Viscous Incompressible Liquid Research Articles

Численный эксперимент в задаче о распространении малых возмущений в круглой трубе

Flow of viscous incompressible isothermal liquid in round pipe at moderate Reynolds numbers corresponding to laminar and turbulent modes is considered. The problem is solved in a non-stationary formulation, the spatial coordinate system of Euler is used. As a mathematical algorithm, quasi-hydrodynamic (QHD) equations are used. Within the framework of this algorithm, it was first shown that random perturbations of the input speed in the channel attenuate for small Reynolds numbers and lead to the formation of a turbulent mode for large Reynolds numbers.

Studying the influence of a gyrostatic moment on the motion of a charged rigid body containing a viscous incompressible liquid

The rotational motion of a charged rigid body (RB) is examined. The RB has a spherical cavity that contains an incompressible viscous liquid. The influence of a gyrostatic moment (GM), constant torques at the body-connected axes, and the action of the torque of a resistant force, due to the shape of the liquid, are considered. Assuming the liquid has a sufficiently high velocity, the Reynolds number does indeed have a small value. The regulating system of motion is derived in an appropriate formulation through Euler's equations of motion. The averaging method is used to approach a suitable form of the motion's governing system. In addition to using Taylor’s method to reach a solution for the averaged equations of motion of the RB, some initial conditions are considered to approach the required results. The asymptotic approach of the averaged system besides the numerical analysis enables us to obtain the appropriate results of the problem. To draw attention to the beneficial effects of the different values of the body’s parameter on the motion's behavior, these results are graphed through a computer program along with the associated phase plane curves. These diagrams illustrate the influence of several values respected to the GM, charge, body-constant torques, and resistive force torque. The stability of the RB's motion has also been discussed through the represented phase plane diagrams. These results are viewed as a generalization of prior ones, which have been reported for the scenario of an uncharged body or the absence case of the GM. The significance of the obtained results is due to its numerous real-world applications in life, such as for spaceships and wagons carrying liquid fuel.

Upscaling of dispersion in gas-liquid absorption on an inclined surface.

We extend the Taylor-Aris dispersion theory to upscale the gas absorption into a viscous incompressible liquid flowing along an inclined surface. A reduced-order model of advection-dispersion-reaction is developed with the aid of Reynolds decomposition and cross-sectional averaging techniques. The upscaled model allowed evaluation of the dispersion, advection, and absorption kinetics as a function of the Peclet number (Pe) and the Damköhler number (Da). The transport and kinetics parameters for the limiting cases of nonabsorption and absorption dominant are also evaluated. The upscaled model is solved analytically, and the obtained solution is used to evaluate the upscaled mass transfer between the gas and liquid. The results for the overall Sherwood number identify three regions: (i) advection dominant, (ii) transition where both advection and absorption play a role, and (iii) absorption dominant. The scaling relation between the Sherwood number (Sh) and the Da for the last region was determined to follow Sh∼Da^{1/2}. It is also revealed that in the first two regions, the Sherwood number versus the Peclet number exhibits a bell-shaped (or Gaussian) behavior, suggesting an optimal Pe that maximizes mass transfer between gas and liquid in these regions. The model and insights presented have the potential to be applied in a wide range of industrial separation processes involving the interaction of a gas exposed to a liquid flowing downward on an inclined surface under gravity.

Self-Propulsion of Two Contacting Bubbles Due to the Radiation Interaction Force.

In this paper, we consider a new bubble-based microswimmer composed of two contacting bubbles. Under the action of an acoustic field, both bubbles are oscillating, and locomotion of the two-bubble system is observed. A theory is developed that allows one to calculate the acoustic radiation interaction forces between two gas bubbles in an incompressible viscous liquid for any small separation distance between the bubbles. This theory is used to demonstrate that two acoustically excited bubbles can create a self-propelled microswimmer due to a nonzero net force experienced by the bubbles when they come in contact. Experimental evidence of the creation of such a swimmer and of its motion is provided.

