Equations Of Fluid Motion Research Articles

The Catastrophe of Rapidly Rotating Fluids

In astrophysics, when studying rotating fluid systems (such as stars or gas giant planets), the rotation effect is most often neglected. This is explained by the fact that the various stars that populate our universe revolve around themselves at relatively low speeds. However, when trying to describe the internal structure of an astrophysical body, the speed of rotation of the fluid contained in the core of the star can reach very high values. It, therefore, becomes impossible to neglect centrifugal forces in the equation of fluid motion. In this work, we carry out a simplified but above all general study of rapidly rotating fluid systems. Euler’s equation then contains a centrifugal force term. The resolution of this equation leads to a solution that reveals a very particular property of this type of system: “the catastrophe of rapidly rotating fluids”.

Power-law generalization of Newton’s formula for shear stress in a liquid in the form of a tensor rheological relation

A generalization of Newton’s formula for the shear stress in a fluid is carried out by giving it a power-law form and the corresponding rheological relation is written in tensor form. Depending on the exponent in this rheological ratio, one can come to a description of either a laminar or turbulent flow regime. In the latter case, there is a system of differential equations with the no-slip boundary condition. The proposed system of equations for turbulent fluid motion can be useful, at least, for obtaining preliminary, estimated characteristics of turbulent flow before starting numerical modeling using modern differential turbulence models. For some values of the exponent, this system can be used to describe the behavior of power-law fluids, as well as fluids with small additives of polymers in the manifestation of the Toms effect.

The problem of hydraulic calculation of pressure distribution pipelines

Most production technologies require a uniform flow path of liquid from pressure distribution pipelines. To achieve this goal, it is proposed to introduce polymer additives into the liquid flow or to use converging distribution pipelines with a continuous longitudinal slot in the wall. To reduce the uneven operation of the distribution pipeline during discrete liquid dispensing, it is proposed to use cylindrical output rotary nozzles with a lateral orthogonal entry of the jet into the nozzle. The problem is the lack of methods for accurate hydraulic calculation of the operation of distribution pipelines. Adequate calculation methods are based on differential equations. Finding the exact solution of the differential equation of fluid motion with variable path flow rate for perforated distribution pipelines is urgent, because it still does not exist. The available calculation methods take into account only the right angles of separation of the jets from the flow in the distribution pipeline. These methods are based on the assumption that the coefficient of hydraulic friction and the coefficient of resistance of the outlets are constant along the flow. A calculation method is proposed that takes into account the change in the values of these resistance coefficients along the distribution pipeline. The kinematic and physical characteristics of the flow outside the distribution pipeline are also taken into account. The accuracy of calculating the value of the flow rate of water distributed from the distribution pipeline has been experimentally verified. The error in calculating the water consumption by the method assuming that the values of the resistance coefficients are unchanged along the distribution pipeline reaches 18.75 %. According to the proposed calculation method, this error does not exceed 6.25 %. However, both methods are suitable for the design of pressure distribution pipelines, provided that the jet separation angles are straight. Taking into account the change from 90° to 360° of the angle of separation of the jets from the flow in the distribution pipeline will expand the scope and accuracy of calculation methods.

