Assumption Of Incompressibility Research Articles

An experimental study on the contact problem of spherical indentation on the anisotropic elastic half-space based on imaging

The contact problem of spherical indentation on the anisotropic elastic half-space is considered based on an experimental method in this work. The mechanical anisotropy is simulated by uniaxially stretching isotropic material made of silica gel. A special optical system is adopted to capture the contact images during indentation tests. The experimental results further prove the reasonability of the assumption that the contact area of spherical indentation on the anisotropic half-space is elliptical. And the results also show that the direction of strengthening by the uniaxial tension is consistent with the minor axis of the elliptical contact area. Such results validate the associated theory. Besides, the error analysis is made to examine the reliability of the fitting results, where the relative errors are less than 0.8% under the influence of given drawbacks. The indentation modulus can be determined through experimental data. Results show that the indentation modulus monotonically increases with the increase of the prestrain value. In the case of isotropy, the initial elastic modulus is determined under the assumption of incompressibility. The result by indentation test is in a good agreement with that by uniaxial tensile test.

Hydrostatic vertical velocity and incompressibility in the Northern Hemisphere

The vertical velocity w is evaluated for the Northern Hemisphere from reanalysis data and two forms of the Richardson equation. This equation is based on the hydrostatic assumption and the thermodynamic energy equation. The standard form of the Richardson equation allows one to quantify the contributions to the vertical velocity of the horizontal divergence δ, the vertical pressure velocity ω and heating, and to test the incompressibility assumption underlying many dynamic models and theories. However, there are cancellations between two important terms. This shortcoming is substantially reduced in a further version of this equation where one term dominates. This version is the backbone of the data evaluation.The vertical velocities resulting from the Richardson equation in the troposphere are in good agreement with those obtained directly from the reanalysis data. It is found that the assumption of incompressibility provides a good estimate for w in the mid troposphere, even above Greenland and the Tibetan Plateau, both for the annual mean and the standard deviation of w, but is less acceptable in the upper troposphere and almost useless in the lower stratosphere. The contribution of heating to w is small.

Variational modeling of plane-strain hyperelastic thin beams with thickness-stretching effect

Soft materials and structures have recently attracted lots of research interests as they provide paramount potential applications in diverse fields including soft robotics, wearable devices, stretchable electronics and biomedical engineering. In the present work, we develop a new Euler–Bernoulli finite strain beam model for soft thin structures subject to stiff constraint in the width direction. The beam model assumes plane-strain deformation and accounts for the thickness-stretching effect, which is very significant for soft structures under large strain. With the Euler–Bernoulli kinematic hypothesis and incompressibility assumption, the finite deformation of the beam is analyzed, constitutive equations are deduced from the theory of finite elasticity, and by employing the standard variational method, the equilibrium equations and associated boundary conditions are derived. Based on the new beam model, analytical formulae are given for uniform stretching and pure bending, which cannot be accurately described by any finite strain beam model with the rigid cross-section assumption. Furthermore, numerical solution procedures and results are presented for cantilevered beams and simply supported beams with movable or immovable ends. The developed beam model is expected to benefit the modeling and simulation of soft robots and soft devices.

Experimental investigation on the atomization of a spherical droplet induced by Faraday instability

Faraday instability plays an important role to realize liquid atomization which has been widely applied in various fields. The spherical Faraday instability has different characteristics from the planar one. It is of great importance to understand the fundamental mechanisms of the surface deformation evolution and atomization of a spherical droplet induced by Faraday instability. In this paper, we first experimentally recorded the deformation and fragmentation process of a spherical droplet on a vertically vibrated plate with high speed camera. The results showed that the zonal, meridional and approximately circular standing waves in proper order exist on the surface of the spherical droplet with the increase of excitation amplitude. When Bo<3.0, the modes of low (zonal) and high (meridional) order m of the spherical harmonic occur more easily, and as Bo is increased, the order m becomes closer to the intermediate value of spherical mode number l. The mechanism of droplet atomization is that a spike forms first on the droplet surface due to the impingement of the liquid flowing from the neighboring trough portions, and then a neck appears because of the velocity difference between the head and bottom of the spike, and the velocity difference determines whether the head liquid can form a sub-droplet ejected from the surface of the parent droplet. The Mathieu equation was derived by a linear theoretical analysis with inviscid and incompressible assumptions for the spherical Faraday instability, in which the parameters have different definitions from its planar counterpart. In the instability diagram, the iso-curves of larger linear growth rate deviate further from the ordinate axis and finally disappear. It is also validated that Lang’s equation is applicable to the spherical Faraday instability in low frequency.