О вибрационной неустойчивости Кельвина-Гельмгольца для жидкостей сравнимых вязкостей

We study the behavior of a two-dimensional system of immiscible incompressible viscous liquids in a rectangular cavity under horizontal harmonic high-frequency vibrations. In this system, influ-ence of pulsating motions of the liquids causes the development of Kelvin-Helmholtz instability leading to the formation of a quasi-stationary relief on the interfacial surface in the form of a frozen wave. A direct numerical modeling of the relief formation was carried out with the use of AN-SYS Fluent software; the influence of the viscosity ratio on the parameters of the frozen wave and the flow generated by vibrations near the interface was investigated. The dependencies of the char-acteristics of the wave relief on supercriticality and vibration frequency are plotted for different ra-tios of liquid viscosities. In the cases of vibrations with sufficiently high frequency, the obtained numerical results are consistent with analytical theory developed for ideal liquids. For lower fre-quencies, the observed relief wavelength turns out to be noticeably longer than the theoretical pre-dictions. For high vibration frequencies, with increasing supercriticality the wavelength increases monotonically, while for low frequencies it decreases monotonically. The relief height increases as the intensity of vibration influence increases; with significant supercriticality, it exceeds the theo-retical values obtained in the ideal fluid approximation. The generation of reverse averaged flow by vibrations near the interface is found when viscosity of the upper fluid is low. In this case, the flows in the upper and lower liquids are directed opposite to each other and a noticeable decrease in the average flow generation intensity is observed.

The Tip of Dendritic Crystal in an Inclined Viscous Flow

We study the flow around the tip of a dendritic crystal by an inclined stream of viscous incompressible liquid. The tip shape is chosen accordingly to recent theory [Phil. Trans. R. Soc. A 2020, 378, 20190243] confirmed by a number of experiments and computations [Phil. Trans. R. Soc. A 2021, 379, 20200326]. Our simulations have been carried out for a 0, 30, 60, and 90-degree flow slope to the dendrite axis. We show that the stream inclination has a significant effect on the hydrodynamic flow and shear stress. In particular, a transition from laminar to turbulent currents on the upstream side of the dendritic crystal may occur in an inclined hydrodynamic flow. This leads to the fact that the heat and mass transfer mechanisms on the upstream and downstream sides of a growing dendritic crystal may be different.

Numerical Simulation of Convective Diffusion of Point Particles in a Laminar Flow Past a Row of Profiled Hollow Fibers

The numerical modeling of transverse laminar flow past a new type of hollow-fiber membranes with external profiling has been performed. A model system of parallel fibers with symmetrical parallel protrusion obstacles or grooves is considered. The absorption of point particles (solute or gas molecules) from a laminar transverse flow of a viscous incompressible liquid (gas) is calculated for a row of fibers, and the dependences of the efficiency of retention of particles by fibers on the Peclet (Pe), Reynolds (Re), and Schmidt (Sc) numbers and on the distance between neighbor fibers in a row are determined. The flow velocity and concentration fields are calculated by numerical solution of the Navier–Stokes equations and the convective diffusion equation in a wide range of Peclet numbers Pe = 0.1 − 105 for Sc = 1, 10, 1000 and Re ≤ 100.

Guided bubbles and wet foam for realistic whitewater simulation

We present a method for enhancing fluid simulations with realistic bubble and foam detail. We treat bubbles as discrete air particles, two-way coupled with a sparse volumetric Euler flow, as first suggested in [Stomakhin et al. 2020]. We elaborate further on their scheme and introduce a bubble inertia correction term for improved convergence. We also show how one can add bubbles to an already existing fluid simulation using our novel guiding technique, which performs local re-simulation of fluid to achieve more interesting bubble dynamics through coupling. As bubbles reach the surface, they are converted into foam and simulated separately. Our foam is discretized with smoothed particle hydrodynamics (SPH), and we replace forces normal to the fluid surface with a fluid surface manifold advection constraint to achieve more robust and stable results. The SPH forces are derived through proper constitutive modeling of an incompressible viscous liquid, and we explain why this choice is appropriate for "wet" types of foam. This allows us to produce believable dynamics from close-up scenarios to large oceans, with just a few parameters that work intuitively across a variety of scales. Additionally, we present relevant research on air entrainment metrics and bubble distributions that have been used in this work.