The shape of wave-packets in a three-layer hydrodynamic system

The article is devoted to the problem of wave-packet propagation in a three - layer hydrodynamic system "layer with a hard bottom - layer - layer with a cover stratified by density. The current research on selected topics is reviewed. The mathematical formulation of the problem is given in dimensionless form and contains the equations of fluid motion, kinematic and dynamic conditions on the contact surfaces, as well as the boundary conditions on the lid and on the bottom. Using the method of multiscale developments, the first three approximations of the studied problem are obtained, of which the first two are given in the article, because the third approximation has a cumbersome analytical form. The solutions of the first approximation and the variance relation are presented. The evolution equations of the circumferential wave-packets on the contact surfaces are derived in the form of the nonlinear Schrodinger equation on the basis of the variance relation and the conditions for the solvability of the second and third approximations. A partial solution of the nonlinear Schrodinger equation is obtained after the transition to a system moving with group velocity. For the first and second approximations, the formulas for the deviations of the contact surfaces are derived, taking into account the solution of the nonlinear Schrodinger equation. The conditions under which the shape of wave-packets on the upper and lower contact surfaces changes are derived. The regions of familiarity of the coefficients for the second harmonics on the upper and lower contact surfaces for both frequency pairs, which are the roots of the variance relation, are presented and analyzed. Also, for both frequency pairs, different cases of superimposition of maxima and minima of the first and second harmonics, in which there is an asymmetry in the shape of wave packets, are graphically illustrated and analyzed. All results are illustrated graphically. Analytical transformations, calculations and graphical representation of results were performed using a package of symbolic calculations and computer algebra Maple.

Nonlinear Analysis of Tropical Waves and Cyclogenesis Excited by Pressure Disturbance in Atmosphere

The horizontal equations of motion for an inviscid homogeneous fluid under the influence of pressure disturbance and waves are applied to investigate the nonlinear process of solitary waves and cyclone genesis forced by a moving pressure disturbance in atmosphere. Based on the reductive perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies the Korteweg–de Vries equation with a forcing term (fKdV equation for short), which describes the physics of a shallow layer of fluid subject to external pressure forcing. Then, with the help of Hirota’s direct method, the analytic solutions of the fKdV equation are studied and some exact vortex solutions are given as examples, from which one can see that the solitary waves and vortex multi-pole structures can be excited by external pressure forcing in atmosphere, such as pressure perturbation and waves. It is worthy to point out that cyclone and waves can be excited by different type of moving atmospheric pressure forcing source.

Analytical modeling and dimensionless characteristics of open wet clutches in consideration of gravity

Wet clutches are widely used in power transmission, but lack of the fact of an energy loss in open state condition. The flow conditions in the fluid flow of an open wet clutches are analyzed by analytical means. The requisite simplifications that result in an analytically integrateable solution are stated in detail. Special emphasis is put on the role of gravitation in the equations of fluid motion. This force component leads to a slightly earlier aeration than stated in earlier conditions. The simplifications and the resultant solutions are considered by means of dimensionless quantities. Despite the actual geometric parameters the drag torque can be described as zeta_{mathrm{m}}=pi/mathrm{Re}_{mathrm{l}}. An additional aeration condition is introduced, which is based on the back flow of the radial velocity. This quantity can be described as non-dimensional volumetric flow rate Q^{*}. With these equations at hand the theoretical considerations are transferred to an evaluation with grooves, where a backward curved groove appears as beneficial for further investigations.

Power-law fluid micromixing enhancement using surface acoustic waves

This study is aimed to numerically investigate the mixing efficiency of non-Newtonian fluid flow based on surface acoustic waves. The standing ultrasonic waves have been used to investigate their effect on the mixing efficiency of shear-thinning fluids, as well as shear-thickening ones that flow through a microchannel. Due to differences in time scales of acoustic physics and the mixing process and nonlinearity of the problem, the perturbation theory is employed. The fluid’s general motion is changed by the acoustic radiation force resulting from standing acoustic waves and the drag force resulting from acoustic streaming to overcome this challenge. The governing equations of fluid motion are divided into three categories: zeroth-order background flow, first-order acoustic field, and second-order acoustic streaming. In addition, power-law fluid with different indices is considered to investigate the impact of the shear-thinning, as well as the shear-thickening behavior on the mixing efficiency. Based on the obtained results, it is found that the acting fluid input velocity and power-law index significantly affect the efficiency of mixing. Moreover, it is revealed that although shear-thickening fluids experience a lower mixing efficiency at any given flow rate, the rate of increase in mixing efficiency at the pressure of acoustic field is much higher for highly shear-thickening fluids in comparison with shear-thickening fluids. The findings also show an increase of 71% in mixing efficiency for the shear-thickening fluid when the power-law index is 1.2, with an inlet velocity of 0.005 m/s and an actuation frequency of 1.97 MHz.

