Incompressible Medium Research Articles

Behavior of entropy in non-classical heat conduction of incompressible media

Abstract The behavior of entropy of a model isolated system in which there is non-classical heat conduction is considered. Within the thermodynamic formal-ism developed, expressions are obtained for the entropy flux and sourceand also for the total entropy of the isolated system. The approach pro-posed leads to a strictly monotonic dependence of the total entropy on timefor the model system considered. 1 Introduction Non-classical heat transfer is well known for metals, porous media, andcomposite materials [1, 2]. When moving media is considered (e.g., thesurface of a rotating cylinder subject to laser irradiation, which is used inthe fabrication of microstructures and synthesis of new materials), the heattransfer equation is given by [3]:ddtdTdtCdTdtD 2r T; (1)where d=dt(D @=@t C v r ) is the material time derivative, v is the mediumvelocity, T is the temperature, is the relaxation time, D =.c/ is thethermal diffusivity, is the thermal conductivity, c is the heat capacity perunit mass, is the mass density. It is assumed that density and all thecoefficients are constant values.

A poroelastic model valid in large strains with applications to perfusion in cardiac modeling

This paper is motivated by the modeling of blood flows through the beating myocardium, namely cardiac perfusion. As in other works, perfusion is modeled here as a flow through a poroelastic medium. The main contribution of this study is the derivation of a general poroelastic model valid for a nearly incompressible medium which experiences finite deformations. A numerical procedure is proposed to iteratively solve the porous flow and the nonlinear poroviscoelastic problems. Three-dimensional numerical experiments are presented to illustrate the model. The first test cases consist of typical poroelastic configurations: swelling and complete drainage. Finally, a simulation of cardiac perfusion is presented in an idealized left ventricle embedded with active fibers. Results show the complex temporal and spatial interactions of the muscle and blood, reproducing several key phenomena observed in cardiac perfusion.

Optimal control in a model of the motion of a viscoelastic medium with objective derivative

In this paper we consider the Jeffreys model of the motion of a viscoelastic incompressible medium with the Yaumann derivative. Within this model we study the optimal control problem for the right-hand sides of the initial boundary value problem. We prove the existence of the optimal strong solution.

Differential constitutive equations of incompressible media with finite deformations

A method is proposed for constructing a system of constitutive equations of an incompressible medium with nonlinear dissipative properties with finite deformations. A scheme of the mechanical behavior of a material is used, in which the points are connected by horizontally aligned elastic, viscous, plastic, and transmission elements. The properties of each element of the scheme are described with the use of known equations of the nonlinear elasticity theory, the theory of nonlinear viscous fluids, and the theory of plastic flow of the material under conditions of finite deformations of the medium. The system of constitutive equations is closed by equations that express the relation between the deformation rate tensor of the material and the deformation rate tensor of the plastic element. Transmission elements are used to take into account a significant difference between macroscopic deformations of the material and deformations of elements of the medium at the structural level.

A universal programmable fiber architecture for the representation of a general incompressible linearly elastic material as a fiber-reinforced fluid

Biological materials typically consist of elastic fibers immersed in an incompressible aqueous milieu. We consider the generality of an elastic material expressed as a fiber-reinforced incompressible fluid. We show that, in the linear regime, any (possibly inhomogeneous and/or anisotropic) incompressible elastic material can be represented as a collection of fifteen families of straight, parallel elastic fibers embedded in an incompressible medium. We can choose these fiber directions to correspond to the fifteen diagonals of an icosahedron that connect the midpoints of its antipodal edges. This fiber architecture, together with the incompressible medium in which it is immersed, is universal and programmable in the sense that its elastic constants can be chosen to model any linear incompressible elastic material, without having to adapt the fiber architecture to the actual microstructure of the material. An explicit algorithm is given to compute the local elastic constants for each fiber direction in terms of the local components of the elasticity tensor. Optimality properties of the icosahedral fiber architecture are conjectured, and numerical evidence in support of these conjectures is presented.