A thermodynamically consistent phase‐field model for viscous sintering

AbstractA thermodynamically consistent phase‐field model for viscous sintering is proposed. It is based on an energetic variational formulation that allows the governing equations to be analytically derived from a defined energy law. The conservation of mass is satisfied through the incompressibility assumption and the assumption that mass density is uniform initially within the particle compact while the balance of linear momentum is formulated from an energy dissipation law. The morphological changes of particles are described by the temporal and spatial evolution of a phase‐field variable governed by a modified Cahn‐Hilliard equation, and the motion of viscous mass flow is controlled by the Stokes equation incorporating the surface tension effect. The application of the phase‐field model is illustrated by examining the effect of particle shape, initial contact angle and rearrangement effects on viscous sintering.

Two-dimensional affine model-based estimators for principal strain vascular ultrasound elastography with compound plane wave and transverse oscillation beamforming

Polar strain (radial and circumferential) estimations can suffer from artifacts because the center of a nonsymmetrical carotid atherosclerotic artery, defining the coordinate system in cross-sectional view, can be misregistered. Principal strains are able to remove coordinate dependency to visualize vascular strain components (i.e., axial and lateral strains and shears). This paper presents two affine model-based estimators, the affine phase-based estimator (APBE) developed in the framework of transverse oscillation (TO) beamforming, and the Lagrangian speckle model estimator (LSME). These estimators solve simultaneously the translation (axial and lateral displacements) and deformation (axial and lateral strains and shears) components that were then used to compute principal strains. To improve performance, the implemented APBE was also tested by introducing a time-ensemble estimation approach. Both APBE and LSME were tested with and without the plane strain incompressibility assumption. These algorithms were evaluated on coherent plane wave compounded (CPWC) images considering TO. LSME without TO but implemented with the time-ensemble and incompressibility constraint (Porée et al., 2015) served as benchmark comparisons. The APBE provided better principal strain estimations with the time-ensemble and incompressibility constraint, for both simulations and in vitro experiments. With a few exceptions, TO did not improve principal strain estimates for the LSME. With simulations, the smallest errors compared with ground true measures were obtained with the LSME considering time-ensemble and the incompressibility constraint. This latter estimator also provided the highest elastogram signal-to-noise ratios (SNRs) for in vitro experiments on a homogeneous vascular phantom without any inclusion, for applied strains varying from 0.07% to 4.5%. It also allowed the highest contrast-to-noise ratios (CNRs) for a heterogeneous vascular phantom with a soft inclusion, at applied strains from 0.07% to 3.6%. In summary, the LSME outperformed the implemented APBE, and the incompressibility constraint improved performances of both estimators.

Viscoelastic modeling of the fusion of multicellular tumor spheroids in growth phase

Background. Since several decades, the experiments have highlighted the analogy of fusing cell aggregates with liquid droplets. The physical macroscopic models have been derived under incompressible assumptions. The aim of this paper is to provide a 3D model of growing spheroids, which is more relevant regarding embryo cell aggregates or tumor cell spheroids.Methods. We extend the past approach to a compressible 3D framework in order to account for the tumor spheroid growth. We exhibit the crucial importance of the effective surface tension, and of the inner pressure of the spheroid to describe precisely the fusion. The experimental data were obtained on spheroids of colon carcinoma human cells (HCT116 cell line). After 3 or 6 days of culture, two identical spheroids were transferred in one well and their fusion was monitored by live videomicroscopy acquisition each 2 h during 72 h. From these images the neck radius and the diameter of the assembly of the fusing spheroids are extracted.Results. The numerical model is fitted with the experiments. It is worth noting that the time evolution of both neck radius and spheroid diameter are quantitatively obtained. The interesting feature lies in the fact that such measurements characterise the macroscopic rheological properties of the tumor spheroids.Conclusions. The experimental determination of the kinetics of neck radius and overall diameter during spheroids fusion characterises the rheological properties of the spheroids. The consistency of the model is shown by fitting the model with two different experiments, enhancing the importance of both surface tension and cell proliferation.General significance. The paper sheds new light on the macroscopic rheological properties of tumor spheroids. It emphasizes the role of the surface tension and the inner pressure in the fusion of growing spheroid. Under geometrical assumptions, the model reduces to a 2–parameter differential equation fit with experimental measurements. The 3–D partial differential system makes it possible to study the fusion of spheroids in non-symmetrical or more general frameworks.

