Impermeable Boundary Conditions Research Articles

On steady solutions to the MHD equations with inhomogeneous generalized impermeability boundary conditions for the magnetic field

The paper deals with the steady MHD equations for a viscous incompressible fluid in a bounded and generally multiply connected domain with the no‐slip boundary condition for the velocity and inhomogeneous generalized impermeability boundary conditions for the magnetic field . The main results concern an appropriate definition of the weak solution , its existence, continuous dependence on the data when the data tend to zero, and uniqueness in a “small” neighborhood of the zero solution .

Stabilized immersed isogeometric analysis for the Navier–Stokes–Cahn–Hilliard equations, with applications to binary-fluid flow through porous media

Binary-fluid flows can be modeled using the Navier–Stokes–Cahn–Hilliard equations, which represent the boundary between the fluid constituents by a diffuse interface. The diffuse-interface model allows for complex geometries and topological changes of the binary-fluid interface. In this work, we propose an immersed isogeometric analysis framework to solve the Navier–Stokes–Cahn–Hilliard equations on domains with geometrically complex external binary-fluid boundaries. The use of optimal-regularity B-splines results in a computationally efficient higher-order method. The key features of the proposed framework are a generalized Navier-slip boundary condition for the tangential velocity components, Nitsche’s method for the convective impermeability boundary condition, and skeleton- and ghost-penalties to guarantee stability. A binary-fluid Taylor–Couette flow is considered for benchmarking. Porous medium simulations demonstrate the ability of the immersed isogeometric analysis framework to model complex binary-fluid flow phenomena such as break-up and coalescence in complex geometries.

Modeling impedance boundary conditions and acoustic barriers using the immersed boundary method: The one-dimensional case

Immersed boundary methods are heavily used in computational fluid dynamics, as an alternative to volumetric meshing, when a problem contains irregular geometric features. In wave-based architectural and room acoustics, the dynamics are simplified, but boundary conditions and acoustic barriers are usually described in terms of frequency-dependent impedance and transmittance functions. In this article, a formulation of the immersed boundary method is developed in the informative special case of one-dimensional linear acoustics. It relies on dual driving terms applied to the conservation of mass and momentum equations separately and is directly tunable against boundary impedances and barrier transmittances. It is shown how the driving terms may be combined to model either an impermeable frequency-dependent boundary condition or a barrier with a given transmittance. An explicit time-domain numerical method of finite-difference time-domain type is presented, and it is shown how the immersed boundary condition may be included, at minimal additional computational cost. Special attention is paid to the discrete approximation of the Dirac delta function, necessary in immersed boundary methods, as well as the discretisation strategy for frequency-dependent boundary and barrier conditions. Numerical results are presented. A complete derivation of numerical stability conditions for this immersed boundary method appears in an appendix.

Performance assessment of internal porous structures on liquid sloshing in various 3D tanks by multi-domain IGABEM

An isogeometric boundary element method (IGABEM) is extended in this work to study liquid sloshing characteristics in various 3D tanks with arbitrary internal bottom-mounted porous structures. Unlike the classical boundary element method (BEM), the present method uses the non-uniform rational B-splines(NURBS) instead of the piecewise Lagrange polynomials as the shape function to approximate both the domain boundary and field variables, it completely inherits the advantages of the traditional BEM and the isogeometric analysis (IGA), such as the properties of only boundary discretization needing, higher order continuity, self-adaptability and so on. All the information required in the IGABEM for the mesh generation can be obtained from the CAD software, which may evidently reduce the time and memory consumptions of preprocessing, meanwhile, since the same basis functions (NURBS) are used for both the IGABEM and CAD models, the proposed method can exactly reconstruct the geometry of the analysis domain without any error, and this substantively contributes to the high calculation accuracy of IGABEM. In this paper, by using a zoning method the entire domains of fluid are divided into several sub-domains with the consideration of compatibility boundary conditions besides the porous structures. Owning to the constant cross section of the container, impermeable boundary condition at the tank bottom and the linear boundary condition at the free surface, the 3D Laplace equation (which governs the 3D sloshing problem) is transformed into a couple of Helmholtz equation and modified Helmholtz equations (corresponding to the propagating and evanescent modes, respectively). Green's theorem and the divergence theorem of Gauss are used in this paper to derive the IGABEM system equations for the two types of equations. Moreover, by introducing a series of eigen-functions associated with the vertical coordinate, the potential and velocity boundary conditions are expanded into those associated with the propagating and evanescent modes, which can be applied upon the Helmholtz equation and modified Helmholtz equations. Finally, the present problem can be solved by solving these equations and combining the results linearly. Accuracy and convergence of the present IGABEM technique are verified through numerical tests by comparing the obtained results with analytical, experimental solutions or those calculated with other numerical method, thereafter, liquid sloshing problems in the cubic or circular cylindrical vessels with the coaxial or eccentric elliptical and circular cylindrical porous structures are analyzed. The main influences of the geometric parameter, porous-effect parameter, number and arrangement of the porous structures are investigated in detail, some conclusions are outlined in the end of this paper.