Ray expansions of solutions around fronts as a shock-fitting tool for shock loading simulation

In shock loading computations based on an implicit finite-difference scheme, the surfaces of velocity discontinuity and the discontinuity sizes are determined by computing an asymptotic (ray) expansion of the solution behind the front surfaces at every time step. The method for constructing ray expansions is based on a recurrence formulation of the geometric and kinematic consistency conditions for discontinuities of the derivatives of functions that are discontinuous on a moving surface. The algorithm is illustrated by computing a simple example of the out-of-plane motion of an incompressible elastic medium.

Exact solutions of the equations of motion for an incompressible viscoelastic Maxwell medium

The unsteady plane-parallel motion of a incompressible viscoelastic Maxwell medium with constant relaxation time is considered. The equations of motion of the medium and the rheological relation admit an extended Galilean group. The class of solutions of this system which are partially invariant with respect to the subgroup of the indicated group generated by translation and Galilean translation along one of the coordinate axes is studied. The system does not have invariant solutions, and the set of partially invariant solutions is very narrow. A method for extending the set of exact solutions is proposed which allows finding solutions with a nontrivial dependence of the stress tensor elements on spatial coordinates. Among the solutions obtained by this method, the solutions describing the deformation of a viscoelastic strip with free boundaries is of special interest from a point of view of physics.

Hydrodynamic stability of flows in vessels with convoluted phase movement

The flow stability of a viscous incompressible medium moving between two permeable cylindrical surfaces toward the axis of convolution, which are rotating at arbitrary angular velocities, is examined in general form. It is established as a result of analysis that the convoluted flow loses stability in two cases: firstly, when the radial flow is of low intensity (on the boundary between the central and peripheral vertices), and, secondly, when the internal layers of the convoluted flow begin to rotate more rapidly than the external layers, which is characteristic of a peripheral vortex, where the flow may lose stability.

Investigation into the fluid—structure interaction of a hydraulically damped rubber mount on the basis of finite element analysis

This paper investigates the working process simulation of a hydraulically damped rubber mount (HDM) based on finite element (FE) analysis of the fluid—structure interaction (FSI). The direct computing method of coupled FSI formulation in arbitrary Lagrangian—Eulerian coordinates and the mixed displacement—pressure FE formulation for the incompressible medium, rubber, are adopted. The constitutive law of rubber is determined by FE analysis of incompressible material. A proportion mesh control algorithm helps to overcome fluid mesh distortion. Comparison of the predicted and experimental static elasticities verifies the effectiveness of the presented modelling approach of an HDM. Analysis of fluid pressure—velocity fields and deformation—stress fields of rubber components under typical working conditions clarifies the working process of the HDM, which can help to evaluate the carrying capacity, to identify the chamber volumetric characteristics, and to determine the fluid field distribution and stress distribution of rubber parts for the structural design of the HDM.

TopSpot® Vario: a novel microarrayer system for highly flexible and highly parallel picoliter dispensing

The standard TopSpot technology has been successfully used in the recent years for the highly parallel nanoliter dispensing of bio-chemical substances for microarray printing. It is based on a pneumatically actuated printhead which enables non-contact microarray fabrication at a pitch of typically 500 microm. This paper reports on a new and improved way of operating the printheads termed TopSpot Vario technology, using an incompressible material between the piezo actuator and the dispensing medium. The advantage of the incompressible medium is the direct relation between the displacement amplitude and the ejected liquid volume. Earlier reports stated that the filling of the printheads is a key issue. In this paper we report on the implementation and characterization of a new printhead design including microchannels for bubble free priming of blind channels. Microarray fabrication was successfully conducted with four different types of protein.

