Incompressible Medium Research Articles

Going with the Flow

Given a sequence of poses of a body we study the motion resulting when the body is immersed in a (possibly) moving, incompressible medium. With the poses given, say, by an animator, the governing second-order ordinary differential equations are those of a rigid body with time-dependent inertia acted upon by various forces. Some of these forces, like lift and drag, depend on the motion of the body in the surrounding medium. Additionally, the inertia must encode the effect of the medium through its added mass. We derive the corresponding dynamics equations which generalize the standard rigid body dynamics equations. All forces are based on local computations using only physical parameters such as mass density. Notably, we approximate the effect of the medium on the body through local computations avoiding any global simulation of the medium. Consequently, the system of equations we must integrate in time is only 6 dimensional (rotation and translation). Our proposed algorithm displays linear complexity and captures intricate natural phenomena that depend on body-fluid interactions.

Математическая модель процесса СВЧ и конвективной выпечки хлеба из сбивного теста

Whipped yeast-free bakery products require effective energy supply to dough in order to optimize energy consumption, baking time, and quality. This article introduces a verified mathematical model of microwave and convective baking for whipped bread based on heat and mass exchange equations.
 A full-scale experiment to verify the calculations involved dough samples with a humidity of 56 ± 1%. The samples underwent microwave and convective processing until the temperature in the crumb center reached 98 ± 1°C.
 The mathematical model was formalized as energy and mass conservation equations, which made it possible to consider baking as a non-stationary process of heat and mass transfer of moisture in an isotropic incompressible continuous medium in the diffusion approximation. The equation took into account the unstable phase transition boundary. The practical verification showed the mean error for microwave baking as 14.5% in temperature and 18.2% in moisture content. For convective baking, the results included 12.6% in temperature and 9.7% in moisture content. The mathematical model proved adequate to the real processes of heat and mass transfer. The error in calculating the temperature and moisture content fields was sufficient tooptimize the process.
 The physical and mathematical model of the baking process made it possible to evaluate the effect of technological variables on the temperature and moisture concentration fields in the dough samples. The mathematical model and the computational experiment can be used to identify static and dynamic characteristics of baking as an object of automatic control, i.e., to identify optimal control channels and actions, as well as to adjust the automatic control system to specific quality indicators.

Quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation with the sharp decay rates

We analyze the asymptotic stability of the quasi-linearly stratified densities in the 2D inviscid incompressible porous medium equation on R2 with respect to the buoyancy frequency N. Our target density of stratification is the sum of the large background linear profile with its slope N and the small perturbation that could be both non-linear and non-monotone. Quantification in N will be performed not only on how large the initial density disturbance is allowed to be but also on how much the target densities can deviate from the purely linear density stratification without losing their stability.For the purely linear density stratification, our method robustly applies to the three fundamental domains R2, T2, and T×[−1,1], improving both the previous result by Elgindi (2017) [18] on R2 and T2, and the study by Castro et al. (2019) [12] on T×[−1,1]. The obtained temporal decay rates to the stratified density on R2 and to the newly found asymptotic density profiles on T2 and T×[−1,1] are all sharp, fully realizing the level of the linearized system. We require the initial disturbance to be small in Hm for any integer m≥4, which we even relax to any positive number m>3 via a suitable anisotropic commutator estimate.

Peculiarities of Pulse Dynamics in Stratified Incompressible Media

Peculiarities of Pulse Dynamics in Stratified Incompressible Media

Evaluation of the Fresnel approximation for the longitudinal particle displacement in shear wave beams

The longitudinal particle displacement in shear wave beams generated with disks that vibrate longitudinally at several hundred hertz is used to assess stiffness of soft tissue in vivo. Nonlinear elasticity of tissue, although less commonly used than linear stiffness, may be a more sensitive or specific biomarker of disease. A mathematical model for linear propagation of both transverse and longitudinal motion in a shear wave beam in an isotropic elastic half-space, and comparisons with measurements in tissue-mimicking phantoms, was published earlier this year [Archer et al., JASA 153, 1591 (2023)]. When including elastic nonlinearity in a propagation model, the complexity is reduced substantially by deriving an evolution equation based on the Fresnel approximation [Zabolotskaya, Sov. Phys. Acoust. 32, 296 (1986)]. For application to elastography, the parameter space in which the Fresnel approximation provides an accurate description of the longitudinal particle displacement must be established. In this presentation, the accuracy of the Fresnel approximation is assessed for linear shear wave beams, both unfocused and focused, by comparison with the existing mathematical model. Attention is devoted primarily to the longitudinal particle displacement in nearly incompressible media. [PGK is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Three-Dimensional Numerical Transfer Model Using a Monotonized Z-Scheme