Numerical algorithm based on regularized equations for incompressible flow modeling and its implementation in OpenFOAM

Various aspects of incompressible viscous flow modeling using regularized equations are considered: from the formulation of quasi hydrodynamic (QHD) equations for fluid motion to the demonstration of the capabilities of the associated OpenFOAM solver QHDFoam.The numerical algorithm for fluid flow simulations is devised using the QHD approach. Unlike prominent operator splitting techniques (PISO, SIMPLE and their variants), the QHD approach provides an equation for pressure explicitly. Practical issues of QHD approximation and stability criteria for the numerical scheme are presented.Several problems are used to validate the developed solver and to demonstrate its capabilities: backward facing step, skewed cavity flow, natural convection and generation of internal gravity waves attractor. Results of simulations for the last case are obtained using 3 different methods to show peculiarities of the quasi hydrodynamic equations approach in comparison with the standard PISO algorithm and finite-volume method and with the solution based on the spectral element method.The source code of the developed OpenFOAM solver(s) can be downloaded from the GitHub repository: https://github.com/unicfdlab/QGDsolver.

Significant Production of Thermal Energy in Partially Ionized Hyperbolic Tangent Material Based on Ternary Hybrid Nanomaterials

Nanoparticles are frequently used to enhance the thermal performance of numerous materials. This study has many practical applications for activities that have to minimize losses of energy due to several impacts. This study investigates the inclusion of ternary hybrid nanoparticles in a partially ionized hyperbolic tangent liquid passed over a stretched melting surface. The fluid motion equation is presented by considering the rotation effect. The thermal energy expression is derived by the contribution of Joule heat and viscous dissipation. Flow equations were modeled by using the concept of boundary layer theory, which occurs in the form of a coupled system of partial differential equations (PDEs). To reduce the complexity, the derived PDEs (partial differential equations) were transformed into a set of ordinary differential equations (ODEs) by engaging in similarity transformations. Afterwards, the converted ODEs were handled via a finite element procedure. The utilization and effectiveness of the methodology are demonstrated by listing the mesh-free survey and comparative analysis. Several important graphs were prepared to show the contribution of emerging parameters on fluid velocity and temperature profile. The findings show that the finite element method is a powerful tool for handling the complex coupled ordinary differential equation system, arising in fluid mechanics and other related dissipation applications in applied science. Furthermore, enhancements in the Forchheimer parameter and the Weissenberg number are necessary to control the fluid velocity.

Determination of Boussinesq and Coriolis coefficients for pipelines operating with discharge connection along the path

Peculiarities of the diagrams of averaged fluid velocities in the cross-sections of pressure collecting perforated pipelines were determined on the basis of the experimental studies conducted by the authors.
 The most characteristic typical diagrams of averaged velocities in the pipelines cross-sections with their different design characteristics were given. A comparative analysis of the obtained diagrams with the diagrams of velocities that occur during uniform motion in pressure pipelines with solid walls was carried out. It is shown that the main difference between them occurs in the flow zones, which are located near the pipeline walls. It was explained by the connected liquid jets effect on the main flow. The degree of diagrams deformation was estimated by the value of the momentum coefficient α0 (Boussinesq coefficient) and the coefficient of kinetic energy α (Coriolis coefficient). It was determined that in the general case these coefficients will be variable in magnitude along the length of the studied pipes. Nevertheless, these coefficients are recommended to be constant in magnitude in engineering calculations. The limits of the structural characteristics of collecting perforated pipes for which this non-uniformity of the diagrams must be taken into account, and for which it can be neglected were determined on the basis of the analysis of the equation of fluid motion with a variable flow rate.