Aerodynamic effects of compressibility for wind turbines at high tip speeds

In the present work two dimensional airfoil computations are used to investigate the effects of compressibility in the tip region of large scale wind turbines of 20 MW+ size. In the past application of incompressible CFD solvers have been wide spread for wind turbine aerodynamics, due to their efficiency and robustness at the near incompressible conditions experienced near the rotor center. With the increasing size of modern wind turbines and the desire to approach high tip speeds, the incompressible assumption might be violated in the tip region of the turbine.To investigate the effects of compressibility and the possibility of correcting incompressible flow solutions using explicit compressibility corrections, a CFD study of 2D airfoil aerodynamics at conditions of a large scale wind turbine is performed. The present study show that classical compressibility corrections can be successfully applied as a post-processing step to incompressible solutions, reducing the error in the predicted lift and drag to within a few percent for attached flow conditions where viscous effects are limited at Mach numbers upto 0.3.

Condition for transition to an unstable state (necking) of a specimen in tension

On the stress-strain curve obtained in simple tension experiments, there is a region of instability due to the neck formation. In the theory of plasticity, conditions for the transition to an unstable state and appearance of the maximum point on the stress-strain diagram are determined. These condition were derived under the assumption of the material incompressibility. However, this assumption cannot be justified, since in the neck region, there are numerous damages (pores, micro-cracks), i.e., the material is compressible. With the current coefficient of lateral deformation, the authors formulate a condition for transition to an unstable state in a compressible plastic medium.

Numerical analysis of the liquid ejection due to the gaseous jet impact through computational fluid dynamics

Abstract Metal droplets generated by an impinging jet, play an important role in metal refining processes, mainly in oxygen steelmaking, where the droplets are ejected into the slag phase. Since the available interfacial area of droplets is very high in this process, the generated droplets enhance the rates of heat transfer and chemical reactions. Therefore, knowledge of the metal droplet generation rate, size distribution and residence time in the slag are of industrial relevance. In this work, the isothermal, transient flow of an incompressible air jet impinging onto an air/water interface at room temperature has been simulated to obtain a better understanding of the droplet ejection phenomenon. The interface was tracked throughout time using the volume of fluid (VOF) technique. The governing equations formulated for mass and momentum conservation and the k-e turbulence model are solved in the axisymmetric computational domain using the commercial code FLUENT. The droplet ejection rates calculated with computational fluid dynamics model are compared to experimental data reported in literature, showing partial agreement, being the incompressibility assumption the probable reason for the deviation observed, which was as far pronounced as the great jet velocity. Nevertheless, the model presented shows itself as a relatively good starting point for the construction of more complex ones (with less simplifying assumptions) which should be able to offer a means to increase the understanding of the droplet ejection phenomena.

A new high accuracy meshfree method to directly simulate fluid dynamics and heat transfer of weakly compressible fluids

A Tayler series expansion based high accuracy meshfree method (TS-HAMM) is developed to calculate first and second derivatives of property values stored in dispersed particle clouds. Compared with other meshfree methods such as smoothed particle hydrodynamics (SPH), TS-HAMM can (a) systematically increase accuracies in gradient, derivative and Laplacian calculations with similar computational cost; (b) provide boundary conditions directly from conservation equations at the fluid/solid interfaces with guaranteed order of accuracy; (c) simulate fluid dynamics and heat transfer of low viscous weakly compressible fluid (e.g. water) directly without making incompressible assumptions. Through numerical verifications with water-like low viscous low compressible fluid in Couette flow, lid-driven flow inside a cavity, and natural convection inside an enclosure, we justify the validity of TS-HAMM in simulating fluid dynamics and heat transfer of water-like low viscous weakly compressible fluid, which are challenging with other meshfree methods such as smoothed particle hydrodynamics (SPH).

Numerical study on the fluid dynamical aspects of atomic layer deposition process

Computational fluid dynamics investigations on the mixing process of gases inside an atomic layer deposition (ALD) reactor are carried out. A test case involving a real ALD reactor geometry is investigated under nonreacting, incompressible flow assumption. The relatively low Reynolds number (Re) of the test reactor, often being in the laminar regime, advocates the usage of scale-resolving simulations. The authors investigate mixing of two precursors in two different injection configurations for 40 &amp;lt; Re &amp;lt; 2400. The feasibility of the approach is shown and discussed. The results illustrate how both Reynolds number and injection configurations influence the precursor distribution in the ALD reactor. The authors also carry out a set of experiments in the same ALD reactor and discuss them in light of the simulations.

