Effect of Compressible Foam Properties on Pressure Amplification During Shock Wave Impact

A comprehensive study is made of the influence of the physical properties of compressible open-cell foam blocks exposed to shock-wave loading, and particularly on the pressure distribution on the shock tube walls. Seven different foams are used, with three different shock Mach numbers, and three different slab lengths. Foam properties examined include permeability, density, stiffness, tortuosity and cell characteristics. The investigations concentrate on both side-wall and back-wall pressures, and the peak pressures achieved, as well as collapse velocities of the front face and the strength and nature of the reflected shock wave. The consequences of deviations from one-dimensionality are identified; primarily those due to wall friction and side-wall leakage. The results presented are the most comprehensive and wide ranging series conducted in a single facility and are thus a significant resource for comparison with theoretical and numerical studies. The different foams show significant differences in behavior, both in terms of peak pressure and duration, depending primarily on their density and permeability.This paper was based on work presented at the 2nd International Symposium on Interdisciplinary Shock Wave Research, Sendai, Japan on March 1–3, 2005.     

The Pressure Singularity for Compressible Stokes Flows in a Concave Polygon

A compressible Stokes system is studied in a polygon with one concave vertex. A corner singularity expansion is obtained up to second order. The expansion contains the usual corner singularity functions for the velocity plus an “associated” velocity singular function, and a pressure singular function. In particular the singularity of pressure is not local but occurs along the streamline emanating from the incoming concave vertex. It is observed that certain first derivatives of the pressure become infinite along the streamline of the ambient flow emanating from the concave vertex. Higher order regularity is shown for the remainder. This work was supported by the Com2MaC-SRC/ERC program of MOST/KOSEF (grant R11-1999-054), and by the U.S. National Science Foundation.     

Pulsatory Turbulent Compressible Fluid Flow and Propagation of Pressure Waves in a Channel

The properties of the damping coefficient and phase velocity of propagation of small-amplitude pressure waves as functions of the oscillation frequency are investigated for the turbulent flow of a weakly compressible fluid in a circular pipe. The wall friction is found by solving numerically the equation of motion and the relaxation equations for the turbulent shear stress and viscosity which provide the basis for a turbulent transfer model developed for unsteady conditions. The properties are explained in terms of an analysis of the calculated data on turbulent transfer. The results obtained are compared with experiments.     

Nonlinear Pressure Diffusion in Flow of Compressible Liquids Through Porous Media

In the flow of liquids through porous media, nonlinear effects arise from the dependence of the fluid density, porosity, and permeability on pore pressure, which are commonly approximated by simple exponential functions. The resulting flow equation contains a squared gradient term and an exponential dependence of the hydraulic diffusivity on pressure. In the limiting case where the porosity and permeability moduli are comparable, the diffusivity is constant, and the squared gradient term can be removed by introducing a new variable y, depending exponentially on pressure. The published transformations that have been used for this purpose are shown to be special cases of the Cole–Hopf transformation, differing in the choice of integration constants. Application of Laplace transformation to the linear diffusion equation satisfied by y is considered, with particular reference to the effects of the transformation on the boundary conditions. The minimum fluid compressibilities at which nonlinear effects become significant are determined for steady flow between parallel planes and cylinders at constant pressure. Calculations show that the liquid densities obtained from the simple compressibility equation of state agree to within 1% with those obtained from the highly accurate Wagner-Pru?  equation of state at pressures to 20 MPa and temperatures approaching 600 K, suggesting possible applications to some geothermal systems.     

Asymptotic Properties of Linearized Equations of Low Compressible Fluid Motion

Initial-boundary value problem for linearized equations of viscous barotropic fluid motion in a bounded domain is considered. Existence, uniqueness and estimates of weak solutions to this problem are derived. Convergence of the solutions towards the incompressible limit when compressibility tends to zero is studied.     

Mechanical Properties of Microcapsules Used in a Self-Healing Polymer

The elastic modulus and failure behavior of poly(urea-formaldehyde) shelled microcapsules were determined through single-capsule compression tests. Capsules were tested both dry and immersed in a fluid isotonic with the encapsulent. The testing of capsules immersed in a fluid had little influence on mechanical behavior in the elastic regime. Elastic modulus of the capsule shell wall was extracted by comparison with a shell theory model for the compression of a fluid filled microcapsule. Average capsule shell wall modulus was 3.7 GPa, regardless of whether the capsule was tested immersed or dry. Microcapsule diameter was found to have a significant effect on failure strength, with smaller capsules sustaining higher loads before failure. Capsule size had no effect on the modulus value determined from comparison with theory.     