Experimental Study of Prefabricated Crack Propagation in Coal Briquettes under the Action of a CO2 Gas Explosion.

CO2 deep-hole presplit explosions are an important technology for enhancing gas drainage in low-permeability coal seams. In the process of a CO2 gas explosion, the initial burst crack generated by the shock wave expands the crack tip under the splitting action of high-pressure CO2 gas. To explore the effects of CO2 gas explosions on crack tips, we constructed an analytical model of gas pressure attenuation at different positions based on fluid motion equations, proposed equations for crack opening and growth rates, and inverted the energy field of the whole process of CO2 blasting. We used a test platform for the independent development of CO2 gas explosions under experimental conditions of 1 MPa axial pressure and 2 MPa CO2 gas pressure and a VIC-3D measurement system. We conducted the gas explosion experiments on prefabricated cracked samples with a crack length of 10 mm and width of 0.2 mm to analyze the dynamic response of the crack tip. The results showed that there were three stages in the propagation of a prefabricated crack under the action of a CO2 gas explosion. The first stage, from 0 to 290 ms, included energy storage at the crack tip and a maximum opening rate increment of 0.0043 m/s. The second stage, from 291 to 295 ms, was rapid crack propagation with maximum opening rate increment and propagation rates of 0.1865 and 5.35 m/s, respectively. In the third stage, from 296 to 309 ms, the crack tip propagated slowly, the maximum opening rate increment and growth rates were 0.0969 and 5.81 m/s, respectively, and the crack arrest coordinates were 4.57 and 35.28 mm. The experimental study verified the accuracy of the calculation model, proved that CO2 gas promotes the growth of crack tips, explained the spatiotemporal evolution mechanism of the CO2 explosion process, and provided experimental support for subsequent research related to explosions.

Leveraging Elasticity to Uncover the Role of Rabinowitsch Suspension through a Wavelike Conduit: Consolidated Blood Suspension Application

The present work presents a mathematical investigation of a Rabinowitsch suspension fluid through elastic walls with heat transfer under the effect of electroosmotic forces (EOFs). The governing equations contain empirical stress-strain equations of the Rabinowitsch fluid model and equations of fluid motion along with heat transfer. It is of interest in this work to study the effects of EOFs, which are rigid spherical particles that are suspended in the Rabinowitsch fluid, the Grashof parameter, heat source, and elasticity on the shear stress of the Rabinowitsch fluid model and flow quantities. The solutions are achieved by taking long wavelength approximation with the creeping flow system. A comparison is set between the effect of pseudoplasticity and dilatation on the behaviour of shear stress, axial velocity, and pressure rise. Physical behaviours have been graphically discussed. It was found that the Rabinowitsch and electroosmotic parameters enhance the shear stress while they reduce the pressure gradient. A biomedical application to the problem is presented. The present analysis is particularly important in biomedicine and physiology.

SIMULATION OF WASTE WATER TREATMENT BASED ON CFD MODEL: EXPRESS CALCULATION

Purpose. Development of CFD model to evaluate the efficiency of wastewater treatment in a horizontal settler. The CFD model can be used to calculate flow hydrodynamics and mass transfer in settlers with complex geometric shape in the area of wastewater flow. Methodology. For numerical simulation of the process of wastewater movement in a horizontal settler, two mathematical models are used. The first model is based on the motion equations of a viscous incompressible fluid – the Navier-Stokes equations. The Navier-Stokes equations are written in the variables «vorticity - flow function». A two-dimensional mass transfer equation is used to calculate the concentration of a pollutant in a horizontal settler. To numerically integrate the two-dimensional mass transfer equation, a finite-difference splitting scheme is used. The splitting of the modeling equation of mass transfer is carried out so that at each fractional step to determine the unknown value of the pollutant concentration by an explicit formula. For numerical integration of the vortex transfer equation and the equation for the flow function (the Navier-Stokes system of equations), finite-difference splitting schemes are used. Findings. Based on the developed CFD model, a complex of computer programs has been developed, which makes it possible to determine the efficiency of water treatment in a horizontal settler with additional elements. The results of a computational experiment to assess the efficiency of water treatment in a horizontal settler with additional elements in the form of plates are presented. Originality. An efficient CFD model has been created, which allows to quickly evaluate the efficiency of wastewater treatment in a horizontal settler with additional elements. The developed CFD model takes into account the geometric shape of the facility and the most significant physical factors, that influence the efficiency of the waste water treatment in horizontal settler: non-uniform flow velocity, diffusion, different position of inlet and outlet openings. Practical value. The developed CFD model belongs to the class of «diagnostic models» and can be used to assess the efficiency of treatment facilities at the stage of their preliminary design.