THE STATE OF THE PROBLEM OF THE JOINT MOVEMENT OF FLUID IN THE PORE SPACE

This article discusses the problems of studying the issue of joint motion of liquids in the porous space. The article provides the construction of a mathematical model of the theory of ltration, which describes phase transitions. The main diculty in constructing this model is associated with the fact that free interphase boundaries create regions that change over time, and it is required to nd the temperature or concentration elds of substances in them. In this case, the coordinates of the considered phase boundaries are not initially specied and must be calculated already in the process of solving. For this, a derivation of the averaged equation for the problem of nding the rupture surface during the movement of two incompressible viscous liquids in the pores of the soil skeleton was proposed. The article deals with the case when the skeleton is an absolutely rigid body. The rationale was given for the choice of an averaged ltration model instead of a microscopic one. The main research methods are classical methods of mathematical physics, functional analysis and computation methods of the theory of partial dierential equations, as well as dierence methods. The formulation of the problem is given, and the denition of a generalized solution for solving the problem is provided. Next, an averaged model is derived and the existence of at least one generalized solution to the problem is proved.

Non-Linear Oscillations of the Channel Wall Filled with a Viscous Liquid Induced by the Vibrating Foundation

The paper considers the issues of mathematical modelling nonlinear vibrations for the wall of a narrow parallel walled channel filled with a viscous incompressible liquid. The upper channel wall is a rigid rectangular plate supported by nonlinear spring with a cubic restoring force, while the bottom one is a fixed rigid rectangular plate. The case of nonlinear oscillations of the upper channel wall due to channel’s foundation vibration has been investigated in the frame of hydroelasticity problem. The main attention is paid to the consideration of steady-state nonlinear oscillations for the upper channel wall and the creeping motion of the liquid in the channel. The liquid layer reaction acting on the upper channel wall is found and the channel’s wall nonlinear oscillation equation is obtained taking into account the energy dissipation due to the liquid viscosity. It was shown that this equation coincided with the Duffing one. The solution of the nonlinear oscillation equation was carried out by the harmonic balance method. Based on the found solution, the hydroelastic response of the channel wall on a nonlinear elastic suspension to the channel foundation vibration was determined.

Evaluation of Nusselt Number for a Flow in a Parallel Plates Using Magnetohydrodynamics Second-Order Slip Model

Abstract This study is the examination of magnetohydrodinamics (MHD) flow and heat transfer between microparallel plates with constant wall heat flux using two different boundary condition coefficients, widely known in the literature, in a second-order slip flow and temperature jump boundary condition model. The stream of incompressible viscous liquid between rigid microtwin plates with electrical conductivity was assumed to be steady, laminar, hydrodynamically and thermally fully improved. The effects of dilution, viscous dissipation, and axial transmission, which are significant at the microscale, are also contained in the analysis. Momentum and energy equations are solved analytically under second-order slip velocity and temperature jump boundary conditions. Closed type answers for the temperature field and the fully developed Nusselt number are derived as a function of the magnetic parameter (MHD), Knudsen number and modified Brinkman number. Particular attention has been paid to the effects of the magnetic field on the second-order slip flow conditions. The use of the second-order boundary condition coefficients proposed in the study varies according to the proposed boundary conditions and tends to increase and decrease the heat transfer. Otherwise, MHD increases the heat transfer for both boundary conditions.

