Incompressible Equations Research Articles

Energy conservation for the nonhomogeneous incompressible Hall-MHD equations in a bounded domain

In this paper, we study the energy conservation of the nonhomogeneous incompressible Hall-magnetohydrodynamic equations in a bounded domain Ω⊂Rd with d=2,3. By exploring the special structure of the nonlinear terms in the Hall-magnetohydrodynamic equations, we obtain the sufficient conditions for the regularity of the weak solutions for energy conservation. The treatment of the boundary terms mainly relies on the coarea formula.

Bayesian Estimation of Adsorption and Desorption Parameters for Pore Scale Transport

Stochastic parameter estimation and inversion have become increasingly popular in recent years. Nowadays, it is computationally reasonable and regular to solve complex inverse problems within the Bayesian framework. Applications of Bayesian inferences for inverse problems require investigation of the posterior distribution, which usually has a complex landscape and is highly dimensional. In these cases, Markov chain Monte Carlo methods (MCMC) are often used. This paper discusses a Bayesian approach for identifying adsorption and desorption rates in combination with a pore-scale reactive flow. Markov chain Monte Carlo sampling is used to estimate adsorption and desorption rates. The reactive transport in porous media is governed by incompressible Stokes equations, coupled with convection–diffusion equation for species’ transport. Adsorption and desorption are accounted via Robin boundary conditions. The Henry isotherm is considered for describing the reaction terms. The measured concentration at the outlet boundary is provided as additional information for the identification procedure. Metropolis–Hastings and Adaptive Metropolis algorithms are implemented. Credible intervals have been plotted from sampled posterior distributions for both algorithms. The impact of the noise in the measurements and influence of several measurements for Bayesian identification procedure is studied. Sample analysis using the autocorrelation function and acceptance rate is performed to estimate mixing of the Markov chain. As result, we conclude that MCMC sampling algorithm within the Bayesian framework is good enough to determine an admissible set of parameters via credible intervals.

Global smooth solutions to the 3D generalized Hall-magnetohydrodynamic equations with large data

In this paper, based on a perturbation argument and fully exploring the nonlinear structure of the considered system, we establish the global existence to the three-dimensional generalized incompressible Hall-MHD equations for a class of large initial data, whose norms can be arbitrarily large. Our obtained result improves considerably the recent result in Li et al. [A class large solution of the 3D Hall-magnetohydrodynamic equations. J Differ Equ. 2020;268(10):5811–5822] to the generalized Hall-MHD system with equipping with different viscosity and magnetic diffusion coefficients.

Numerical Modeling of Flow through the Vertical Rectangular Microchannel for Drug Screening Applications

A numerical study of flow in the rectangular vertical microchannel for drug screening applications is described. The flow characteristics through the microchannel are presented by incompressible Naiver-Stroke equations. Flow rate is evaluated from numerical modeling to simulate the drug flow through the microchannel. Here in this study, computational fluid dynamics based finite element method commercial software COMSOL is used for simulation of drug flow through the artery for cancer drug screening applications. The simulation work also presents the diffusion velocity at the intersection area between the drug and cancer cells..

The research on current rush to air mass in delivery pipeline system with YOUNGS-VOF method

Water hammer with gas often occurs in long delivery pipeline system lager than ordinary water hammer, surface tracking between air and water is important and difficult. In this paper, flow rush to interception air-mass in long water conveyance pipe system is researched on base on surface in between water and gas tracking method of staggered grid dispersed with YOUNGS-VOF, the limited compressibility method of viscous incompressible fluid control equations, and numerical method of ideal gas state equation. The numerical results show that YOUNGS-VOF method can be used to deal with the interface and water hammer is consistent with the theory analysis.

Exact Solutions to the Navier–Stokes Equations with Couple Stresses

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time.

On 3D Hall-MHD Equations with Fractional Laplacians: Global Well-Posedness

The Cauchy problem for 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with fractional Laplacians \((-\Delta )^{\frac{1}{2}}\) is studied. The well-posedness of 3D incompressible Hall-MHD equations remains an open problem with fractional diffusion \((-\Delta )^{\frac{1}{2}}\). First, global well-posedness of small-energy solutions with general initial data in \(H^s\), \(s>\frac{5}{2}\), is proved. Second, a special class of large-energy initial data is constructed, with which the Cauchy problem is globally well-posed. The proofs rely upon a new global bound of energy estimates involving Littlewood–Paley decomposition and Sobolev inequalities, which enables one to overcome the \(\frac{1}{2}\)-order derivative loss of the magnetic field.

Clebsch gauge fluid

We propose a novel gauge fluid solver based on Clebsch wave functions to solve incompressible fluid equations. Our method combines the expressive power of Clebsch wave functions to represent coherent vortical structures and the generality of gauge methods to accommodate a broad array of fluid phenomena. By evolving a transformed wave function as the system's gauge variable enhanced by an additional projection step to enforce pressure jumps on the free boundaries, our method can significantly improve the vorticity generation and preservation ability for a broad range of gaseous and liquid phenomena. Our approach can be easily implemented by modifying a standard grid-based fluid simulator. It can be used to solve various fluid dynamics, including complex vortex filament dynamics, fluids with different obstacles, and surface-tension flow.