Energy release rate for cracks in hydrogels undergoing finite deformations

The rupture of hydrogels and swellable elastomers involves large deformations, and there exists a large literature devoted to their experimental characterization including methods for measuring and enhancing fracture toughness. Analytical investigations of the fracture of hydrogels have recognized the importance of large deformations and the contributions of liquid flow, but they have largely been restricted to plane-strain formulations that ignore through-thickness effects. In this paper, boundary–initial value problems for cracked specimens are solved in plane-strain and three dimensions for both permeable and impermeable boundary conditions and various rates of loading. The transient stress, strain and chemical potential fields near the crack tip/front are found to be notably different than the asymptotic solutions of linear poroelasticity and large deformation plane-strain formulations. The energy release rate is computed using a poroelastic path-independent integral ( J ∗ ), and generally that also is a function of the liquid flow. It is shown for moderately thin three-dimensional specimens that liquid flow is largely in the out-of-plane direction under permeable boundaries and largely in-plane for impermeable boundaries; thus, liquid flow makes larger contributions to the energy release in the latter. In agreement with experiments, the energy release rate tends to be larger at higher loading rates due to the contributions of liquid flow. Finally, criteria for crack growth based on the critical stretch ahead of the crack are adopted to predict the critical energy release rate as a function of solid volume fraction, and the possibility of a non-monotone dependence of energy release on solid volume fraction is uncovered. The methods presented in this paper can be utilized to analyze a wide variety of problems in the rupture of hydrogels including applications to soft tissues and fibrous gels.

Multiplicity in an optimised kinematic dynamo

Multiplicity in optimal kinematic dynamos exists for certain types of symmetry classes and boundary conditions, at least near the lowest dynamo onset . Here we investigate the NNT type dynamo generated by steady flows with impermeable boundary conditions in a cube, where the letter N or T stands for pseudo-vacuum or superconducting boundary conditions along x, y, z directions, respectively. We find the top two of the three branches in the neighbourhood of have their growth rates crossed over at . Within each branch, the spatial structure of the optimal velocity field gradually shifts with respect to . At about above , the original optimal branch has developed distinct combinations of dominant Fourier modes. In contrast, the first suboptimal branch shows the least change in structure. We then follow the evolution of selected optimised solutions when varies until it becomes unstable. Specific modes in the flow that can destabilise the dynamo are identified. Within the range surveyed, we find that there can be one to two dynamo windows. All three branches generate a steady dynamo near , but the first suboptimal branch can generate an oscillatory dynamo at about eight times , and for both suboptimal branches, the growth rate reaches saturation approximately at . We find the two suboptimal branches create a more robust dynamo action supercritically.

Wave-induced thermal flux and scattering of P waves in a medium with aligned circular cracks

High temperature affects the seismic properties of cracked and faulted reservoirs and can be an indicator for their detection. To this purpose, the authors study the wave-induced thermal flux (WITF) and develop two exact solutions for the scattering of compressional waves by a circular crack filled with a compressible fluid, in which the approach is based on thermally permeable and impermeable boundary conditions. The authors obtained the phase velocity and attenuation as a function of frequency, which found that there are two loss mechanisms, i.e., thermoelastic dissipation at low frequencies and elastic scattering at high frequencies. Basically, when the crack size is comparable to the thermal and elastic wavelengths, there are substantial dispersion and attenuation (anelasticity) in the WITF and scattering frequency ranges, respectively. This means that the spatial inhomogeneity scale for inducing WITF is much smaller than that of scattering and the two mechanisms can be discriminated. The dependence of the compressional-wave velocity and attenuation on the compressibility and thermal expansion of the crack-filling fluid are different depending on the thermal diffusion rates at the crack interface. The anelasticity is much higher in the fully permeable case. This model has the potential to evaluate thermoelastic properties and heterogeneity at different scales from seismic responses.

