain and Darcy's law in the porous part, or on Brinkman's equation. In both cases, difficulties arise at the boundary between the two parts because the equations used in the porous part are not of Stokes type. In this paper, an alternative to Darcy's law is proposed for modeling flows in porous media. This alternative relies on equations of Stokes type where the permeability tensor is replaced by force and stress derivative tensors. Numerical procedures are presented to compute these tensors from simulations at pore scale. Simulations in domains containing both fluid and porous parts are finally conducted simply assuming continuity of velocity and pressure and hence without imposing any condition at the boundary between the two parts. Results show that the proposed method is accurate and hence a promising alternative to Darcy's law for problems involving both fluid and porous parts. • Alternative to Darcy's law relying on equations of Stokes type for modeling flows in porous media. • Permeability tensor replaced by force and stress derivatives tensors computed from simulations at pore scale. • Flows in domains containing both fluid and porous parts modeled without any boundary conditions between the two parts. [ABSTRACT FROM AUTHOR]

The Arnold-Winther element successfully discretizes the Hellinger-Reissner variational formulation of linear elasticity; its development was one of the key early breakthroughs of the finite element exterior calculus. Despite its great utility, it is not available in standard finite element software, because its degrees of freedom are not preserved under the standard Piola push-forward. In this work we apply the novel transformation theory recently developed by Kirby [SMAI J. Comput. Math., 4:197-224, 2018] to devise the correct map for transforming the basis on a reference cell to a generic physical triangle. This enables the use of the Arnold-Winther elements, both conforming and nonconforming, in the widely-used Firedrake finite element software, composing with its advanced symbolic code generation and geometric multigrid functionality. Similar results also enable the correct transformation of the Mardal-Tai-Winther element for incompressible fluid flow. We present numerical results for both elements, verifying the correctness of our theory. [ABSTRACT FROM AUTHOR]

The Keller-Segel-Stokes system

In this paper, we consider a viscous droplet migrating in a viscous fluid of a different viscosity. Further, we assume that the surface of the droplet is partially contaminated with a stagnant layer of surfactant (surface active agent which reduces the interfacial tension). We analyze the effects of the following phenomena associated with the thermocapillary migration of a droplet in a transient Stokes flow. The first is the influence of surfactant cap for an arbitrary cap angle which is partially coated on the droplet surface for both high and low surface Péclet number cases. The second is the influence of the energy changes associated with stretching and shrinkage of the interfacial area elements, when the droplet is in motion. It can be noted that for the vanishing cap angle, both high and low surface Péclet number limits reduce to the case of a pure thermocapillary migration of a droplet in a transient Stokes flow. For a given ambient flow, the migration of the droplet is controlled by the magnitude of the ambient velocity and the surface tension variations due to temperature and surfactant concentration. In particular, these surface tension variations balance the tangential stress balance. Considering axisymmetric transient Stokes flow, we obtain analytical solutions in two limiting cases, namely low and high surface Péclet number. This work considers linear variation of interfacial tension on both thermal and surfactant gradients. The main contribution is pertaining to the capillary drift and the corresponding surfactant transport on the droplet for an axisymmetric hydrodynamic as well as thermal and surfactant fields. We have analyzed the level curves corresponding to stream function and temperature fields, i.e., streamlines and isotherms for various parameters in order to develop a realistic picture of the migration pattern and the influence of thermal fields. We observe that the streamlines in the vicinity of rear end of the droplet show asymmetry due to the surfactant accumulation at that region. Increasing cap angle breaks the symmetry of the induced stream. It is seen that increasing values of nondimensional parameter that accounts for the stretching and shrinkage of the droplet surface immobilizes the surface and offers retardation to the migrating droplet. The variation of migration velocity with time suggests a control mechanism for the migration of the drop under external/surface gradients and hence may serve as a useful tool in applications like targeted drug delivery systems. [ABSTRACT FROM AUTHOR]

In this paper we introduce and analyze, up to our knowledge for the first time, Banach spaces-based mixed variational formulations for nearly incompressible linear elasticity and Stokes models. Our interest in this subject is motivated by the respective need that arises from the solvability studies of nonlinear coupled problems in continuum mechanics that involve these equations. We consider pseudostress-based approaches in both cases and apply a suitable integration by parts formula for ad-hoc Sobolev spaces to derive the corresponding continuous schemes. We utilize known and new preliminary results, along with the Babuška-Brezzi theory in Banach spaces, to establish the well-posedness of the formulations for a particular range of the indexes of the Lebesgue spaces involved. Among the aforementioned new results from us, we highlight the construction of a particular operator mapping a tensor Lebesgue space into itself, and the generalization of a classical estimate in L 2 for deviatoric tensors, which plays a key role in the Hilbertian analysis of linear elasticity, to arbitrary Lebesgue spaces. No discrete analysis is performed in this work. [ABSTRACT FROM AUTHOR]