Mass-conserving advection–diffusion Lattice Boltzmann model for multi-species reacting flows

Given the complex geometries usually found in practical applications, the Lattice Boltzmann (LB) method is becoming increasingly attractive. In addition to the simple treatment of intricate geometrical configurations, LB solvers can be implemented on very large parallel clusters with excellent scalability. However, reacting flows and especially combustion lead to additional challenges and have seldom been studied by LB methods. Indeed, overall mass conservation is a pressing issue in modeling multi-component flows. The classical advection–diffusion LB model recovers the species transport equations with the generalized Fick approximation under the assumption of an incompressible flow. However, for flows involving multiple species with different diffusion coefficients and density fluctuations – as is the case with weakly compressible solvers like Lattice Boltzmann –, this approximation is known not to conserve overall mass. In classical CFD, as the Fick approximation does not satisfy the overall mass conservation constraint a diffusion correction velocity is usually introduced. In the present work, a local expression is first derived for this correction velocity in a LB framework. In a second step, the error due to the incompressibility assumption is also accounted for through a modified equilibrium distribution function. Theoretical analyses and simulations show that the proposed scheme performs much better than the conventional advection–diffusion Lattice Boltzmann model in terms of overall mass conservation.

Aerodynamic Modelling and Analysis of a Novel Mechanism for Chord and Camber Morphing Wing

This paper presents the aerodynamic modelling and analysis of surfaces created by a novel deployable mechanism, which is composed of a four-bar linkage and a scissor-structural mechanism (SSM) which contains several scissor-like elements (SLEs). With the help of that mechanism, which is located inside the trailing portion of wing section, continuous adjustment of the airfoil is possible. In order to highlight the advantageous aerodynamic characteristics of newly created airfoil geometries via proposed SSM, several aerodynamic analyses have been performed. The flow characteristics used for the analyses are determined by the flight envelope of an intended generic UAV. Since the maximum speed range of the sample aircraft is well below Mach 0.3, incompressible flow assumption is made throughout the solutions and conservation laws of Reynolds Transport Theorem are employed.

Static trace free Einstein equations and stellar distributions

We construct models of static spherical distributions of perfect fluid in trace--free Einstein gravity theory. The equations governing the gravitational field are equivalent to the standard Einstein's equations however, their presentation is manifestly different which motivates the question whether new information would emerge due to the nonlinearity of the field equations. The incompressible fluid assumption does not lead to the well known Schwarzschild interior metric of Einstein gravity and a term denoting the presence of a cosmological constant is present on account of the integration process. The Schwarzschild interior is regained as a special case of a richer geometry. On the other hand, when the Schwarzschild geometry is prescribed, a constant density fluid emerges consistent with the standard equations. A complete model of an isothermal fluid sphere with pressure and density obeying the inverse square law is obtained. Corrections to the model previously presented in the literature by Saslaw {\it {et al}} are exhibited. The isothermal ansatz does not yield a constant gravitational potential in general but both potentials are position dependent. Conversely, it is shown that assuming a constant $g_{rr}$ gravitational potential does not yield an isothermal fluid in general as is the case in standard general relativity. The results of the standard Einstein equations are special cases of the models reported here. Noteworthy is the fact that whereas the previously reported isothermal solution was only of cosmological interest, the solution reported herein admit compact objects by virtue of the fact that a pressure-free hypersurface exists. Finally we analyze the consequences of selecting the Finch--Skea metric as the seed solution.

Optimum Autofrettage Pressure for a High Pressure Cylinder of a Waterjet Intensifier Pump

This study presents analytical approach for the determination of optimum autofrettage pressure of a thick walled cylinder which is commercially used as a high pressure cylinder of a waterjet intensifier pump. Bilinear kinematic hardening model is used in the derivation of analytical model based on plane strain, Von-Mises yield criteria and incompressible volume assumptions. Elasto-plastic radius is dependent on autofrettage pressure and optimum value of autofrettage pressure is found as a value that will give minimum value of maximum Von-Mises stress on the wall of the cylinder under operating pressure. Hence, considering elasto-plastic radius as the design variable, its optimum value is determined for the specified operating pressure. Reverse yielding owing to Bauschinger effect is used as the only non-linear constraint, Matlab© optimtool finds optimum value of the elasto-plastic radius and the corresponding autofrettage pressure is then calculated. By using ANSYS 15©, stress analysis of the autofrettaged cylinder has been performed and the results obtained from both methods are found to be almost the same.