Electroconvection of a weakly conducting liquid subjected to unipolar injection and heated from above

The problem of a weakly conducting viscous incompressible liquid placed in an infinite horizontal capacitor is solved numerically by the two-field method using finite difference schemes. It is assumed that a free charge arises in the liquid (heated from above) due to a homogeneous unipolar autonomous injection from the cathode. In mechanical equilibrium, the Coulomb force acting on the charge injected from the cathode and the buoyancy force are oppositely directed, and the oscillatory instability and wave supercritical modes of convection occur. The problem is investigated in full statement, i.e., the electric field redistribution created by the charge redistribution is taken into account. Periodic boundary conditions are used on the lateral boundaries of the studied area, which makes it possible to detect and analyze not only the mode of stationary electroconvection, but also the solutions in the form of traveling waves. A mixed convection regime that arises as a result of a direct Hopf bifurcation from the state of mechanical equilibrium was found and investigated. In this regime, the phases of standing and traveling waves happen one after another. Traveling waves, modulated traveling waves and stationary convection successively replace each other with an increase in the control parameter (the electric Rayleigh number) proportional to the voltage on the capacitor plates. A bifurcation diagram that characterizes the intensity and phase velocity of supercritical fluid flow modes is plotted. The intensity of the stationary electroconvective flow is an order of magnitude higher than the intensity of the flow in the traveling wave mode. The influence of the grid scale on the observed regimes is studied.

Обоснование математической модели многослойного сдвигового течения бетонной смеси по наклонной поверхности откоса канала

In assessing the hydrodynamic parameters of concrete mixtures placed in monolithic channel linings, the most important point is the development of mathematical models that describe the processes of concrete mixture slippage, taking into account the dynamics of speed reduction and increase in the viscosity of the concrete mixture over time. The main regularities of the hydrodynamic multilayer shear flow of a concrete mixture on the surface of a flat channel slope, where a multilayer plane-parallel viscous flow moves under the action of gravity, are investigated. The boundary value problem of a two-layer flow of a concrete mixture is solved, a system of two layers of viscous incompressible liquids is considered with the properties of each, respectively, the thickness of the bottom and upper layers, taking into account the sliding velocity of particles of a viscous liquid located one above the other. The obtained analytical solution of the problem in the stationary regime of a two-layer shear flow of two different immiscible viscous fluids with different properties makes it possible to formulate the main laws of such flows and describe the processes of sliding of a multilayer concrete mixture, taking into account the dynamics of speed reduction and increase in the viscosity of the concrete mixture over time. A mathematical model has been developed where the dynamic and kinematic structures of a multilayer shear flow in both viscous layers are presented separately.

THREE-DIMENSIONAL MODEL FOR NUMERICAL STUDY OF THE MOVEMENT OF A VISCOUS LIQUID ON THE ROUNDING OF AN OPEN CHANNEL

The article considers the statistically stationary flow of a viscous incompressible liquid at the turn of an open inclined channel of a rectangular cross-section with large Reynolds numbers. A three-dimensional numerical model with a periodic boundary condition along the curvilinear axis of the main flow is proposed. Three-layer boundary conditions are placed on the wall and near the channel wall, and zero tangential voltage conditions are placed on the free surface. To solve the problem, the predictor-corrector scheme of the method of fractional steps by horizontal coordinates and with a cosine-oidal spectral function by vertical coordinate are used. A general algorithm for solving the problem with an isoperimetrically optimized cross-section parameter is compiled.

Evolution equation for nonlinear Lucassen waves, with application to a threshold phenomenon.