Subsonic source and doublet panel methods

This work presents a source and doublet surface panel method for solving unsteady subsonic flows around wing and bodies. The approach is based on Morino’s theory but features three modifications. The impermeability boundary condition is defined as zero mass flux normal to the surface instead of zero normal velocity, a new methodology is introduced to calculate the aerodynamic influence coefficient matrices and the pressure distribution around the surface is evaluated using a nonlinear unsteady Bernoulli equation. The method is demonstrated on three experimental test cases from the literature; it is shown that it can predict more accurately the steady and oscillating pressure distribution around the surface than Morino’s original linear approach. Crucially, the modified approach can represent asymmetries in the pressure distribution due to both asymmetric motion and shape. While higher fidelity alternatives exist for solving such flows, the present method has very low computational cost and would be suitable for optimization procedures during preliminary design, as long as the flow remains subcritical and attached at all times.

On the Effect of Fast Rotation and Vertical Viscosity on the Lifespan of the 3D Primitive Equations

We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation |Omega |, we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only L^2 in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large |Omega |, we show that the existence time of the solution can be prolonged, with “well-prepared” initial data. Finally, in the case of two spatial dimensions with Omega =0, we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables.

On the flow-acoustic coupling of fan blades with over-the-rotor liner

Over-the-rotor liner exhibits the potential to further attenuate turbofan noise, but the physics involved remain to be explored. In this paper, a three-dimensional coupled singularity method is proposed to investigate the flow-acoustic coupling effects of axially overlapping annular rotor and finite-length liner in subsonic flow. The formulation adopts the orthogonal basis expansion of the generated disturbances in terms of the hard-walled duct modes. The sound scatterings at the rotor and the liner are then characterized, respectively, by the underdetermined dipole and monopole distributions. We derive a simultaneous solution to the coupled unsteady rotor and liner responses, which ensures that the resultant perturbed field satisfies both the impermeable boundary condition on the blade surfaces and the impedance boundary condition on the lined wall. The effect of a perforated porous-material liner on the wake–rotor interaction tones is investigated. The analysis reveals that for the sound field of varying mode and frequency characteristics, moving the inlet liner to the over-the-rotor location generally leads to limited loss or even an increase of upstream sound absorption, along with additional acoustic benefits in the aft duct. The flow-acoustic coupling between the axially overlapping rotor and liner is shown to alleviate significantly the unsteady blade loading and meanwhile intensify the fluid particle oscillation through the acoustically treated wall. Sound source reduction and sound dissipation enhancement are thus identified as the governing noise attenuation mechanisms. Finally, we extend the analysis to provide insights into the effectiveness of over-the-rotor acoustic treatment with shortened axial length.

An improved impermeable solid boundary scheme for Meshless Local Petrov–Galerkin method

Meshless methods have become an essential numerical tool for simulating a wide range of flow–structure interaction problems. However, the way by which the impermeable solid boundary condition is implemented can significantly affect the accuracy of the results and computational cost. This paper develops an improved boundary scheme through a weak formulation for the boundary particles based on Pressure Poisson’s Equation (PPE). In this scheme, the wall boundary particles simultaneously satisfy the PPE in the local integration domain by adopting the Meshless Local Petrov–Galerkin method with the Rankine source solution (MLPG_R) integration scheme (Ma, 2005b) and the Neumann boundary condition, i.e., normal pressure gradient condition, on the wall boundary which truncates the local integration domain. The new weak formulation vanishes the derivatives of the unknown pressure at wall particles and is discretized in the truncated support domain without extra artificial treatment. This improved boundary scheme is validated by analytical solutions, numerical benchmarks, and experimental data in the cases of patch tests, lid-driven cavity, flow over a cylinder and monochromic wave generation. Second-order convergent rate is achieved even for disordered particle distributions. The results show higher accuracy in pressure and velocity, especially near the boundary, compared to the existing boundary treatment methods that directly discretize the pressure Neumann boundary condition.