Electro-viscoelastic response of an acrylic elastomer analysed by digital image correlation

Experimental investigations are presented with respect to the electromechanically coupled and time-dependent behaviour of an acrylic elastomer, VHB 4910. For the electromechanically coupled experiments different biaxial pre-stretches have been considered and full-field measurements were made using three-dimensional surface digital image correlation. Both equi-biaxial and non equi-biaxial pre-stretches as well as both circular and non-circular electrodes are investigated. The experimental data provide new insights into the complex material behaviour of VHB 4910 and will enable enhanced calibration and development of constitutive models. Special emphasis lies on the quantification of out-of-plane deformation, i.e. the thickness change, of biaxially pre-stretched specimens based on an assumption of material incompressibility. The level of pre-stretch and thickness of the membrane plays a critical role in view of electromechanical instabilities. For completeness, cyclic uniaxial tests of the purely mechanical response are also presented and compared to similar experiments found in the literature.

Numerical investigation of fluid flow and heat transfer characteristics in a helically-finned tube

In order to investigate the characteristics of flow and heat transfer rate in a Helically-finned tub (HFT), we used continuity, momentum and energy equations under a steady, three-dimensional and incompressible fluid flow assumptions. For the performance metrics, we considered the Darcy friction factor, Colburn j-factor, volume goodness factor and area goodness factor of the HFT. We could also evaluate the effect of geometry parameters on the results of local pressure coefficient, fluid vorticity and Nusselt number of the HFT. We carried out the CFD calculation for a range of laminar flow (Re = 100) and turbulent flow (Re = 2000 and 10000). In a laminar and turbulent flow regime, the friction factor increases with increasing the each geometric parameter. While the Colburn j-factor decreases as increasing these geometric parameters. Consequently, the thermal performance of HFT is poorer than that of single straight circular tube type because of having a small volume and area goodness factor as increasing the Reynolds numbers.

Magneto-Rayleigh–Taylor instability in compressible Z-pinch liner plasmas**Project supported by the National Natural Science Foundation of China (Grant Nos. 11475027, 11274051, 11105017, and 11275030) and the National Basic Research Program of China (Grants No. 2013CB834100).

In this paper, the characteristics of magneto-Rayleigh–Taylor (MRT) instability of liner plasmas in MagLIF is theoretically investigated. A three-region slab model, based on ideal MHD equations, is used to derive the dispersion relation of MRT instability. The effect of compressibility on the development of MRT instability is specially examined. It is shown that the growth rate of MRT instability in compressible condition is generally lower than that in incompressible condition in the presence of magnetic field. In the case of zero magnetic field, the growth rate in compressible assumption is approximately the same as that in incompressible assumption. Generally, MRT instability in (x, y) plane can be remarkably mitigated due to the presence of magnetic field especially for short-wavelength perturbations. Perturbations may be nearly completely mitigated when the magnetic field is increased to over 1000 T during liner implosions. The feedthrough of MRT instability in liner outer surface on inner surface is also discussed.

Numerical investigation of the pressure-time method

The pressure-time method is a discharge measurement technique commonly used to estimate the flow rate in hydro-power plants to assess the turbine hydraulic efficiency. In this paper two-dimensional Reynolds-averaged Navier-Stokes equations with the low-Reynolds k-ω SST turbulence model are used to model and evaluate the pressure-time method. The flow rate decrease is modeled by valve closure and use of the dynamic mesh technique. The results are compared with experimental data showing the close trend. Moreover, the affecting parameters in the pressure-time method including compressibility, calculation of the losses and finding the upper time limit of integration of the pressure-time diagram are studied in detail. Furthermore, the flow characteristics with both compressible and incompressible assumptions are investigated. The obtained results demonstrated the compressibility effect even before the complete valve closure. The physics of the flow, in the closed conduit in the pressure-time method, before the complete valve closure is studied based on the time variation of the wall shear stress and the γ parameter. The γ parameter represents the difference between the turbulence structure in a transient accelerating or decelerating flow and the one in the quasi-steady condition. It is demonstrated that for the pressure-time method, some part of the flow decrease excursion can be characterized as quasi-steady and the rest is unsteady. The dominance of inertia and turbulence dynamics is investigated to evaluate the wall shear stress in the part of the excursion with the unsteady assumption. It is found that the inertia has a dominant effect during the excursion. The flow rate is calculated by evaluating all the terms of losses, e.g., wall shear stress, normal stresses, Reynolds stresses and density variation. Based on the relative magnitude of each term the results indicate that the wall shear stress is a good approximation of the losses for the flow rate calculation and the other terms can be neglected. Due to the primary role of losses in the flow rate calculation other methods in losses calculation are also considered for comparison (e.g., quasi-steady friction or constant friction factor). The available methods to find the upper time limit of the flow rate approximation integral still have significant errors. The flow rate reduction is calculated at each time step before complete valve closure using CFD methods.