Incompressible Medium Research Articles

On Projectors to Subspaces of Vector-Valued Functions Subject to Conditions of the Divergence-Free Type

We study operators that project a vector-valued function υ ∈ W1,2(Ω, ℝd) to subspaces formed by the condition that the divergence is orthogonal to a certain amount (finite or infinite) of test functions. The condition that the divergence is equal to zero almost everywhere presents the first (narrowest) limit case while the integral condition of zero mean divergence generates the other (widest) case. Estimates of the distance between υ and the respective projection on such a subspace are important for analysis of various mathematical models related to incompressible media problems (especially in the context of a posteriori error estimates. We establish different forms of such estimates, which contain only local constants associated with the stability (LBB) inequalities for subdomains. The approach developed in the paper also yields two-sided bounds of the inf-sup (LBB) constant.

Electromagnetic internal gravity waves in an ideally conducting incompressible medium

Properties and peculiarities of linear 3D - propagation of electromagnetic internal gravity waves in an ideally conducting incompressible medium embedded in a uniform magnetic field are studied. Local and non-local (plane waves) approaches are applied. It is shown that ordinary internal gravity waves couple with Alfven waves. Associated partial differential equations and dispersion relations are obtained. New branches of oscillations are revealed. The results obtained may be applicable to the Earth's ionosphere and solar atmosphere.

О единственности классического решения начально-краевой задачи течения вязкоупругой среды Максвелла с нулевыми скоростями на границе

Рассматривается система уравнений с вращательной производной Яуманна, описывающая движения вязкоупругой несжимаемой среды Максвелла в ограниченной области трехмерного пространства в приближении Стокса (без конвективных слагаемых как в уравнении движения, так и в реологическом соотношении). Выбор объективной производной Яуманна в реологическом соотношении обусловлен наличием энергетического тождества, справедливость которого не удается доказать для случаев использования верхней и нижней конвективных производных в качестве объективных производных. Доказывается единственность классического решения начально-краевой задачи с условием прилипания на границе области течения в классе достаточно гладких функций в предположении, что система обладает четырьмя различными вещественными звуковыми характеристиками. Доказательство основано на применении интегральных оценок и использовании условия гиперболичности системы. Для наглядности выкладки проводятся для двумерного случая, однако нигде не используются свойства двухмерности. Отличия от трехмерного случая носят чисто технический характер. Отмечается, что наличие конвективных членов в уравнениях движения не препятствует доказательству единственности. Единственность первоначально доказывается для малого промежутка времени, затем стандартным образом утверждение распространяется на промежуток произвольной длины. DOI 10.14258/izvasu(2018)4-17

Bending and Deformation Waves on the Surface of a Volumetrically Charged Cylindrical Jet of Dielectric Liquid Moving in Material Medium

Analytical research into the stability of axially asymmetric capillary waves on the surface of a volumetrically charged cylindrical jet of an ideal incompressible dielectric liquid that moves in an ideal incompressible dielectric medium made it possible to specify the conditions for realization of instability of waves with azimuthal numbers 1 and 2, which corresponds to excitement of bending and deformation waves. It was found that realization of the Kelvin-Helmholtz instability occurs on the media interface because of a tangential jump of the field of velocities on the jet surface for the axially asymmetric waves. The linear dimensions of wave numbers' ranges of unstable waves and the sizes of increments of instability are defined by values of the charging Weber’s parameters. The variation of dielectric permeability of a jet and medium affects the characteristics of wave motions, including those in the range of instability realization.

A modified formulation of quasi-linear viscoelasticity for transversely isotropic materials under finite deformation.

The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in TI, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.

On momentum transfer and external work done to clamped elasto-plastic beams in an air blast

This paper revisits a classical fluid-structure interaction problem on the momentum transfer and external work done to structures by a blast wave propagating in a nonlinear compressible medium (air). A fully-clamped ductile beam system is considered here, in place of the rigid free-standing plate used in the KNR theory (Kambouchev et al., 2006 [2], 2007 [3]), which takes into account limits to material deformation, boundary compliance and boundary failure. The model is used to critically re-evaluate the impulse imparted to the beam, and the subsequent external work done by a blast wave, for different modes of the beam deformation compared to its free-standing counterpart of identical mass per unit area (considered previously by others). It will be shown that the maximum beam deflection, transmitted impulse and external work done by the interface pressure are insensitive to FSI effects if loading is sufficient severe to induce boundary failure. Comparisons are also made between the structural performance of the beam system during fluid structure interaction in a compressible and incompressible fluid medium.