A chemotaxis‐Stokes system modeling coral fertilization under the effects of p‐Laplace diffusion is investigated in spatially three‐dimensional setting. By making appropriate use of certain conditional estimates for chemotactic signal gradient and the fluid velocity, we prove global existence and boundedness under a mild assumption that p>2, which extends previous results derived in [8]. [ABSTRACT FROM AUTHOR]

In this paper we analyse geometric multiscale models arising in the description of physiological flows such as blood flow in arteries or air flow in the bronchial tree. The geometrical complexity of the networks in which air/blood flows lead to a classical decomposition in two areas: a truncated 3D geometry corresponding to the largest contribution of the domain, and a 0D part connected to the 3D part, modelling air/blood flows in smaller airways/vessels. The fluid in the 3D part is described by the Stokes or the Navier–Stokes system which is coupled to 0D models or so-called Windkessel models. The resulting Navier–Stokes–Windkessel coupled system involves Neumann non-local boundary conditions that depends on the considered applications. We first show that the different types of Windkessel models share a similar formalism. Next we derive existence results and stability estimates for the continuous coupled Stokes–Windkessel or Navier–Stokes–Windkessel problem as well as stability estimates for the semi-discretized systems with either implicit or explicit treatment of the boundary conditions. In all the calculations, we pay a special attention to the dependency of the various constants and smallness conditions on the data with respect to the physical and numerical parameters. In particular we exhibit different kinds of behavior depending on the considered 0D model. Moreover even if no energy estimates can be derived in energy norms for the Navier–Stokes–Windkessel system, leading to possible and observed numerical instabilities for large applied pressures, we show that stability estimates for both the continuous and semi-discrete problems, can be obtained in appropriate norms for small enough data by introducing a new well chosen Stokes-like operator. These sufficient stability conditions on the data may give a hint on the order of magnitude of the data enabling stable computations without stabilization method for the problem. Numerical simulations illustrate some of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n S (| ∇ c | 2) ∇ c) − n m , c t + u ⋅ ∇ c = Δ c − c + m , m t + u ⋅ ∇ m = Δ m − m n , u t = Δ u + ∇ P + (n + m) ∇ Φ , ∇ ⋅ u = 0 which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where S is a given function fulfilling | S (σ) | ≤ K S (1 + σ) − θ 2 , σ ≥ 0 with some K S > 0 . Based on conditional estimates of the quantity c and the gradients thereof, a relatively compressed argument as compared to that proceeding in related precedents shows that if θ > 0 , then for any initial data with proper regularity an associated initial-boundary problem under no-flux/no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded, and which also enjoys the stabilization features in the sense that ∥ n (⋅ , t) − n ∞ ∥ L ∞ (Ω) + ∥ c (⋅ , t) − m ∞ ∥ W 1 , ∞ (Ω) + ∥ m (⋅ , t) − m ∞ ∥ W 1 , ∞ (Ω) + ∥ u (⋅ , t) ∥ L ∞ (Ω) → 0 as t → ∞ with n ∞ : = 1 | Ω | { ∫ Ω n 0 − ∫ Ω m 0 } + and m ∞ : = 1 | Ω | { ∫ Ω m 0 − ∫ Ω n 0 } + . [ABSTRACT FROM AUTHOR]

The treatment of exhaust gas from large diesel engines is an important factor to affect environment. Because of its advantages in reaction efficiency and heat transfer efficiency, circulating fluidized bed (CFB) has shown great potential with regard to save energy and purify emissions from large diesel engines. This paper mainly investigated a three-dimensional modeling of the CFB, and a rolling function is applied to simulate its unsteady working conditions, which is called the rolling circulation fluidized bed (RCFB). Two-fluid model based on Euler-Euler approach is utilized to numerically simulate the gas-solid two phases flow within the RCFB. In order to give the comparable results between the present simulation and the previous experiment, the effect of Stokes number on particle phase distribution behaviors in the RCFB is investigated. By simulating the overall swing condition of the RCFB based on a dynamic grid, the effect of the swing condition on particle radial distribution uniformity in the riser is obtained. The reliability of quantitative modeling is verified based on the experimental results of the extant literature. Based on this simulation, the influence of the Stokes number on the radial distribution of particles as well as the relationship between the uniformity of the distribution and particle size is obtained. [ABSTRACT FROM AUTHOR]