The aim of this work is to investigate a three-dimensional second-order monotone finite-difference scheme for the transport equation. The investigation is conducted for model three-dimensional transport equations of an incompressible medium. The properties of the three-dimensional extension of the Z-scheme with nonlinear correction are studied in this work. This study is an extension of the author’s previous works, where a nonlinear correction of one-dimensional transport equations was constructed. The proposed scheme uses “skew differences” containing values from different time layers instead of fluxes for the correction. The monotonicity of the obtained nonlinear scheme is numerically studied for a family of limiter functions for both smooth and non-smooth continuous solutions. The constructed scheme is absolutely stable but loses the monotonicity property when the Courant step is exceeded. The distinctive feature of the proposed finite-difference scheme is the minimalism of its template. The constructed numerical scheme is designed for models of plasma instabilities of various scales in the low-latitude ionospheric plasma of the Earth. One of the real problems that arise in solving such equations is the numerical modeling of strongly non-stationary medium- and small-scale processes in the low-latitude ionosphere of the Earth under conditions of the occurrence of the Rayleigh–Taylor instability and other types of instabilities, leading to the phenomenon of F‑scattering. Due to the fact that transport processes in the ionospheric plasma are controlled by the magnetic field, it is assumed that the plasma is incompressible in the direction perpendicular to the magnetic field.

Experimental observations of Scholte waves propagating in an incompressible soft solid

Due to the heterogeneous structure of the soft biological tissue, such as the brain, surface waves might be important to elucidate the biomechanics of injury formation from impacts. In this context, surface waves generate a wavelength on the order of the centimeter with a typical penetration length of the same order. Therefore, investigating surface waves at depth is crucial for understanding their relationship with the physics of soft tissue injuries. Planar surface waves produce particle motion along two dimensions, the direction of propagation and the depth direction, making them more challenging to measure when compared to polarized shear waves that only produce motion in one direction. This study presents an experimental setup capable of generating Scholte wave propagating in a soft solid–liquid interface. In particular, we studied a tissue-mimicking phantom material, such as gelatin, under a layer of water. Ultrasound imaging techniques, operating at 8600 frames per second, and a one-dimensional cross-correlation algorithm were used to independently estimate the two components of the wave’s particle displacement. We conducted experiments sweeping frequencies between 50 and 500 Hz for different gelatin stiffness, finding a surface wave speed of 0.86 times the shear wave speed and a penetration distance of 0.35 times the wavelength. These results agree with the theory of Scholte waves propagating in an incompressible semi-infinite elastic medium covered by an incompressible fluid.

Hydrodynamic Modeling of Calender Forming (Pressing) of Sheet and Roll Composites

Statement of the problem. The problem of production of sheet and roll composites necessary for the construction industry in the conditions of roll-calender processing and pressing of polymer materials and reinforcing tape is investigated. The solution of this problem is carried out within the framework of a hydrodynamic model of a stationary, isothermal flow of a viscous, incompressible medium in a flat symmetrical layer between two ideal cylinders (calenders) rotating in opposite directions with equal speeds. The purpose of such a statement is to analyze the stress-strain obtained under calendering conditions, as well as to obtain at the design stage of the molding process the results of calculating a complex of energy-power parameters that allow controlling the calendering process. Results and conclusions. Using the approximate Reynolds equations in the part of the deformable medium in direct contact with the cylinders, assuming that its rheological properties correspond to a Newtonian fluid, analytical expressions are obtained for the force and kinematic parameters of pressing, depending on the thickness of the formed layer and the speed of rotation of the rollers. Based on the equilibrium condition, an expression is obtained for the magnitude of the thrust force that ensures the molding process. It is shown that the value of the rolling friction coefficient decreases with a decrease in the thickness of the layer, the specific spacer force, as well as the angular velocity of rotation.