Response of an Elastic Body whose Heat Conduction Is Pressure Dependent

The properties of many real materials such as the viscosity, thermal and electrical conductivity, specific heat, relaxation time, as well as optical properties, depend upon the pressure to which the body is subject. For instance, the viscosity of fluids can vary by several orders of magnitude due to the variation in the pressure. In this paper we investigate the change in the response of an elastic solid due to the thermal conductivity being pressure dependent. It is well known that higher pressure leads to reduced molecular mobility, in rubber-like materials, leading in turn to higher cross-linking reaction rates. We find that the response of the solid is quite different from the classical response that is obtained by using Fourier??s law of heat conduction. The theoretical predictions according to the assumption that the thermal conductivity is pressure dependent, are in keeping with experimental results concerning the vulcanization of rubbers wherein one observes the conduction to be dependent on the pressure. To our knowledge, this is the first theoretical study that evaluates the response of non-linear elastic solids due the thermal conductivity depending on the pressure.     

The Stability of a Compressible Rayleigh Layer

The aim of this work is to determine the linear stability of a compressible Rayleigh layer and to ascertain what role unsteady effects play. A Rayleigh layer is formed when an infinite flat plate is impulsively set in motion in its own plane with constant velocity beneath an initially quiescent fluid. When the fluid is compressible there is a motion both parallel and normal to the plate. The classical boundary-layer scaling is employed to determine solutions which are expressed in terms of a similarity variable and are valid for a large range of Mach, Prandtl and Reynolds numbers. Solutions are presented for both an adiabatic and iso-thermal temperature boundary condition at the plate. The temporal stability of the flow is considered by solving an Orr–Sommerfeld system: here the underlying flow is calculated at a certain time and the instantaneous stability to viscous travelling waves is determined. The stability is seen to be altered by changing the Mach number (an increase of which decreases the stability of the flow), and also by cooling and heating the wall. These results are limited by the fact that the growth of the layer in time is not taken into account. To include this we consider the large Reynolds number limit and use a triple-deck structure to determine the modes characteristics. The triple-deck approach is used to determine an asymptote to the lower branch of the neutral curve and unsteady effects can be included in a consistent manner. For the upper branch, however, a five-deck structure is required due to the fact that the critical layer is now distinct from the viscous sublayer. The upper-branch stability is only calculated to the first order which is sufficient to give an insight into the stability characteristics.     

Surface Waves for a Compressible Viscous Fluid

In this paper, we are concerned with free boundary problem for compressible viscous isotropic Newtonian fluid. Our problem is to find the three-dimensional domain occupied by the fluid which is bounded below by the fixed bottom and above by the free surface together with the density, the velocity vector field and the absolute temperature of the fluid satisfying the system of Navier-Stokes equations and the initial-boundary conditions. The Navier-Stokes equations consist of the conservations of mass, momentum under the gravitational field in a downward direction and energy. The effect of the surface tension on the free surface is taken into account. The purpose of this paper is to establish two existence theorems to the problem mentioned above: the first concerns with the temporary local solvability in anisotropic Sobolev-Slobodetskiĭ spaces and the second the global solvability near the equilibrium rest state. Here the equilibrium rest state (heat conductive state) means that the temperature distribution is a linear function with respect to a vertical direction and the density is determined by an ordinary differential equation which involves equation of state. For the proof, we rely on the methods due to Solonnikov in the case of incompressible fluid with some modifications, since our problem is hyperbolic-parabolic coupled system. Dedicated to Professors Takaaki Nishida and Masayasu Mimura on their sixtieth birthdays     

Heating of a Compressible Liquid by a Constant Heat Flux

Several variants of the problem of heating a compressible liquid by a timeindependent heat flux are numerically studied. It is shown that, after a certain time, the pressure everywhere behind the shock wave differs only little from some constant value. Approximate analytical formulas are obtained, which demonstrate independence of pressure of thermal conductivity and some other features of the relation between the pressure and the heatflux intensity. Several examples are given, which confirm the adequacy of formulas to numerical solutions of the problem.     

On the Mathematical Modelling of a Compressible Viscoelastic Fluid

Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved.     

Temperature Dependent Viscoelastic Properties of Polymers Investigated by Small-Scale Dynamic Mechanical Analysis

The temperature-dependent viscoelastic properties of polymers were investigated by small-scale dynamic mechanical analysis in the range of −100°C to 200°C. The polymers tested included glassy polymer (atactic polystyrene), semicrystalline polymer (high-density polyethylene) and rubbery polymer (polyisobutylene). The small-scale dynamic mechanical analyses were performed by using a flat-tip indenter with an oscillating displacement of amplitude 36 nm. The force amplitude and phase angle were measured, from which the storage modulus E′ and loss tangent tanδ were calculated. The results obtained from indentation experiments are consistent with those obtained from conventional dynamic mechanical analyzer (DMA). It is thus demonstrated that the indentation technique can quantitatively measure the temperature-dependent viscoelastic properties of polymers at small dimensions.     