A nonlinear, fractional, surface wave equation with a spatial derivative of second order was developed by Kappler, Shrivastava, Schneider, and Netz [Phys. Rev. Fluids 2, 114804 (2017)] for propagation along an elastic interface coupled to a viscous incompressible liquid. Linear theory for the attenuation and dispersion was developed originally by Lucassen [Trans. Faraday Soc. 64, 2221 (1968)]. Kappler et al. introduced a fractional time derivative to account for the Lucassen wave attenuation and dispersion, and they included quadratic and cubic nonlinearity associated with compression of the elastic interface. Presented here is an integrated form of their time domain equation for progressive waves that is first order in the spatial derivative. Solutions of this evolution equation capture the main features of waveforms predicted by the full model equation of Kappler et al., especially the formation and propagation of shocks, while the evolution equation can be solved numerically with substantially less computational cost. Approximate analytical expressions obtained from the evolution equation for the nonlinear propagation speed and attenuation of a compression pulse reveal that a threshold phenomenon discussed by Kappler et al. is due to competition between quadratic and cubic nonlinearity associated with a lipid monolayer interface.

Numerical Solution of One Problem of Carbon Dioxide Injection into the Rock

The paper considers a two-dimensional mathematical model of filtration of a viscous incompressible liquid or gas in a porous medium. A unique feature of the model under consideration is the incorporation of poroelastic properties of the solid skeleton. From a mathematical point of view, the equations of mass conservation for liquid / gaseous and solid phases, Darcy's law, the rheological ratio for a porous medium, and the conservation law of the balance of forces are considered. The work is aimed at numerical study of the model initial-boundary value problem of carbon dioxide injection into the rock with minimum initial porosity. Also, it is necessary to find out the parameters at which the porosity will increase in the upper layers of the rock and, as a result, the gas will come to the surface. Section 1 contains a statement of the problem and a brief review of scientific papers related to this topic. In Section 2, the original system of constitutive equations is transformed. In the case of slow flows, when the convective term can be neglected, a system arises that consists of a second-order parabolic equation for the effective pressure of the medium and a first-order equation for porosity. Section 3 presents the results and conclusions of a numerical study of the initial-boundary value problem.

Numerical simulations of liquid-solid flows with free surface by coupling IMPS and DEM

Liquid-solid two-phase flow with free surface is a significant issue in nature and engineering fields. In present study, an in-house solver MPSDEM-SJTU based on a fully Lagrangian coupled method is developed to solve the problems of liquid-solid two-phase flows. The improved moving particle semi-implicit (IMPS) method is applied for the simulation of incompressible viscous liquid flows while the discrete element method (DEM) is used to model the interaction among the solid particles. The coupling strategy is based on the local averaging technique. Multiple time-step algorithm is adopted for those two methods to balance the stability and the efficiency. The free surface detection and the solid-fluid neighbor search are modified. A simulation of multi-balls collapse is conducted to verify the DEM program of the present solver. The distributions of solid balls at different instants are in good agreement with the experimental results. Then, this solver is applied for two typical problems of two-phase flows, including the two-phase dam-break and the liquid-solid flows in a rotating cylindrical tank. The flow front, flow patterns and the shape of solid beds all agree well with experimental observations.