Forces and stresses acting on fusion pore membrane during secretion

To assess the forces and stresses present in fusion pore during secretion the stationary convective flux of lipid through a fusion pore connecting two planar membranes under different tensions was investigated through computer simulations. The physics of the problem is described by Navier–Stokes equations, and the convective flux of lipid was evaluated using finite element method. Each of the membrane monolayer is considered separately as an isotropic, homogeneous and incompressible viscous medium with the same viscosity. The difference in membrane tensions, which is simulated as the pressure difference at two ends of each monolayer, is the driving force of the lipid flow. The two monolayers interact by sliding past each other with inter-monolayer frictional viscosity. Fluid velocity, pressure, shear and normal stresses, viscous and frictional dissipations and forces were calculated to evaluate where the fusion pore will deform, extend (or compress) and dilate. The pressure changes little in the planar sections, whereas in the toroidal section the change is rapid. The magnitude of lipid velocity peaks at the pore neck. The radial lipid velocity is zero at the neck, has two peaks one on each side of the pore neck, and diminishes without going to zero in planar parts of two monolayers. The peaks are of opposite signs due to the change of direction of lipid flow. The axial velocity is confined to the toroidal section, peaks at the neck and is clearly greater in the outer monolayer. As a result of the spatially highly uneven lipid flow the membrane is under a significant stress, shear and normal. The shear stress, which indicates where the membrane will deform without changing the volume, has two peaks placed symmetrically about the neck. The normal stress shows where the membrane may extend or compress. Both, the radial and axial normal stresses are negative (extensive) in the upper toroidal section and positive (compressive) in the lower toroidal section. The pressure difference determines lipid velocity and velocity dependent variables (shear as well as normal axial and radial stresses), but also contributes directly to the force on the membranes and critically influences where and to what extent the membrane will deform, extend or dilate. The viscosity coefficient (due to friction of one element of lipid against neighboring ones), and frictional coefficient (due to friction between two monolayers sliding past each other) further modulate some variables. Lipid velocity rises as pressure difference increases, diminishes as the viscosity coefficient rises but is unaffected by the frictional coefficient. The shear and normal stresses rise as pressure difference increases, but the change of the viscosity coefficients has no effect. Both the viscous dissipation (which has two peaks placed symmetrically about the neck) and much smaller frictional dissipation (which peaks at the pore neck) rise with pressure and diminish if the viscosity coefficient rises, but only the frictional dissipation increases if the frictional coefficient increases. Finally, the radial force causing pore dilatation, and which is significant only in the planar section of the vesicular membrane, is governed almost entirely by the pressure, whereas the viscosity and frictional coefficients have only a marginal effect. Many variables are altered during pore dilatation. The lipid velocity and dissipations (viscous and frictional) rise approximately linearly with pore radius, whereas the lipid mass flow increases supra-linearly owing to the combined effects of the changes in pore radius and greater lipid velocity. Interestingly the radial force on the vesicular membrane increases only marginally.

On some stationary mathematical models for Tornados and other funnel‐shaped rotating liquid and gas media

AbstractBased on Navier‐Stokes stationary equations a mathematical model is constructed for liquid and gas media with funnel‐shaped rotation observed in atmospheric and sea tornados. Both compressible and incompressible media are considered. The differential equations corresponding to the mathematical model are integrated in elementary functions and the solutions are represented by seven rotationally symmetrical orthogonal curvilinear coordinates applicable to different shapes of funnels.

A multi-grain model for migration recrystallization in polar ice

In the paper a multi-grain model for a migration recrystallization phenomenon in polar ice is presented. A single crystal of ice is treated as a transversely isotropic and incompressible medium which deforms by viscous creep. The anisotropic viscous behaviour of the ice crystal is described by a constitutive law that includes three microscopic viscosity parameters, and the macroscopic behaviour of the polycrystal is derived by adopting the Taylor–Voigt approximation of the velocity gradient homogeneity in the material. It is assumed that recrystallize, that is gradually disappear, these crystals which are most stressed, and at their expense new crystals are nucleated with the orientations that enhance the deformation of the polycrystal. The model predictions are illustrated by results of numerical simulations of simple flows, showing the evolution of the oriented structure of ice and the variation of macroscopic viscosities with increasing strains.

