Isentropic Approximation¶of the Compressible Euler System¶in One Space Dimension

Using the stability results of Bressan & Colombo [BC] for strictly hyperbolic 2 × 2 systems in one space dimension, we prove that the solutions of isentropic and non-isentropic Euler equations in one space dimension with the respective initial data (ρ₀, u ₀) and (ρ₀, u ₀, &\theta;₀=ρ₀ ^{γ− 1}) remain close as soon as the total variation of (ρ₀, u ₀) is sufficiently small. Accepted April 25, 2000?Published online November 24, 2000     

Constitutive Relations for Monatomic Gases¶Based on a¶Generalized Rational Approximation¶to the Sum of the Chapman-Enskog Expansion

. We study the asymptotic behavior as time goes to infinity of solutions to the initial‐boundary‐value problem on the half space for a one‐dimensional model system for the isentropic flow of a compressible viscous gas, the so‐called p‐system with viscosity. As boundary conditions, we prescribe the constant state at infinity and require that the velocity be zero at the boundary . When the velocity at infinity is negative and satisfies a condition on the magnitude, we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile, which is suitably far from the boundary, then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity. The proof is given by an elementary energy method. (Accepted March 2, 1998)     

An Existence Theorem¶for the Navier-Stokes Flow¶in the Exterior of a Rotating Obstacle

We consider the three-dimensional Navier-Stokes initial value problem in the exterior of a rotating obstacle. It is proved that a unique solution exists locally in time if the initial velocity possesses the regularity L^1/2. This regularity assumption is the same as that in the famous paper of Fujita &; Kato. An essential step for the proof is the deduction of a certain smoothing property together with estimates near t˸ of the semigroup, which is not an analytic one, generated by the operator \Cal Lu = -P[\De u+(\om×x)·\na u-\om×u]\Cal Lu= -P\left[\De u+(\om\times x)\cdot\na u-\om\times u\right] in the space L², where y stands for the angular velocity of the rotating obstacle and P denotes the projection associated with the Helmholtz decomposition.     

Invariant Properties for Finding Distance in Space of?Elasticity Tensors

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor; hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogonal group. These expressions are stated in terms of the eigenvalues of both the given tensor and the projected one. The process of minimization is facilitated by the fact that, as we prove, the traces of the corresponding Voigt and dilatation tensors are invariant under these orthogonal projections. For isotropy, cubic symmetry and transverse isotropy, we formulate algorithms to find both the orientation and the eigenvalues of the elasticity tensor endowed with a particular symmetry and closest to the given elasticity tensor.      

The Oberbeck–Boussinesq Approximation as a Singular Limit of the Full Navier–Stokes–Fourier System

We consider the asymptotic limit for the complete Navier–Stokes–Fourier system as both Mach and Froude numbers tend to zero. The limit is investigated in the context of weak variational solutions on an arbitrary large time interval and for the ill-prepared initial data. The convergence to the Oberbeck–Boussinesq system is shown.      

The Mechanical Properties of the Slurry and the Resistance Force of a Sphere Moving With Uniform Velocity in the Slurry

It is commonly considered that the mechanical properties of the slurryare different from that of ordinary Newtonian fluid,and can be describedby that of Bingham fluid.Hence its shearing stress should be described bythe formula of the shearing stress of Bingham fluid.But the author holdsthe contrary opinion and firmly believes that the slurry is a highly viscousfluid with very long relaxation time such as those of asphalt,glass,etc.In this article,we have discussed the mechanical properties of the slurryand the reslstance of a sphere moving with uniform veloclty in the slurry.In the process of discussion.we use Stokes solution of the vlscous fluidpassing around a sphere.When the sphere is in equilibrium under theaction of gravitational force,the force of buoyancy and the resistance,we get the velocity of sedimentation.When the velocity of sedimentationis equal to zero,we get the relation between the yield stress of Binghamfluid and the diameter of the particles which will not sink.The theoreticalresults     

Effect of Allowance for Small‐Scale Perturbations of the Carrier Phase on Properties of the System of Equations of a Two‐Fluid Flow with Incompressible Phases

The effect of the fluctuating components of kinetic energy and stress tensor of the carrier phase, which were previously obtained by the cell technique, on the properties of the system of equations of a gas–liquid flow with incompressible phases is considered. It is shown that the characteristic properties of this system and also the possibility of modeling the Zuber–Findlay empirical relation are determined by the tensor of fluctuating stresses of the carrier phase.     

Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation

We show that a smooth solution u ⁰ of the Euler boundary value problem on a time interval (0, T ₀) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T ₀). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “naturally inhomogeneous” if we deal with the strong solutions. We provide information on the rate of convergence of the solutions of the Navier–Stokes problem to the solution of the Euler problem for ν → 0. We also discuss possibilities when Navier’s boundary condition becomes homogeneous.     

Homogenization of a Hele–Shaw Problem in Periodic and Random Media

We investigate the homogenization limit of a free boundary problem with space-dependent free boundary velocities. The problem under consideration has a well-known obstacle problem transformation, formally obtained by integrating with respect to the time variable. By making rigorous the link between these two problems, we are able to derive an explicit formula for the homogenized free boundary velocity, and we establish the uniform convergence of the free boundaries.     

The Motion of a Fluid–Rigid Disc System at the Zero Limit of the Rigid Disc Radius

We consider the two-dimensional motion of the coupled system of a viscous incompressible fluid and a rigid disc moving with the fluid, in the whole plane. The fluid motion is described by the Navier–Stokes equations and the motion of the rigid body by conservation laws of linear and angular momentum. We show that, assuming that the rigid disc is not allowed to rotate, as the radius of the disc goes to zero, the solution of this system converges, in an appropriate sense, to the solution of the Navier–Stokes equations describing the motion of only fluid in the whole plane. We also prove that the trajectory of the centre of the disc, at the zero limit of its radius, coincides with a fluid particle trajectory.     

The Ring-Pull Assay for Mechanical Properties of Fibrous Soft Tissues – an Analysis of the Uniaxial Approximation and a Correction for Nonlinear Thick-Walled Tissues

Background: The ring-pull test, where a ring of tissue is hooked via two pins and stretched, is a popular biomechanical test, especially for small arteries. Although convenient and reliable, the ring test produces inhomogeneous strain, making determination of material parameters non-trivial. Objective: To determine correction factors between ring-pull-estimated and true tissue properties. Methods: A finite-element model of ring pulling was constructed for a sample with nonlinear, anisotropic mechanical behavior typical of arteries. The pin force and sample cross-section were used to compute an apparent modulus at small and large strain, which were compared to the specified properties. The resulting corrections were validated with experiments on porcine and ovine arteries. The correction was further applied to experiments on mouse aortic rings to determine material and failure properties. Results: Calculating strain based on centerline stretch rather than inner-wall or outer-wall stretch afforded better estimation of tissue properties. Additional correction factors were developed based on ring wall thickness H, centerline ring radius R_c, and pin radius a. The corrected estimates for tissue properties were in good agreement with uniaxial stretch experiments. Conclusions: The computed corrections improved estimation of tissue material properties for both the small-strain (toe) modulus and the large-strain (lockout) modulus. When measuring tensile strength, one should minimize H/a to ensure that peak stress occurs at the sample midplane rather than near the pin. In this scenario, tensile strength can be estimated accurately by using inner-wall stretch at the midplane and the corrected properties.

     

Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions to a Riemann Problem

We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.     

Methods of Small and Large δ in the Nonlinear Dynamics – A Comparative Analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x ⁿ are proposed and analyzed. Two natural limiting cases are studied: n 1 + , 1 and n . In the firstcase, the 'small method' (SM)is used and its applicability for dynamical problems with the nonlinearterm sin as well as the usefulness of the SMfor the problem with small denominators are outlined. For n , a new asymptotic approach is proposed(conditionally we call it the 'large method' –LM). Error estimations lead to the followingconclusion: the LM may be used, even for smalln, whereas the SM has a narrow application area. Both of the discussed approaches overlap all values ofthe parameter n.     

Properties of the Eulerian–Lagrangian method using linear interpolators in a three-dimensional shallow water model using z-level coordinates

We investigate the impact of the inexact interpolation on the Eulerian–Lagrangian solution of the advection equation by combining numerical experiments and formal analysis. The simulations, respectively, using the Eulerian–Lagrangian method (ELM) and the upwind scheme are compared. The artificial resistance of the ELM is observed which is characterised by the higher free-surface elevation and the distorted turbulent properties at a smaller time step. Through analysis, we find that the abnormalities are caused by the fact the conventional linear interpolation does not adapt well to the nonlinear velocity distribution, which produces an advection computation error that increases with a decreasing time step. The phenomena are explained and an improved method ELM is proposed based on the illustrations and analysis. The new method combines the face-controlled interpolation and the adjustable sub time steps to skip the large computation error domain in the backtracking, and it is validated by the original test case.     