Fluid-structure interaction (FSI) analysis of stent-graft for aortic endovascular aneurysm repair (EVAR): Material and structural considerations

The effect of hemodynamic load on various stent-graft designs used for endovascular aneurysm repair (EVAR) in cardiovascular treatments is studied using a numerical fluid-structure interaction (FSI) analysis that couples computational fluid dynamics (CFD) and finite element analysis (FEA). Radial displacements, mechanical stresses, wall shear stress and wall compliance quantities are evaluated for four stent materials and one graft material. The strut thickness is varied from 0.3 mm to 1 mm. The materials are assumed linearly elastic and isotropic while blood is assumed as a Newtonian and incompressible medium with a pulsatile and turbulent flow profile. Time dependent pressure conditions are assumed at the inlet and outlet. Results are benchmarked against a study taken from the literature and indicate that the stent material and the strut thickness greatly influence the mechanical behavior of the structure. This computational study will serve as an additional tool to vascular surgeons when assessing the choice of material and design for stent-graft recipients.

Symmetries in equations of incompressible viscoelastic Maxwell medium*

We consider unsteady flows of incompressible viscoelastic Maxwell medium with upper, low, and Jaumann convective derivatives in the rheological constitutive law. We give characteristics of a system of equations that describe a three-dimensional motion of such a medium for all three types of convective derivative. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. We study group properties of this system in the two-dimensional case. On this basis, we choose submodels of the Maxwell model that can be reduced to hyperbolic ones. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. We also present concrete examples of invariant solutions.

Mathematical model for axisymmetric steady flow within layered structures with finite or infinitely high bulk viscosity

A mathematical model was developed for viscous incompressible medium; this model comprises a small parameter for regularization of ill-posed problem. Starting from the analogy with strain problems of axisymmetric laminar structures which include the elastic orthotropic and elastic volume-incompressible layers with a connection to viscous laminar fluid flow, we have developed a method providing a single algorithm for computation of velocity field, stresses, and other parameters of laminar composite material. The example of calculation for the binding agent flow during pultrusion formation is presented.

Computational fluid–structure interaction analysis of blood flow on patient-specific reconstructed aortic anatomy and aneurysm treatment with Dacron graft

We use the fluid–structure interaction (FSI) capability in Abaqus to evaluate radial displacements, von Mises stresses and wall shear stresses (WSS) on the human aorta in response to the blood flowing through it. Complications arise when aneurysm is detected and causes the wall to thinner so much that rupture may result. We use the Materialise suite, a specialty software to reconstruct a three-dimensional geometry of the aorta from two-dimensional computerized axial tomography (CAT) images. Results are compared to those obtained on a healthy individual. Blood is assumed to be a Newtonian and incompressible medium and the blood flow is taken as pulsatile, fully developed and turbulent. The model used for aorta is Holzapfel–Gasser–Ogden (HGO), a sophisticated hyperelastic model that can describe biological tissues. We also study the behavior of Dacron, a polyester fabric used as graft in aortic surgical repair. Here, Dacron is represented by a neo-Hookean (isotropic) hyperelastic model. Time-dependent pressure conditions are assumed at the inlet and outlet of the resulting structure. Results indicate that using a patient-specific geometry for the aorta yields additional insight on the state of the stresses applied on the aortic walls. In addition, stress contours on the Dacron are comparable to those obtained on a healthy patient and stresses evaluated at the interface of the biological tissues and the fabric, provide useful information regarding the suture strength needed during surgery. In the case of aneurysm, our simulation results agree well with experimental data taken from the literature particularly with regard to WSS which can be used to assess the seriousness of the aneurysm condition. If an idealized cylindrical shell is used in place of the reconstructed anatomy, the von Mises stress values do not differ much but it underestimates the values of WSS which could interpreted as the presence of an aneurysm when there is not. Among the novel contributions, the present FSI model has the ability to predict the Dacron’s response to realistic hemodynamic loading conditions.

On exact analytical solutions of equations of Maxwell incompressible viscoelastic medium

Unsteady two-dimensional flows of incompressible viscoelastic Maxwell medium with upper, low and corotational convective derivatives in the rheological constitutive law are considered. A class of partially invariant solutions is analyzed. Using transition to Lagrangian coordinates, an exact solution of the problem of unsteady flow near free-stagnation point was constructed. For the model with Johnson–Segalman convected derivative and special linear dependence of the vertical component of velocity, the general solutions were derived.