Uzawa method on semi-staggered grids for unsteady Bingham media flows

A finite difference scheme on semi-staggered grids is used for numerical simulation of unsteady flows of an incompressible viscoplastic Bingham medium. The Duvaut–Lions variational inequality is considered as a mathematical model. The convergence of the Uzawa method is proved. The efficiency of the method is demonstrated on a model problem with a known exact solution (plane Poiseuille flow) and on the unsteady driven cavity problem.

Small and large-amplitude gravitational instability of an elastically compressible viscoelastic Maxwell solid overlying an inviscid incompressible fluid: Dependence of growth rates on wave number and elastic constants at low Deborah numbers

To understand how elasticity affects convective instability of viscoelastic fluids near the viscous limit, we carried out numerical experiments of Rayleigh–Taylor instability of compressible viscoelastic fluids overlying inviscid inelastic substrata (hence with infinite viscosity and elasticity ratios). Unlike incompressible viscous fluids, for which growth becomes super-exponential when perturbations to the thickness of the unstable layer grow to several tens of percent of the thickness, for compressible viscoelastic fluids, super-exponential growth does not appear to develop at relatively low Deborah numbers, De = Δρgh/η ∼ 10 − 6 –10 − 4 (where Δρ is the density difference between unstable layer and substratum, g is gravity, h is the thickness of the unstable layer, and η is its viscosity) and for very high (> 10 6) ratios of viscosity between layer and substratum, which characterize large-scale geodynamic systems. This behavior differs from that of viscoelastic two-layer systems with higher Deborah numbers (> 10 − 4 ) and with smaller viscosity ratios (< 10 4) because, instead of accelerating the instability, as for incompressible media, elastic deformation may also retard growth in its most rapid phase as the amplitude of flow increases. For small De (< 10 − 3 –10 − 4 ), retardation of growth manifests itself in three ways: (1) while perturbations remain small, the commonly observed exponential growth is delayed, (2) during exponential growth, the growth rate decreases monotonically with decreasing De, and (3) when perturbations grow to large amplitude (> 100%), the exponential growth rate decreases, due to the formation of a compressible viscoelastic drop that has a distinguishable drop head and a stretched filament. Values of De appropriate for Earth's mantle suggest that in most circumstances elasticity will not affect the growth of lithospheric instabilities, but for high density contrasts (e.g. atypically warm lithosphere) elasticity may retard their growth. By contrast, for relatively large Deborah numbers (> 10 − 3 ), finite viscosity ratios (< 10 7), and small amplitudes of perturbations, elasticity accelerates growth of the instability.

Dynamic simulation of viscoelastic soft tissues in harmonic motion imaging application

A finite element model was built to simulate the dynamic behavior of soft tissues subjected to sinusoidal excitation during harmonic motion imaging. In this study, soft tissues and tissue-like phantoms were modeled as isotropic, viscoelastic, and nearly incompressible media. A 3D incompressible mixed u– p element of eight nodes, S1P0, was developed to accurately calculate the stiffness matrix for soft tissues. The finite element equations of motion were solved using the Newmark method. The Voigt description for tissue viscosity was applied to estimate the relative viscous coefficient from the phase shift between the response and excitation in a harmonic case. After validating our model via ANSYS simulation and experiments, a MATLAB finite element program was then employed to explore the effect of excitation location, viscosity, and multiple frequencies on the dynamic displacement at the frequency of interest.