СРАВНЕНИЕ МОДЕЛЕЙ ДВИЖЕНИЯ ИЗМЕЛЬЧАЕМОГО МАТЕРИАЛА В КОРПУСЕ ЦЕНТРОБЕЖНОЙ МЕЛЬНИЦЫ ВЕРТИКАЛЬНОГО ТИПА

Introduction. Improving the performance, increasing productivity, reducing the metal consumption of grinding equipment and other mining machines is usually a very expensive process. It requires a large amount of development work, the production of prototype machines, and a large amount of experimental research. In this regard, one of the most important tasks is to simulate the movement of bulk material in operations for processing minerals in various equipment. In such modeling, the discrete element method (DEM) is widely used. The purpose of the research is to compare the models of the movement of the crushed material in the body of a vertical centrifugal mill. Research methodology The motion of the bulk medium in a vertical centrifugal mill was modeled using two models. In the first model, the cylindrical body of the centrifugal mill was assumed to be stationary, and on its surface and on the entire surface of the rotor, conditions were set for the absence of a relative speed of movement of the crushed material. In the second model, a hydrodynamic model was used to describe the motion of a granular material as a viscous incompressible liquid with a compression ratio that depends on the pres-sure. In this model, the viscosity coefficient is represented as consisting of two terms: a constant (analogous to dynamic viscosity) and an excess pressure over hydrostatic pressure. Research results It is established that both models give the same character of the movement of the material in the mill body. It is determined that the absolute velocity of the material movement near the walls and near the mill rotor is approximately the same for both models, but in the data obtained using the hydrodynamic model, as the material moves away from the walls and the rotor, it slows down more than for the model using the discrete element method. It is revealed that the absolute velocity of the material movement near the walls and at the axis of the mill rotor is approximately the same for both models, but in the data obtained using the hydrodynamic model, as it moves away from the walls and the rotor, the material slows down significantly more than for the model using the discrete element method. Based on the simulation results, it can be concluded that for a more accurate simulation of the processes occurring during the rapid movement of bulk material in the grinding equipment, it is preferable to use a model using the discrete element method. It is advisable to use the hydrodynamic model for conducting a large number of search dawns or as a predicate model that will allow you to quickly set the initial velocity values for particles in a model using the discrete element method. Conclusions 1. A hydrodynamic model of the motion of a bulk medium in a vertical centrifugal mill, represented as a viscous incompressible liquid with a compression coefficient depending on the pressure has been developed. 2. It is established that for a more correct simulation of the processes occurring during the rapid movement of bulk material in the grinding equipment, it is preferable to use a model using the discrete element method. At the same time, it is advisable to use the hydrodynamic model for conducting a large number of search calculations or as a predicate model that will allow you to quickly set the initial velocity values for particles in a model using the discrete element method.

Tectonic depressions on the East-European and Siberian platforms: numerical modeling of convection beneath the Eurasian continent

In modern concepts, the upper mantle of the Earth is a highly viscous incompressible liquid, and its flow is described using the Navier – Stokes equations in the Oberbeck – Boussinesq and geodynamic approximations. Convective flows in the upper mantle play a decisive role in the kinematics of lithospheric plates and the geological history of continental regions. Mathematical modeling is a basic method for studying convective processes in the mantle. Our paper presents a numerical model of convection, which is based on the implicit artificial compressibility method. This model is tested in detail by comparing our calculation results with the results of a well-known international test. It is demonstrated that the Fedorenko grids sequence method is highly efficient and reduces the computing time almost by a factor of eight. The numerical model is generalized in order to state the problem in a spherical system of coordinates. It is used to analyse the distribution of convective flows in the upper mantle underneath the Eurasian continent. The analysis shows that the thickness and geometrical parameters of the lithospheric blocks are the factors of significant influence on the distribution of convective flows in the upper mantle. The resulting structure of convective flows is manifested in the surface topography of large platform areas wherein the lithosphere thickness is increased. Thus, the locations of extended downward convection flows under the East European and Siberian platforms are clearly comparable to syneclises observed in the study area.

Passive Balancing of the Rotor with an Auto-Balancing Device with a Viscous Incompressible Liquid

The use of liquid auto-balancers to compensate the operational changes in the imbalance of rotary systems without stopping them is of interest because of the relative structural simplicity of these devices, which are passive direct-acting regulators that do not require power supply and control systems to move correction masses. The experience of the study of passive auto-balancing devices (SBD) indicates that the existing theory (statements) of passive automatic balancing of the fluid is idealized and inaccurately describes the processes that occur with the working bodies during their operation. In particular, the lack of studies on the effect of liquid viscosity on the efficiency of self-balancing and the reasonableness of the selection of liquid during the development of fluid SBD demanded to analyse the operation of liquid SBD in the real system, taking into account the influence of liquid properties on the efficiency of the vertical rotor balancing process. It is shown that the efficiency of balancing increases with the approach of the angular velocity to the critical one and with the increase of the external resistance. The massive forces of the working fluid have less effect on the balancing efficiency than the viscosity for liquid SBDs.