Combined effect of anisotropy and uncertainty on the safe mud pressure window of horizontal wellbore drilled in anisotropic saturated rock

The investigation of safe mud pressure window of horizontal wellbore drilled in the saturated rock by accounting for the combined effect of anisotropy and uncertainty is the main purpose of this work. To this aim, the deterministic solution of collapse and fracture initiation pressures are firstly presented in three cases that describe the behavior of wellbore: (1) immediately after drilling (i.e., undrained problem); (2) at long term due to the steady-state fluid flow (permeable boundary case) and (3) by neglecting the variation of initial pore pressure (i.e., impermeable boundary condition case). Based on these deterministic solutions, the key parameters of different sources of anisotropy (initial stress state, poro-elastic and strength properties of rock mass) are highlighted through sensitivity analysis. Then, the famous Monte Carlo Simulation (MCS) is undertaken to quantify the uncertainty effect on the probability of success of safe mud pressure window of the wellbore. We also present an adaptation of the Kriging metamodeling technique to study the stability of wellbore. In comparison with the referent solution of MCS, the Kriging metamodel provides a high accuracy and can be used as a performant tool for the probabilistic assessment of wellbore. The consideration of anisotropy combined with uncertainty, and the hydraulically boundary condition around wellbore in this work allows us to complete the contributions in the literature and confirm their strong effect on the design of wellbore.

General analytical solutions for the equal‐strain consolidation of prefabricated vertical drain foundation in unsaturated soils under time‐dependent loading

AbstractExternal disturbances were scarcely considered in the consolidation of unsaturated vertical drain foundation, in which the ideally permeable or impermeable boundary conditions were always adopted to investigated the consolidation patterns. In view of this, by taking the flows along the vertical and radial orientations simultaneously, this study proposes the general analytical solutions for the axisymmetric consolidation of prefabricated vertical drain foundations in unsaturated soils which consider drain resistance, smear effect, and time‐dependent loading under the equal‐strain assumption. Through theories related to consolidation in unsaturated soils, the governing equations of air and water phases in the matrix form are proposed. Subsequently, Fourier series expansion, matrix analysis and decoupling method are used to acquire the analytical solutions. The separate expressions of smear coefficient and drain resistance factors, which can represent the potential influence of the external boundaries more obviously, are included in the solutions. Degradation validity method is used to verify the credibility of the current solutions, and the vertical flow is found to promote the consolidation process under external conditions. Finally, the solutions obtained in this paper can provide guidance for more complicated consolidation problems in unsaturated soils.

Numerical simulation of the free surface flow passing two hydrofoils

In this paper, a numerical method based on potential flow theory is proposed to investigate the hydrodynamic interaction between two foils with free surface effect. The integral equation is solved using boundary element method. On foil surfaces, the impermeable boundary condition is satisfied. On free surface, nonlinear kinematic and dynamic boundary conditions are satisfied. A robust numerical model is proposed to avoid the complicated vortical interference between the two foils. A fully nonlinear scheme is developed for the numerical simulation. The proposed method is validated by extensive convergence studies. The hydrodynamic loadings on the foils are investigated in time domain. Two major quantitative findings are discussed. Firstly, the free surface elevation could be weakened significantly if the two foils are appropriately arranged. Secondly, the lifting force of the following foil varies periodically with the increase of the horizontal distance between the foils.

On the initial conditions for the problem of flow rising in front of an obstacle

Flowing water encounters a fixed obstacle, which causes a subsequent unsteady process whereby the water level rises. This process is modeled by the Euler equations with natural initial conditions and boundary conditions. If the impermeability boundary conditions of the obstacle do not match the initial data, a curious and unexpected phenomenon occurs: the velocity and pressure almost instantly depart from the initial values, and their new values should be considered as the “forced” initial conditions. The forced initial conditions depend on the real conditions but differ from them in the case of the mismatch mentioned above. The problem analysis assumes a smooth and local deformation of the boundary conditions from the given initial values at the initial time to the real (impermeability) values after the short transition period (and in the limit as the transition period approaches zero). A similar phenomenon may occur for hyperbolic systems of equations in general.

Practical computation of the diffusion MRI signal based on Laplace eigenfunctions: permeable interfaces.

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal, called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the tissue geometry and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. In a previous publication, we presented a simulation framework that we implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for biological cells subject to impermeable membrane boundary conditions. In this work, we extend our simulation framework to include geometries that contain permeable cell membranes. We describe the new computational techniques that allowed this generalization and we analyze the effects of the magnitude of the permeability coefficient on the eigendecomposition of the diffusion and Bloch-Torrey operators. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

Nonlinear effects in memristors with mobile vacancies.