압축공기로 움직이는 관 내부 수중 이동물체의 거동에 미치는 물의 압축성 영향

The motion of a projectile initiated by the release of highly pressurized air is simulated presuming the flow field as a two dimensional one. The effects of water compressibility on projectile movements are investigated, comparing results based on the Fluent VOF model where water is treated as an incompressible medium with those from the presently developed VOF scheme. The present model considers compressibility of both air and water. The Fluent results show that the body moves farther and at higher speeds than the present ones. As time proceeds, the relative difference of speed and displacement between the two results drops substantially, after acoustic waves in water traverse and return the full length of the tube several times. To estimate instantaneous accelerations, however, requires implementation of the water compressibility effect as discrepancies between them do not decrease even after several pressure wave cycles.

Modeling of compressible magnetohydrodynamic turbulence in electrically and heat conducting fluid using large eddy simulation

Many electrically and heat conducting fluid flows cannot be described within the framework of incompressible medium or by compressible magnetohydrodynamic equations on the assumption of polytropic (or adiabatic) process. Therefore, we consider a heat conducting compressible fluid with the use of an energy equation. Application of large eddy simulation approach to heat conducting compressible magnetohydrodynamics is considered. The system of the filtered magnetohydrodynamic equations with the total energy equation using the mass-weighted filtering procedure has been obtained. It is shown that novel subgrid-scale terms arise in the Favre-filtered equations due to the presence of a magnetic field in the total energy equation. Parametrizations of these extra terms are developed. In order to derive these subgrid-scale terms, we use an approach based on generalized central moments. Computations at various Mach numbers are made for decaying compressible magnetohydrodynamic turbulence. The obtained numerical large eddy simulation results are analyzed on the basis of comparison with results of numerical experiments performed by direct numerical simulation. Validity of large eddy simulation method is thus demonstrated. It is shown that consideration of the subgrid-scale terms in the total energy equation scarcely affects the kinetic and the magnetic energy for low and even high Mach number, while for the temperature the presence of subgrid-scale models in the energy equation is an important condition for improvement in calculation accuracy of thermodynamic quantities. The technique with the mass-weighted filtering and with the use of various types of subgrid-scale models provides good calculation accuracy for different problems for compressible fluid in the absence of discontinuities, associated with the appearance of shocks, in other words, when the value of the Mach number is low or moderate (that is, the flow is subsonic). For supersonic magnetohydrodynamic flows, it is necessary to use special numerical methods.

Nonlinear problem of the mechanics of an incompressible continuous medium

An iterative algorithm for solving the problem of deformation of incompressible plastic bodies has been built. The incremental flow theory and the relationship of general form connecting stress intensities with deformation rate intensities have been used.

Construction of approximate solutions of boundary value impact strain problems

The method for constructing approximate solutions of boundary value problems of impact strain dynamics in the form of ray expansions behind the strain discontinuity fronts is generalized to the case of curvilinear and diverging rays. This proposed generalization is illustrated by an example of dynamics of an antiplane motion of an elastic medium. The ray method is one of the methods for constructing approximate solutions of nonstationary boundary value problems of strain dynamics. It was proposed in [1, 2] and then widely used in nonstationary problems of mathematical physics involving surfaces on which the desired function or its derivatives have discontinuities [3–7]. A complete, qualified survey of papers in this direction can be found in [8]. This method is based on the expansion of the solution in a Taylor-type series behind the moving discontinuity surface rather than in a neighborhood of a stationary point. The coefficients of this series are the jumps of the derivatives of the unknown functions, for which, as a consequence of the compatibility conditions, one can obtain ordinary differential equations, i.e., discontinuity damping equations. In the case where the problem with velocity discontinuity surfaces is considered in a nonlinear medium, this method cannot be used directly, because one cannot obtain the damping equation. A modification of this method for the purpose of using it to solve problems of that type was proposed in [9–11], where, as an example, the solutions of several one-dimensional problems were considered. In the present paper, we show how this method can be transferred to the case of multidimensional impact strain problems in which the geometry of the ray is not known in advance and the rays become curvilinear and diverging. By way of example, we consider a simple problem on the antiplane motion of a nonlinearly elastic incompressible medium.