POWER EXPENSES TO DRIVE A TURBINE STAGE UNDER LOW-FLOW MODES

The task of determining the power consumption during the operation of the turbine stage in the low-cost mode is considered. To solve it, a theoreticalexperimental approach was used, based on the one-dimensional theory of the movement of a incompressible working medium and the results of an experimental study of a number of models of large fanning degrees, in which the working medium is air. When operating heating turbines, the lowpressure cylinders work in the area of low-cost modes up to 85 % of the time, both with a partially and completely closed rotary diaphragm of the regulating stage. When the volumetric flow rate of steam in the flow part of the low-pressure cylinder at low-flow modes is reduced, a sleeve separation of the flow develops and a vortex is formed in the inter-crown gap of the stage. The mode in which the power supplied to the impeller is spent on maintaining these currents corresponds to the "pure" ventilation mode. In contrast to the existing methods for determining the ventilation power losses in the stages of large fanning, the paper proposes a dependence based on the geometric configuration of the stages of the low-pressure cylinder of the heating turbines and their operating conditions. Taking into account the processes in the stage and the data obtained on the experimental stand, formulas were determined account the components of power losses - the function of the influence of the flow exit angles from the guide apparatus of the stage; influence of fanning l/Dmid; influence of the relative width of the working blade B/Dmid and the angle of inclination of the peripheral meridional circuit γm. The obtained dependences, which allow determining the coefficients of power consumption for the stage in the ventilation mode and the operation mode before idling, make it possible to calculate power consumption in the entire range of changes in low-flow modes. On the example of the last stage of the T-250/300-240 turbine, a comparison of the results of calculation studies according to the proposed dependence with the results obtained in real conditions of full-scale experiments was performed, which showed that their difference does not exceed 5 %.

Parametric Study on the Sensitivity of Different Properties of Waves Propagating through an Incompressible Medium

This article deals with the study of disturbance that travels through the transversely isotropic medium in the form of waves. The particles of the considered medium have an additional property of small-scale internal rotation along with macroscopic translational deformation. This extra translational freedom causes the medium to be micropolar in nature. Along with this, the medium is incompressible, and the dispersion relation of waves propagating through the medium is obtained under specific plan-strain conditions. From the dispersion relation, we can conclude that because of incompressibility, three transverse waves propagate through the medium. The velocity profile, attenuation coefficient, and specific heat loss for these waves are discussed for a particular medium. Later, the special normalized local sensitivity analysis (NLSA) technique is used to depict the effects of parameters on the outcomes of the mathematical model. The obtained results are represented graphically for a particular medium. The proposed model is used to model the mechanical behavior of complex materials with microstructural heterogeneity, such as composites and biological tissues.

Investigating the dynamics of point helical vortices on a rotating sphere to model tropical cyclones

A general exact weak solution to the non-linear equation of the conservation of the absolute vorticity in a thin layer of an incompressible medium on a rotating sphere is proposed. It takes into account the helicity of the point vortices and the non-uniformity of the depth of the layer. This is used to develop a model of the observed interactions of spiral atmospheric vortices. The fusion of two-point helical vortices (HVs) on the rotating sphere is considered. We also analyze the prognostic applicability of the dynamics of the HVs for modeling the abrupt changes observed in the trajectories of tropical cyclones and their landfall in comparison with the traditional approach. The analytical condition for chiral symmetry violation related to the direction of the movement of the center of a helical cyclone is obtained.

Deviatoric stress waves due to rheology in incompressible thermoviscoelastic solid medium with small strain, small deformation physics

Deviatoric stress waves due to rheology in incompressible thermoviscoelastic solid medium with small strain, small deformation physics

Research on internal flow characteristics of the labyrinth disc regulating valve

As a new type of multi-stage high-performance regulating valve, the labyrinth disc regulating valve takes advantage of the labyrinth channel on the sleeve disc to dissipate the high-pressure differential energy into the multi-stage throttling flow resistance. It can effectively prevent the failure problems such as erosion, cavitation, and noise caused by extreme working conditions such as the high-pressure differential, high-speed flow, and high temperature. In order to study the flow characteristics of the fluid inside a new type of labyrinth disc-type regulating valve in the thermal power process, an experiment on the pressure reduction characteristics of a double-entrance labyrinth flow channel was carried out. Firstly, an experimental device was designed and built, and then the pressure drop values of each horizontal flow channel were measured and compared with the numerical simulation results to verify the accuracy of the numerical model. Aiming at the actual working conditions of the labyrinth disc regulating valve in thermal power generation applications, the Realizable k-- turbulence model and the mixture multiphase flow model are combined with the Zwart-Gerber-Belamri cavitation model to clarify the flow characteristics and cavitation of the incompressible medium water in the labyrinth flow channel inside the labyrinth disc regulating valve and the flow characteristics of a double-entrance labyrinth flow channel under different opening degrees.