A discrete scheme of the fluid motion equation based on the pore-scale SPH method

To study the phenomenon of convection diffusion and solute migration in porous media, the smoothed particle hydrodynamics method is used to solve the N–S equation describing pore water flow and the linear homogeneous second-order differential equation describing the solute diffusion process. The accuracy of the simulation method is verified by solving the one-dimensional definite solution of the convection diffusion equation at a low Peclet (Pe) number. Under the condition of a high Pe number, the simulation experiment of solute breakthrough porous media at the pore scale is simulated. Through the analysis of the velocity distribution of fluid particles in pore channels of porous media and the mechanical dispersion process of solutes in porous media, it is found that the flow velocity presents a parabolic distribution in the pore channel, and the particles with high velocity are mainly concentrated in the middle of the pore channel between the solid particles. The solute dispersion is obvious in porous media, and some fluid particles containing solutes penetrate porous media earlier along fast-flowing pore channels. Finally, the solute breakthrough curve and the motion path length of the particle are obtained through the simulation experiment. The calculation methods of the dispersion degree and tortuosity of porous media are further given, and the dispersion degree of porous media in this paper is 0.000 95 m.

ОБ УСОВЕРШЕНСТВОВАНИИ МАТЕМАТИЧЕСКОЙ МОДЕЛИ ЭЛЕКТРОАКУСТИЧЕСКОГО ПРЕОБРАЗОВАТЕЛЯ ПРИ УСЛОВИИ ТОНКОГО ДВОЙНОГО СЛОЯ В ПОРИСТОЙ СТРУКТУРЕ ТЕЛА ПРЕОБРАЗОВАТЕЛЯ

In the approximation of a thin double layer, simple dependences of the acoustic fields excited in an electrokinetic electroacoustic transducer on the velocity of the Helmholtz—Smolukhovsky osmotic movement are obtained. From the dependencies it follows that in the absence of losses, when the nonlinearity of the fluid motion equation does not yet affect and there is no turbulent mode of fluid motion in the transducer body, the magnitudes of the amplitudes of the acoustic velocity and pressure linearly depend on the magnitude of the Helmholtz—Smolukhovsky electroosmotic motion velocity.

Reconstruction of the MPD system parameters at the gas kicks into well with the measured surface data in situ by the mathematical modeling

One of the main problems of managed pressure drilling (MPD) is detection and monitoring of gas kicks from the formation to the annulus. The model is proposed to estimate the parameters of the gas kick and gas-liquid flow in a well during MPD process based on measurements performed only at the surface. A gas-liquid flow is described by the kinematic wave model. The ability of gaseous hydrocarbon to dissolve in mud, when drilling on an oil based mud (OBM), is taken into account. The hydrodynamic parameters of the flow are estimated by numerically solving the equations of fluid motion jointly with measured input data: the flowrate of the mud at inlet, the flowrate of gas-liquid mixture at outlet, and the surface back pressure (SBP). The control parameters are the change in mass gas and void fraction over the well depth and on time, as well as BHP. To verify the proposed model, the calculation of gas-liquid flow parameters is conducted on base of the detailed measured data of field test during drilling of 16,000-ft deep well using a water based mud (WBM). The pattern of the gas kick into the well from the formation is reconstructed, the change in void fraction during gas rising was traced until it was kicked to the surface, and the total mass of gas entering the well is determined. The error is estimated by comparing deviations of the measured and calculated bottomhole pressure (BHP) value.