SPATIAL VIBRATIONS OF A PIPELINE HAVING A SLIDING SUPPORT UNDER THE ACTION OF VARIABLE INTERNAL PRESSURE

This paper studies spatial vibrations of a pipeline having a sliding 'weightless' support under the action of variable internal pressure. The pipeline vibrations occur around an axis running through two supports, one of which is assumed to be immovable and the other slides freely on an ideally smooth horizontal plane. The pipe filled with an incompressible transport medium is surrounded by a viscous incompressible fluid. The internal pressure in the pipe-line is given by the harmonic law. Gravitational forces, Coriolis inertial forces, Archimedes buoyant force, viscous drag forces and those associated with the acceleration of the pipe transverse motion in the surrounding medium are taken into account. In this case the veloc-ity of the transport medium motion, flow friction and linear inertial forces are not taken into account. Deformations of the pipe due to its exit out of the flexure plane are assumed to be small. That is why, spatial vibrational movements of the pipeline are considered to consist of transferable rotational movements around the axis running through the supports and rela-tive flexural displacements in the flexure plane. As a result of these assumptions, the solution to the problem is reduced to integrating a set of two nonlinear partial differential equations describing rotational and flexural vibrations of the pipeline. The function that satisfies the boundary conditions is taken according to the first basic mode. Then using the Bubnov-Galerkin procedure, this set is reduced to two nonlinear ordinary differential equations regarding the bending angle and deflection at mid-span of the pipe in terms of time. The Runge-Kutt numerical method is applied to integrate the resultant set of equations under specific initial conditions. This numerical solution is then subjected to the discrete Fourier transform and Poincare mapping. Calculations were performed for a steel pipeline filled with fluid and gas. The pipeline executed vibrations in air and water media. The magnitude of the average, the amplitude and the initial phase of the internal pressure variable component took the only one value each, and the frequency of this pressure took three values. The computational results are given as time-dependence graphs of the mid-span deflection dynamic constituent, bending angle and sliding support displacement, their phase pattern, Fourier spectra and Poincare maps. The mid-span trajectories are also constructed to give a more visual representation of the pipeline in space over a given time interval.

Radiative Instability of an Unbounded Jet Flow

The problem of the linear dynamics of disturbances of an unbounded jet flow with a piecewise linear velocity profile is considered. The stable disturbances of a flow in an incompressible medium are so-called flow waves localized near a vorticity jump. In this work, it is shown that the amplitudes of these waves slowly increase in a compressible medium due to acoustic radiation; i.e., an instability appears. The asymptotic solution to the problem for small Mach numbers is represented in terms of Airy functions. An analytic expression for the growth increment of the disturbances is obtained.

Application of Acoustoelasticity to Evaluate Nonlinear Modulus in Ex Vivo Kidneys.

Currently, dynamic elastography techniques estimate the linear elastic shear modulus of different body tissues. New methods that investigate other properties of soft tissues such as anisotropy, viscosity, and shear nonlinearity would provide more information about the structure and function of the tissue and might provide a better contrast than tissue stiffness and hence provide more effective diagnostic tools for some diseases. It has previously been shown that shear wave velocity in a medium changes due to an applied stress, a phenomenon called acoustoelasticity (AE). Applying a stress to compress a medium while measuring the shear wave velocity versus strain provides data with which the third-order nonlinear shear modulus can be estimated. To evaluate the feasibility of estimating , we evaluated ten ex vivo porcine kidneys embedded in 10% porcine gelatin to mimic the case of a transplanted kidney. Under assumptions of an elastic incompressible medium for AE measurements, the shear modulus was quantified at each compression level and the applied strain was assessed by measuring the change in the thickness of the kidney cortex. Finally, was calculated by applying the AE theory. Our results demonstrated that it is possible to estimate a nonlinear shear modulus by monitoring the changes in strain and due to kidney deformation. The magnitudes of are higher when the compression is performed progressively and when using a plate attached to the transducer. Nevertheless, the values obtained for are similar to those previously reported in the literature for breast tissue.

A stabilised mixed meshfree method for incompressible media: Application to linear elasticity and Stokes flow

This paper addresses some problems of modelling incompressible media using meshfree methods with a stabilised mixed formulation. The aims of this paper are twofold: to avoid the volumetric locking that occurs at the incompressible limit; and to circumvent the inf–sup condition while approximating both displacement and pressure with the same set of nodes and shape functions. We employ a modified polynomial pressure projection stabilisation method to stabilise and filter spurious pressure modes, where the gradient is corrected as proposed by Duan et al. (2014). The gradient correction is used to counteract the lack of variational consistency when numerical integration of the rational shape functions is performed, a threat that could undermine the stabilisation effort. The resulting novel method passes the mixed patch tests, and is demonstrated to be effective and optimally accurate in linear elasticity and Stokes flow test cases.