The Toda Flow as a Porous Medium Equation

We describe the geometry of the incompressible porous medium (IPM) equation: we prove that it is a gradient dynamical system on the group of area-preserving diffeomorphisms and has a special double-bracket form. Furthermore, we show its similarities and differences with the dispersionless Toda system. The Toda flow describes an integrable interaction of several particles on a line with an exponential potential between neighbours, while its continuous version is an integrable PDE, whose physical meaning was obscure. Here we show that this continuous Toda flow can be naturally regarded as a special IPM equation, while the key double-bracket property of Toda is shared by all equations of the IPM type, thus manifesting their gradient and non-autonomous Hamiltonian origin. Finally, we comment on Toda and IPM modifications of the QR diagonalization algorithm, as well as describe double-bracket flows in an invariant setting of general Lie groups with arbitrary inertia operators.

The Study of the Lift Coefficient of NACA-0012 Airfoil Subjected to an Incompressible Media by Using the Cartesian Cut Cell Method

In order to shorten product design time, engineers need to create high quality meshes within a few days or hours. Newer automated techniques have been published to tackle this need, the Cartesian Cut Cell is one of them. This study is focused on the effect of different Cut Cell meshing strategies on the accuracy of aerodynamic performance predictions. The method can be described as a methodology in which Cut Cells are applied to the geometry utilizing a process involving rectangular/hexagonal cells on a regular lattice cutting through the geometry. The Cut Cell meshing is a general purpose designed for ANSYS FLUENT, making use of Workbench to construct the airfoil shape and the mesh. The results obtained for NACA-0012 are computed using two models available in ANSYS FLUENT, namely the Eq. 2 k-ω SST and the Eq. 3 k-kl-ω models. The three-dimensional numerical simulations were created for steady incompressible flow around NACA-0012 shape. Lift coefficient, Boundary layer thickness, mesh expansion ratio, and mesh density variation parameters were investigated. For this application both models produce good lift results. k-kl-ω produce better lift and the results are close to the measured data. The Cut Cell method showed a very good agreement between Computational Fluid Dynamics results and experimental data. This work illustrated that the Cartesian Cut Cell method has the ability to generate high quality mesh, which captures the details of the viscous boundary layer easily. The future work is to use more sophisticated turbulence models and mesh refinement for air craft wing with flap.

The role of acoustofluidics and microbubble dynamics for therapeutic applications and drug delivery.

Targeted drug delivery is proposed to reduce the toxic effects of conventional therapeutic methods. For that purpose, nanoparticles are loaded with drugs called nanocarriers and directed toward a specific site. However, biological barriers challenge the nanocarriers to convey the drug to the target site effectively. Different targeting strategies and nanoparticle designs are used to overcome these barriers. Ultrasound is a new, safe, and non-invasive drug targeting method, especially when combined with microbubbles. Microbubbles oscillate under the effect of the ultrasound, which increases the permeability of endothelium, hence, the drug uptake to the target site. Consequently, this new technique reduces the dose of the drug and avoids its side effects. This review aims to describe the biological barriers and the targeting types with the critical features of acoustically driven microbubbles focusing on biomedical applications. The theoretical part covers the historical developments in microbubble models for different conditions: microbubbles in an incompressible and compressible medium and bubbles encapsulated by a shell. The current state and the possible future directions are discussed.