A physics-informed and hierarchically regularized data-driven model for predicting fluid flow through porous media

This paper presents a new deep learning data-driven model for predicting structure dependent pore-fluid velocity fields in rock. The model is based on a Convolutional Auto-Encoder (CAE) artificial neural network capable of learning from image data generated by direct numerical simulations of fluid flow through pore-structures, such as by Lattice Boltzmann or molecular dynamics methods. The main novelty of the model in comparison to previous CAE-based data-driven approaches consists of three parts. The first is a methodology for decomposing the full-domain of the porous media into sub-regions, or “sub-domains”, in order to reduce the overall size of the CAE, batch process the sub-domains in parallel, and enable the CAE to learn local and generalizable nonlinear mappings of pore-fluid velocities. The second consists of embedding the finite difference solutions of the incompressible Navier-Stokes and continuity equations into convolutional layers prior to the CAE in order to provide the CAE with knowledge of fluid dynamics physics (PhyFlow). The third main novelty is that the training of the CAE is regularized with a hierarchical loss function that encourages the learning of fluid flow patterns (in a way similar to ranked modes in principal component analysis), ranking from most to least important. This is shown to increase the stability in learning, reduce over-fitting, and promote interpretability of the CAE neural network layers (HierCAE). The comprehensive new data-driven model, which we call the PhyFlow-HierCAE model, is shown to exhibit improved accuracy and generalizability of flow field predictions over conventional CAE models, attributable to the embedded physical knowledge and the hierarchical regularization, as well as realize orders of magnitude speed-ups in computation times as a surrogate for the direct numerical simulations. Examples of training and forward predictions on unseen pore-structures are provided and evaluated for data from Lattice Boltzmann and molecular dynamics simulations of pore-fluid flow. The model is shown to be a fast and accurate emulator (or “surrogate”) for predicting effective permeability of unseen pore-structures based on learning from relatively small direct numerical simulation datasets.

Space-time decay estimates of solutions to 3D incompressible viscous Camassa-Holm equations

In this paper, based on the parabolic interpolation inequality and inductive argument, we study the space-time decay estimates of higher-order time and spatial derivatives of strong solutions for the 3D incompressible viscous Camassa-Holm equations provided that the initial datum is well localized.

A projection-based stabilized virtual element method for the unsteady incompressible Brinkman equations

This paper is devoted to a stabilized mixed virtual element method (mixed VEM) for the unsteady incompressible Brinkman equations. We employ the pairs of C0-conforming virtual element spaces containing the “equal-order” polynomials to approximate the velocity and pressure variables, and replace the time derivative by a backward Euler difference quotient. The numerical stability is guaranteed with a new projection-based stabilization term, which is simple without the projection of second derivatives or the coupling terms. We also establish the error estimates for both the semi-discrete and fully-discrete schemes with respect to the viscosity coefficient. Finally, we carry out several numerical experiments to validate the theoretical analysis.

Mixing solutions for the Muskat problem

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type $$H^5$$ initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non smooth semiclassical type pseudodifferential operators which is developed.

Global classical solutions to the elastodynamic equations with damping

In this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

Scale dependence and cross-scale transfer of kinetic energy in compressible hydrodynamic turbulence at moderate Reynolds numbers

We investigate properties of the scale dependence and cross-scale transfer of kinetic energy in compressible three-dimensional hydrodynamic turbulence, by means of two direct numerical simulations of decaying turbulence with initial Mach numbers M = 1/3 and M = 1, and with moderate Reynolds numbers, R_lambda ~ 100. The turbulent dynamics is analyzed using compressible and incompressible versions of the dynamic spectral transfer (ST) and the Karman-Howarth-Monin (KHM) equations. We find that the nonlinear coupling leads to a flux of the kinetic energy to small scales where it is dissipated; at the same time, the reversible pressure-dilatation mechanism causes oscillatory exchanges between the kinetic and internal energies with an average zero net energy transfer. While the incompressible KHM and ST equations are not generally valid in the simulations, their compressible counterparts are well satisfied and describe, in a quantitatively similar way, the decay of the kinetic energy on large scales, the cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. There exists a simple relationship between the KHM and ST results through the inverse proportionality between the wave vector k and the spatial separation length l as k l ~ 3^1/2. For a given time the dissipation and pressure-dilatation terms are strong on large scales in the KHM approach whereas the ST terms become dominant on small scales; this is owing to the complementary cumulative behavior of the two methods. The effect of pressure dilatation is weak when averaged over a period of its oscillations and may lead to a transfer of the kinetic energy from large to small scales without a net exchange between the kinetic and internal energies. Our results suggest that for large-enough systems there exists an inertial range for the kinetic energy cascade ...