On a Theory of the Infusion Method of Measuring the “Opening Pressure” in the Urethra

At rest the muscles which control the urethra (urine duct) are contracted and its lumen is practically equal to zero over its entire length. To open the urethra, a mechanical effort (due, for example, to a pressure rise in the bladder) must be applied. Reduced contractile activity of the muscles may be one of the reasons for incontinence (enuresis). A widespread method of estimating the blocking capability of the urethra consists in inserting a catheter with lateral perforations near the end. The catheter enters the bladder and is then removed at a constant velocity while a fluid is constantly pumped (infused) into it by a syringe pump at a steady rate and then flows out through the gap between the catheter and the urethral wall. The pumping pressure is considered to be a local measure of the blocking capability, and its dependence on the location of the catheter is regarded as an important diagnostic characteristic.Below, we will consider the simple, longitudinally homogeneous model system formed by an elastic tube pulled over a catheter segment when the initial stresses in the tube are constant over its length. An incompressible viscous fluid flows out of the perforations and percolates in a thin layer along the catheter. In solving the model problem, we will use the lubricating layer approximation under the assumption of small layer curvature. On the basis of an analysis of the results and a comparison of the model with a practical intraurethral measurement procedure, we discuss, firstly, the relationship between the measured quantities and the real characteristics of the urethra and, secondly, the possible formulation of a more realistic model problem.     

Air–water flow in a vertical pipe: experimental study of air bubbles in the vicinity of the wall

This study deals with the influence of bubbles on a vertical air–water pipe flow, for gas-lift applications. The effect of changing the bubble size is of particular interest as it has been shown to affect the pressure drop over the pipe. Local measurements on the bubbles characteristics in the wall region were performed, using standard techniques, such as high-speed video recording and optical fibre probe, and more specific techniques, such as two-phase hot film anemometry for the wall shear stress and conductivity measurement for the thickness of the liquid film at the wall. The injection of macroscopic air bubbles in a pipe flow was shown to increase the wall shear stress. Bubbles travelling close to the wall create a periodic perturbation. The injection of small bubbles amplifies this effect, because they tend to move in the wall region; hence, more bubbles are travelling close to the wall. A simple analysis based on a two-fluid set of equations emphasised the importance of the local gas fraction fluctuations on the wall shear stress.     

Role of nuclei density as a “hidden” parameter in the formation of anomalous zones in a heavy cavitating magma

The formation of zones with anomalously high values of the basic flow characteristics in decompression waves in a heavy cavitating magma with an intense increase in the density of cavitation nuclei is numerically studied within the framework of a mathematical model of multiphase media with a system of kinetic equations. The basic effects leading to the formation of the anomalous zone are identified.     

Experimental and Numerical Study of a Hypersonic Separated Flow in the Vicinity of a Cone‐Flare Model

An axisymmetric laminar separated flow in the vicinity of a coneflare model is studied experimentally and numerically for a Mach number M = 6. The distributions of pressure and Stanton numbers along the model surface and velocity profiles in the region of shock wave–boundary layer interaction are measured and compared with the calculated data. The influence of the laminar–turbulent transition on flow parameters is studied numerically.     

Modelling the three-dimensional flow of a semi-dilute polymer solution in microfluidics—on the effect of aspect ratio

The flow of polymer solutions in microfluidic devices is inherently three-dimensional, especially in the non-linear flow regime, and often results in flow phenomena that might not even be encountered in macro-devices. Using a multi-mode Phan-Thien–Tanner model, three-dimensional (3-D) simulations of a semi-dilute polyethylene oxide (PEO) solution through 8:1 planar contraction micro-channels with various depths have been carried out to systematically study the effect of the aspect ratio on the flow fields. Vortex dynamics in the upstream flow section and excess pressure drop are quantified in detail. A transition from a salient-corner vortex mechanism to a lip vortex mechanism is observed as the aspect ratio is varied from 1 to 1/4, which corresponds to the elasticity numbers El = 36.3 to 48.3. The numerical results show that varying the aspect ratio of microfluidic channels has similar effects to varying other parameters, such as fluid properties, which influence the elasticity number. Thus, our results support the view that vortex growth mechanism is determined by the elasticity number, which is fixed for a given fluid and geometry. The principle is of significance to the design of new microfluidic chips for a wide range of applications.