Control for well-posedness about a class of non-Newtonian incompressible porous medium fluid equations

Considering the non-Newtonian fluid equation of incompressible porous media, using the properties of operator semigroup and measure space and the principle of squeezed image, Fourier analysis and a priori estimate in the measurement space are used to discuss the non-compressible porous media, the properness of the solution of the equation, its gradual behavior and its topological properties. Through the diffusion regularization method and the compressed limit compact method, we study the overall decay rate of the solution of the equation in a certain space when the initial value is sufficient. The decay estimation of the solution of the incompressible seepage equation is obtained, and the asymptotic behavior of the solution is obtained by using the double regularization model and the Duhamel principle.

Characteristic properties of the system of equations for an incompressible viscoelastic Maxwell medium

Characteristics of a system of equations that describe three-dimensional motion of an incompressible viscoelastic Maxwell medium with the upper and lower convective derivatives and the rotational Jaumann derivative being used in the rheological relation are calculated. An initial-boundary-value problem is formulated for the system linearized in the vicinity of the state at rest, and its unique solvability is established.

Electrostatic instability of the surface of a volume charged jet of dielectric liquid moving relative to the surrounding medium

The laws of implementation of electrostatic instability of the surface of a cylindrical volume charged jet of an ideal incompressible dielectric liquid moving relative to the ideal incompressible dielectric medium and the stability of bending-deformation capillary waves developed on the surface are investigated analytically. It is found that there are thresholds for the critical conditions of implementation of the instability with respect to the jet velocity relative to the medium (Weber number) and with respect to the electric space charge (relative to the ratio of the electrostatic pressure on the jet surface to the Laplace pressure). The critical analytic dependence between these dimensionless parameters is found.

Dense CO Adlayers as Enablers of CO Hydrogenation Turnovers on Ru Surfaces.

High CO* coverages lead to rates much higher than Langmuirian treatments predict because co-adsorbate interactions destabilize relevant transition states less than their bound precursors. This is shown here by kinetic and spectroscopic data-interpreted by rate equations modified for thermodynamically nonideal surfaces-and by DFT treatments of CO-covered Ru clusters and lattice models that mimic adlayer densification. At conditions (0.01-1 kPa CO; 500-600 K) which create low CO* coverages (0.3-0.8 ML from in situ infrared spectra), turnover rates are accurately described by Langmuirian models. Infrared bands indicate that adlayers nearly saturate and then gradually densify as pressure increases above 1 kPa CO, and rates become increasingly larger than those predicted from Langmuir treatments (15-fold at 25 kPa and 70-fold at 1 MPa CO). These strong rate enhancements are described here by adapting formalisms for reactions in nonideal and nearly incompressible media (liquids, ultrahigh-pressure gases) to handle the strong co-adsorbate interactions within the nearly incompressible CO* adlayer. These approaches show that rates are enhanced by densifying CO* adlayers because CO hydrogenation has a negative activation area (calculated by DFT), analogous to how increasing pressure enhances rates for liquid-phase reactions with negative activation volumes. Without these co-adsorbate effects and the negative activation area of CO activation, Fischer-Tropsch synthesis would not occur at practical rates. These findings and conceptual frameworks accurately treat dense surface adlayers and are relevant in the general treatment of surface catalysis as it is typically practiced at conditions leading to saturation coverages of reactants or products.

On the stability of capillary waves on the surface of a space-charged dielectric liquid jet moving in a material medium

We have derived and analyzed the dispersion equation for capillary waves with an arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a space-charged cylindrical jet of an ideal incompressible dielectric liquid moving relative to an ideal incompressible dielectric medium. It has been proved that the existence of a tangential jump of the velocity field on the jet surface leads to a periodic Kelvin–Helmholtz- type instability at the interface between the media and plays a destabilizing role. The wavenumber ranges of unstable waves and the instability increments depend on the squared velocity of the relative motion and increase with the velocity. With increasing volume charge density, the critical value of the velocity for the emergence of instability decreases. The reduction of the permittivity of the liquid in the jet or an increase in the permittivity of the medium narrows the regions of instability and leads to an increase in the increments. The wavenumber of the most unstable wave increases in accordance with a power law upon an increase in the volume charge density and velocity of the jet. The variations in the permittivities of the jet and the medium produce opposite effects on the wavenumber of the most unstable wave.