Transforming Method of Torsional Tuned Liquid Column Damper-Eccentric Structure to Torsional Tuned Mass Damper-Eccentric Structure for Vibration Control

The performance of torsional tuned liquid column dampers (TTLCDs) for mitigating coupled lateral and torsional vibrations of an eccentric structure, which can be modeled as a 3N degree-of-freedom structure, is theoretically investigated. Biaxial horizontal earthquakes are applied at the rigid foundation as a single-point excitation. The equations of relative fluid motion in a TTLCD with different reference points A and inclined angles β of upright columns are derived. Moreover, the interaction forces and moments between the absorber and structure are determined. On the basis of the coupled equations of motion of the main system with the TTLCD, a TTLCD is tuned with respect to a selected natural frequency of the main structure through a geometric transformation in analogy to a torsional tuned mass damper (TTMD). An equivalent structure mass ratio renders the mass of the TTMD and the changed structural modal mass. Thus, the easy implementation of the simulated TTMD by Ansys software can analyze the vibration control of an eccentric structure installed with TTLCDs. Numerical simulations show that a transforming method with an explicit physical definition is feasible and accurate, and can be directly used.

Thermal Charging Optimization of a Wavy-Shaped Nano-Enhanced Thermal Storage Unit.

A wavy shape was used to enhance the thermal heat transfer in a shell-tube latent heat thermal energy storage (LHTES) unit. The thermal storage unit was filled with CuO–coconut oil nano-enhanced phase change material (NePCM). The enthalpy-porosity approach was employed to model the phase change heat transfer in the presence of natural convection effects in the molten NePCM. The finite element method was applied to integrate the governing equations for fluid motion and phase change heat transfer. The impact of wave amplitude and wave number of the heated tube, as well as the volume concertation of nanoparticles on the full-charging time of the LHTES unit, was addressed. The Taguchi optimization method was used to find an optimum design of the LHTES unit. The results showed that an increase in the volume fraction of nanoparticles reduces the charging time. Moreover, the waviness of the tube resists the natural convection flow circulation in the phase change domain and could increase the charging time.

On a hyperbolic system of equations in the problem of unsteady fluid motion

We consider a nonlinear system of the first-order partial differential equations with constant coefficients admitting a solution of the traveling wave type. Such systems are used in applied research to describe the dynamics of unsteady water movement in a pressure pipeline. In this paper, we propose methods for obtaining exact solutions in the case of an inhomogeneous equation of fluid motion under restrictions on the right-hand side in the form of a function. A special feature of the present research is reduction of the problem under study to the equations integrable in elementary functions.

Mathematical model of a micropolar lubricating stuff

The article is devoted to the mathematical analysis model of the radial bearing of infinite length, lubricated with lubricating stuff and molten low-melting metal coating on the surface of the bearing bush, having the general rheological properties in the laminar condition of micro-polar lubrication properties, taking into account the dependence of viscosity rheological properties of micro-polar lubricating stuff, and the permeability of the porous coating from the pressure. The authors on the basis of the motion equations of a viscous incompressible fluid having micro-polar properties, and for “lamina”, of the continuity equation, Darcy equation and determining from the expression for the dissipation rate of mechanical energy, the profile of molten contour of the bearing bush, while taking account of the dependence of the overall rheological properties of the lubricating stuff and melt the low-melting metal coating having the micro-polar properties and the permeability of the porous coating of the pressure found the asymptotic and exact self-similar solution of the differential equation system on the parameter, due to the melt and the dissipation rate of mechanical energy to zero (without melt) and first (including melt) approximation.