Because local concentration of vacancies in any material is bounded, their motion must be accompanied by nonlinear effects. Here, we look for such effects in a simple model for electric field-driven vacancy motion in a memristor, solving the corresponding nonlinear Burgers’ equation with impermeable nonlinear boundary conditions exactly. We find non-monotonous relaxation of the resistance while switching between the stable (‘on’ and ‘off’) states, and qualitatively different dependencies of switching time (under applied current) and relaxation time (under no current) on the memristor length. Our solution can serve as a useful benchmark for simulations of more complex memristor models.

LAMB’S PROBLEM AND CHARACTERISTIC OF ENERGY TRANSMISSION IN UNSATURATED HALF-SPACE

Because unsaturated soil is widely distributed on the earth’s surface, when the traditional saturated two-phase medium theory is used for dynamic analysis, the results are often inconsistent with the actual situation. Aiming at this problem, this paper takes unsaturated elastic half-space as the research object, firstly based on continuum mechanics and porous media theory, and then considers the basic equations of mass conservation equation, momentum conservation equation, constitutive equation and effective stress principle of each phase in unsaturated porous media, and finally, we established a dynamic control equation in which skeleton displacement, pore water pressure and pore gas pressure are basically unknown quantities. Aiming at the dynamic response and energy transmission of the unsaturated half-space surface under the action of vertical concentrated harmonic loads, an axisymmetric calculation model of the classical Lamb problem in the frequency domain is established. The Helmholtz decomposition method is used and the displacement component of the skeleton uses the potential function Φ and Ψ to represent, and combined with the constitutive equation, the analytical solutions of physical quantities such as the displacement field and energy field of the half-space surface under different boundary conditions are obtained. Finally, influencing factors such as load parameters (excitation frequency) and material parameters (saturation, permeability coefficient) are analyzed and discussed through numerical examples. The results show that: (1) An increase in saturation or a decrease in excitation frequency will increase the surface displacement amplitude of the unsaturated half-space; (2) When the permeability coefficient drops to a critical value, the surface displacement amplitude will tend to a limit value, and the influence of permeability coefficient under permeable (gas) boundary and impermeable (gas) boundary conditions shows obvious difference.

Surface effects on a mode-III reinforced nano-elliptical hole embedded in one-dimensional hexagonal piezoelectric quasicrystals

To effectively reduce the field concentration around a hole or crack, an anti-plane shear problem of a nano-elliptical hole or a nano-crack pasting a reinforcement layer in a one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) is investigated subject to remotely mechanical and electrical loadings. The surface effect and dielectric characteristics inside the hole are considered for actuality. By utilizing the technique of conformal mapping and the complex variable method, the phonon stresses, phason stresses, and electric displacements in the matrix and reinforcement layer are exactly derived under both electrically permeable and impermeable boundary conditions. Three size-dependent field intensity factors near the nano-crack tip are further obtained when the nano-elliptical hole is reduced to the nano-crack. Numerical examples are illustrated to show the effects of material properties of the surface layer and reinforced layer, the aspect ratio of the hole, and the thickness of the reinforcing layer on the field concentration of the nano-elliptical hole and the field intensity factors near the nano-crack tip. The results indicate that the properties of the surface layer and reinforcement layer and the electrical boundary conditions have great effects on the field concentration of the nano-hole and nano-crack, which are useful for optimizing and designing the microdevices by PQC nanocomposites in engineering practice.

Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law

<p style="text-indent:20px;">This paper is concerned with the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem and the initial-boundary value problem in the half space with impermeable wall boundary condition for a scalar conservation laws with an artificial heat flux satisfying Cattaneo's law. In our results, although the <inline-formula><tex-math id="M1">\begin{document}$ L^2\cap L^\infty- $\end{document}</tex-math></inline-formula>norm of the initial perturbation is assumed to be small, the <inline-formula><tex-math id="M2">\begin{document}$ H^1- $\end{document}</tex-math></inline-formula>norm of the first order derivative of the initial perturbation with respect to the spatial variable can indeed be large. Moreover the far fields of the artificial heat flux can be different. Our analysis is based on the <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> energy method.</p>