О проблемах при моделировании плоских течений вязкой жидкости при повышенных значениях числа Рейнольдса вихревыми методами в программном комплексе VM2D

Vortex methods of computational fluid dynamics are an efficient tool in engineering practice for estimating hydrodynamic loads acting on bodies placed in a flow. Their usage allows for solving of coupled hydroelastic problems with relatively small computational cost. In many applications, the cross flow around structural elements with large elongation is considered, that allows one to use the flat cross-sections method providing the acceptable accuracy. Thus, flat flows simulation around airfoils is required. Modern modifications of vortex particle methods make it possible to simulate flows of a viscous incompressible medium. Based on the method of viscous vortex domains in 2017-2022 the VM2D code have been developed in Bauman University and Ivannikov Institute for System Programming. This code allows for flow simulating around airfoils with acceptable accuracy at low Reynolds numbers, while for higher Reynolds numbers, correct results are observed only for airfoils with sharp edges and corner points, and only in regimes where the most intensive flow separation takes place at these points. The reason for the error in the results for other regimes is seen in incorrect modeling of the flow separation on smooth airfoil surface line at high Reynolds numbers, which, in turn, is a consequence of incorrect modeling of vorticity evolution in the vicinity of separation points (zones). Some results of flow simulations around different airfoils at different values of the Reynolds number are presented and a hypothesis explaining the reason for the discrepancy between numerical results and experimental data is proposed. It is shown that the kinetic energy spectrum of turbulence corresponds to “two-dimensional turbulence”.

A new trigonometric shear deformation theory for free vibration of graphene reinforced metal–matrix nanocomposite plate submerged in fluid medium

A novel trigonometric shear deformation plate theory (TSDPT) is proposed for the vibrational characteristics of the functionally graded graphene-reinforced metal matrix nanocomposite (FG-GRMMNC) plate immersed in the fluid medium. Combining trigonometric and inverse trigonometric functions, a new shape function is presented to determine the distribution of the transverse shear strains and stresses without using any transverse shear correction factors. Using a combination of the Halpin–Tsai micromechanical model and the rule of mixtures, the properties of three distributions of GRMMNC are investigated. Aluminum serves as the matrix material, and graphene nanoplatelets are utilized as the reinforcing material. Due to the ideal and incompressible fluid medium, the influence of hydrostatic pressures and free surface waves on structures is disregarded. The governing equations are derived using the Hamilton’s principle, taking into account the influences of three plate–fluid interaction models. The Galerkin’s method is then utilized to determine the natural frequencies of the FG-GRMMNC plate with simply supported boundary conditions. To confirm the validity of the proposed theory, the obtained outcomes are compared with those found in the published works. On the vibrational characteristics of the nanocomposite plate, influences of material properties, distribution patterns, geometrical parameters, and fluid environments are also investigated in depth.

Global well-posedness and optimal decay estimate for the incompressible porous medium equation near a nontrivial equilibrium

In this paper, we prove global well-posedness and optimal decay estimate of solution to the 2D inviscid incompressible porous medium equation near a nontrivial equilibrium x2. The obtained decay rate is optimal in sense that this rate coincides with that of the linear system. In particular, the obtained L∞ estimate of u2 improves the associated work in (T.M. Elgindi, Arch. Ration. Mech. Anal. 2017).

Hydrodynamic modeling of the molding process of baking dough and confectionery masses

The process of molding (flat pressing) of bakery and confectionery dough blanks, which are one of the important preparatory elements in the production technology of high-quality finished products, is investigated. Considering that the estimated indicators for predicting the quality of bakery and confectionery products are determined mainly by the rheological properties of the dough, a hydrodynamic model is proposed in which the problem of axisymmetric flow of a nonlinear viscous layer completely filling the space between two parallel absolutely rigid converging disks is solved. It is assumed that the incompressible model medium is described by the well-known Ostvadde de Ville equation, which also corresponds to bakery and flour confectionery semi-finished products by its basic rheological properties. The purpose of the proposed work is to study the flow of test material under axisymmetric compression under the action of constant force of approaching disks. The use of axisymmetric flow is due to its greater realism and ease of reproduction in laboratory conditions. Assuming a linear dependence of the tangential stresses on the radial pressure gradient, as well as the ideal adhesion of the deformable medium at the boundary of the approaching planes, analytical expressions are obtained showing the distribution of the radial flow velocity over the thickness of the deformable layer. Using the continuity equation, the distribution of the molding pressure over the surface of the deforming plates and the total force from the approaching disks are obtained. The obtained ratios describe not only the dynamics of the medium flow, but also allow, under the conditions of a single test, to determine the values of the rheological characteristics of the model medium: the coefficient of consistency (density) and the degree of viscosity. The experimental dependence of the layer thickness on the deformation time makes it possible to control the technological process of axisymmetric molding of flat dough blanks: pizza, cakes, noodles, etc..