This manuscript is concerned with the Keller-Segel-Stokes system (⋆) { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n F (| ∇ c | 2) ∇ c) , c t + u ⋅ ∇ c = Δ c − c + n , u t = Δ u + ∇ P + n ∇ Φ , ∇ ⋅ u = 0 , under no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded three-dimensional domains, with given suitably regular functions F and Φ. Here in accordance with recent developments in the literature on refined modeling of chemotactic migration, the introduction of suitably decaying F is supposed to adequately account for saturation mechanisms that limit cross-diffusive fluxes near regions of large signal gradients. In the context of such nonlinearities which suitably generalize the prototype given by F (ξ) = K F (1 + ξ) − α 2 , ξ ≥ 0 , with K F > 0 , known results addressing a fluid-free parabolic-elliptic simplification of (⋆) have identified the value α c = 1 2 as critical with regard to the occurrence of blow-up in the sense that some exploding solutions can be found when α < 1 2 , whereas all suitably regular initial data give rise to global bounded solutions when α > 1 2. The intention of the present study consists in making sure that the latter feature of blow-up prevention by suitably strong flux limitation persists also in the more complex framework of the fully coupled chemotaxis-fluid system (⋆). To achieve this, as a secondary objective of possibly independent interest the manuscript separately establishes some conditional bounds for corresponding fluid fields and taxis gradients in a fairly general setting that particularly includes the subsystem of (⋆) concerned with the evolution of (c , u , P). These estimates relate respective regularity features to certain integrability properties of associated forcing terms, as in the context of (⋆) essentially represented by the quantity n. The application of this tool to the specific problem under consideration thereafter facilitates the derivation of a result on global existence of bounded classical solutions to (⋆) for widely arbitrary initial data actually within the entire range α > 1 2 , and by means of an argument which appears to be significantly condensed when compared to reasonings pursued in previous works concerned with related problems. [ABSTRACT FROM AUTHOR]

We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes. [Display omitted] • General control-volume finite-element framework. • Construction of overlapping control volumes giving more flexibility. • Hybrid approach using a hierarchical splitting for the MINI element. • Detailed numerical investigations for various test cases. [ABSTRACT FROM AUTHOR]

We present a general finite volume method to solve a coupled Stokes-Darcy problem, we propose two domains corresponding to fluid region and porous region with a physical intersection. At the contact interface between the fluid region and the porous media we impose two conditions; the first one is the normal continuity of the velocity and the second one is the continuity of the pressure. Furthermore, due to the lack of information about both the velocity and the pressure on the interface, we will use Schwarz domain decomposition. In Darcy equations, the tensor of permeability will be considered as variable, since it depends on both the properties of the porous medium and the viscosity of the fluid. Numerical examples are presented to demonstrate the efficiency of the proposed method.

To advance our understanding of the movement of elastic microstructures in a viscous fluid, techniques that utilize available data to estimate model parameters are necessary. Here, we describe a Bayesian uncertainty quantification framework that is highly parallelizable, making parameter estimation tractable for complex fluid–structure interaction models. Using noisy in silico data for swimmers, we demonstrate the methodology’s robustness in estimating fluid and elastic swimmer parameters, along with their uncertainties. We identify correlations between model parameters and gain insight into emergent swimming trajectories of a single swimmer or a pair of swimmers. Our proposed framework can handle data with a spatiotemporal resolution representative of experiments, showing that this framework can be used to aid in the development of artificial micro-swimmers for biomedical applications, as well as gain a fundamental understanding of the range of parameters that allow for certain motility patterns. [ABSTRACT FROM AUTHOR]

The Keller-Segel-Stokes system { nt + u ⋅ ∇n = Δn − ∇ ⋅ (n∇c) + ρn − μnα, ct + u ⋅ ∇c = Δc − c + n, ut = Δu + ∇P − n∇Λ , ∇ ⋅ u = 0 (*), is considered in a bounded domain Ω ⊂ ℝ3 with smooth boundary, with parameters ρ ≥ 0, μ > 0 and α > 1, and with a given gravitational potential Λ ∈ W2,∞(Ω). It is shown that in this general setting, when posed under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u, and for any suitably regular initial data, an associated initial value problem possesses at least one globally defined solution in an appropriate generalized sense. Since it is well-known that in the absence of absorption, already the corresponding fluid-free subsystem with u ≡ 0 and μ = 0 admits some solutions blowing up in finite time, this particularly indicates that any power-type superlinear degradation of the form in (*) goes along with some significant regularizing effect. [ABSTRACT FROM AUTHOR]

The presence of macropores and fractures significantly affects the effective transport properties of porous solids such as concrete and rocks. The dimensions of the fractures are generally large behind that of the initial porosity, so that the problem contains two porosities. The influence of the macroporosity is studied in the homogenization framework by solving at the intermediate scale, that of the macropores, a coupled Darcy/Stokes problem with the Beavers–Joseph–Saffman (BJS) interface condition. We derive analytic expressions of the macroscopic permeability in the case of an isotropic permeable matrix containing spheroidal-shaped macropores. To this aim, we consider a representative volume element (RVE) on which uniform boundary conditions are considered for the velocity and pressure fields. The local problem is written as minimum principles; kinematic and static approaches are developed to derive rigorous bounds for the macroscopic permeability. Closed-form expressions of the longitudinal permeability (along the revolution axe of the spheroid) are determined by considering a simplified RVE constituted of two confocal spheroids. They depend on the volume fraction and the eccentricity of the spheroidal macropores, the scale factor between the two porosities and the slip coefficient of the BJS model. Illustrations show the influence of these parameters. [ABSTRACT FROM AUTHOR]