Nonstationary Axisymmetric Problem for a Half-Space of a Compressible Fluid*

International Applied Mechanics - The exact analytical solution to the axisymmetric problem for a half-space of an ideal compressible fluid under nonstationary pressure is found. The method of...     

Modeling and Numerical Simulations of Immiscible Compressible Two-Phase Flow in Porous Media by the Concept of Global Pressure

A new formulation is presented for the modeling of immiscible compressible two-phase flow in porous media taking into account gravity, capillary effects, and heterogeneity. The formulation is intended for the numerical simulation of multidimensional flows and is fully equivalent to the original equations, contrary to the one introduced in Chavent and Jaffré (Mathematical Models and Finite Elements for Reservoir Simulation, 1986). The main feature of this formulation is the introduction of a global pressure. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic (the global pressure equation) and a nonlinear diffusion–convection one (the saturation equation) which can be efficiently solved numerically. A finite volume method is used to solve the global pressure equation and the saturation equation for the water and gas phase in the context of gas migration through engineered and geological barriers for a deep repository for radioactive waste. Numerical results for the one-dimensional problem are presented. The accuracy of the fully equivalent fractional flow model is demonstrated through comparison with the simplified model already developed in Chavent and Jaffré (Mathematical Models and Finite Elements for Reservoir Simulation, 1986).     

Retrieval of Spacewise Dependent Hydraulic Properties of Anisotropic Rocks from Transient Flow Experiments

An inverse finite-difference method (FDM) is developed to characterise the spatially dependent components of the hydraulic conductivity tensor together with the specific storage for anisotropic materials, using experimental data generated from the direct FDM. This simulated surface data serves as additional information to a genetic algorithm (GA) optimisation procedure, using a modified least squares functional, that minimises the difference between the experimental data and the FDM-predicted boundary pressure and/or average hydraulic flux measurements based on current hydraulic conductivity tensor and specific storage estimates.     

Subsonic Compressible Flow Past a Body with Developed Cavitation

The essence of the Prandtl rule consists in the fact that in a flow at a Mach number M < 1 the transverse dimensions of a sharp-ended slender body must be reduced by a factor of 1/(1 - M²)^1/2 to conserve the same pressure distribution over the body surface as in a flow with the same velocity at M=0.Following Prandtl [1], the derivation is repeated with reference to axisymmetric flow with developed cavitation.     

Compressible Flow in a Half-Space with Navier Boundary Conditions

We prove the global existence of weak solutions of the Navier–Stokes equations of compressible flow in a half-space with the boundary condition proposed by Navier: the velocity on the boundary is proportional to the tangential component of the stress. This boundary condition allows for the determination of the scalar function in the Helmholtz decomposition of the acceleration density, which in turn is crucial in obtaining pointwise bounds for the density. Initial data and solutions are small in energy-norm with nonnegative densities having arbitrarily large sup-norm. These results generalize previous results for solutions in the whole space and are the first for solutions in this intermediate regularity class in a region with a boundary.     

Transient Flow of a Compressible Fluid in a Connected Layered Permeable Medium

We develop a semi-analytical model of transient fluid flow in a 2D layered permeable medium with cross-flow between adjacent layers. It is shown that the pressure satisfies a diffusion equation to leading order, even when the non-linear term and gravity are included in the mathematical model. The solution is based on an analytical expression in the transform domain for the fluid pressure in terms of interfacial flux functions; the algorithm to compute the flux functions accepts an arbitrary number of formation layers. We show some benchmark tests that validate the general model; the model is then applied to an example derived from experiments. Numerical experiments confirm the significance of the cross-flow in a particular scaling of the ratio of permeabilities and quantify the influence of the various physical parameters.     

Universal Deformations for a Class of Compressible Isotropic Hyperelastic Materials

Differential conditions are derived for a smooth deformation to be universal for a class of isotropic hyperelastic materials that we regard as a compressible variant (a notion we make precise) of Mooney–Rivlin’s class, and that includes the materials studied originally by Tolotti in 1943 and later, independently, by Blatz. The collection of all universal deformations for an incompressible material class is shown to contain, modulo a uniform dilation, all the universal deformations for its compressible variants. As an application of this result, by searching the known families of universal deformations for all incompressible isotropic materials, a nontrivial universal deformation for Tolotti materials is found. This revised version was published online in August 2006 with corrections to the Cover Date.