On the Blowup for the 3D Axisymmetric Incompressible Chemotaxis-Euler Equations

In this paper, we investigate the 3D incompressible chemotaxis-Euler equations. Taking advantage of the structure of axisymmetric fluids, we establish the blowup criterion of the system using the Fourier localization method.

Boundary layer problems for the two-dimensional inhomogeneous incompressible magnetohydrodynamics equations

In this paper, we study the well-posedness of boundary layer problems for the inhomogeneous incompressible magnetohydrodynamics (MHD) equations, which are derived from the two-dimensional density-dependent incompressible MHD equations. Under the assumption that initial tangential magnetic field is not zero and density is a small perturbation of the outer constant flow in supernorm, the local-in-time existence and uniqueness of inhomogeneous incompressible MHD boundary layer equations are established in weighted conormal Sobolev space by energy method.

Exact solutions of the Navier–Stokes equations for describing an isobaric one-directional vertical vortex flow of a fluid

The article proposes a family of exact solutions to the Navier–Stokes equations for describing isobaric inhomogeneous unidirectional fluid motions. Due to the incompressibility equation, the velocity of the inhomogeneous Couette flow depends on two coordinates and time. The expression for the velocity field has a wide functional arbitrariness. This exact solution is obtained by the method of separation of variables, and both algebraic operations (additivity and multiplicativity) are used to substantiate the importance of modifying the classical Couette flow. The article contains extensive bibliographic information that makes it possible to trace a change in the exact Couette solution for various areas of the hydrodynamics of a Newtonian incompressible fluid. The fluid flow is described by a polynomial depending on one variable (horizontal coordinate). The coefficients of the polynomial functionally depend on the second (vertical) coordinate and time; they are determined by a chain of the simplest homogeneous and inhomogeneous partial differential parabolic-type equations. The chain of equations is obtained by the method of undetermined coefficients after substituting the exact solution into the Navier–Stokes equation. An algorithm for integrating a system of ordinary differential equations for studying the steady motion of a viscous fluid is presented. In this case, all the functions defining velocity are polynomials. It is shown that the topology of the vorticity vector and shear stresses has a complex structure even without convective mixing (creeping flow).

Global well-posedness of strong solutions to the 2D nonhomogeneous incompressible primitive equations with vacuum

We prove the global well-posedness of the strong solution to the two-dimensional inhomogeneous incompressible primitive equations for initial data with small H12-norm, which also satisfies a natural compatibility condition. A logarithmic type Sobolev embedding inequality for the anisotropic Lx∞Lz2(Ω) norm is established to obtain the global in time a priori H1(Ω)∩W1,6(Ω) estimate of the density, which guarantee the local solution to be a global one.

Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and anisotropic mesh grading

The velocity solution of the incompressible Stokes equations is not affected by changes of the right hand side data in form of gradient fields. Most mixed methods do not replicate this property in the discrete formulation due to a relaxation of the divergence constraint which means that they are not pressure-robust. A recent reconstruction approach for classical methods recovers this invariance property for the discrete solution, by mapping discretely divergence-free test functions to exactly divergence-free functions in the sense of {varvec{H}}({text {div}}). Moreover, the Stokes solution has locally singular behavior in three-dimensional domains near concave edges, which degrades the convergence rates on quasi-uniform meshes and makes anisotropic mesh grading reasonable in order to regain optimal convergence characteristics. Finite element error estimates of optimal order on meshes of tensor-product type with appropriate anisotropic grading are shown for the pressure-robust modified Crouzeix–Raviart method using the reconstruction approach. Numerical examples support the theoretical results.

Application of physics-informed neural networks for self-similar and transient solutions of spontaneous imbibition

Spontaneous imbibition is a natural process where the wetting phase spontaneously replaces the non-wetting phase due to the capillary force in porous media. This multiphase flow mechanism happens frequently in many natural processes such as during nonaqueous phase liquids transport, CO2 storage, soil infiltration, improved oil recovery, etc. Due to the nature of the capillary dominated flow, the imbibition rate is a natural consequence of the rock-fluid interaction and remains unknown from the problem configuration. This has made the solution to spontaneous imbibition unique and differentiates it from the traditional Buckley-Leverett type of displacement problems. While many attempts have been made in the literature to solve the spontaneous imbibition governing equations, most of them are quite mathematically involved even for the self-similar solution governed by an ordinary differential equation (ODE). Only a few more complex solutions exist for the transient spontaneous imbibition problems. In this research, we have adopted the concept and workflow of the deep physics-informed neural networks (PINN) to solve the spontaneous imbibition problems, both self-similar and transient. It focuses on learning the prior knowledge embedded within the governing differential equations rather than depending on input-output data pairs. The one-dimensional unsteady state immiscible, incompressible horizontal flow equations with a saturation-dependent dispersion coefficient under Lagrangian formulation is included in the workflow as the main loss function, along with the physics governed boundary conditions such as capillary end effect. The methodology represents a robust and straightforward function approximator for the solutions of this class, and the effectiveness of the proposed framework is demonstrated through two example calculations.