Equations Of Incompressible Fluid Research Articles

Solitons for the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid via the Pfaffian technique

Fluids are common in nature, the study of which helps the design of the related industries. Under investigation in this letter is the (2[Formula: see text]+[Formula: see text]1)-dimensional Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid. Pfaffian solutions have been obtained based on the Pfaffian technique with the assistance of the real auxiliary function [Formula: see text]. N-soliton solutions with [Formula: see text] are constructed, where y is the scaled space coordinate and [Formula: see text] is the steady stream function in the irrotational incompressible flow. Background shapes of the solutions are affected by [Formula: see text], but the structures of the solutions are affected by the derivative of the log terms in the solutions. Neighborhoods at the origins of the solitons are different in consequence of the different values of the real auxiliary parameter a. One- and two-soliton solutions are illustrated, which are the superpositions of the two kink solitons and different forms of [Formula: see text]. Interactions of the two solitons are presented, from which we see that the velocities, amplitudes and shapes of the two solitons remain unchanged before and after each interaction.

Global well-posedness for the 2D incompressible magneto-micropolar fluid system with partial viscosity

In this paper, we consider an initial-boundary value problem for the 2D incompressible magneto-micropolar fluid equations with zero magnetic diffusion and zero spin viscosity in the horizontally infinite flat layer with Navier-type boundary conditions. We establish the global well-posedness of strong solutions around the equilibrium $(0,e_1,0)$.

Regularity of solutions to non-stationary Navier–Stokes equations

This paper covers a non-stationary system of Navier–Stokes equations for incompressible fluids. A regularized problem is considered that factors in the velocity field being relaxed to a solenoidal field; this problem is used to prove the pressure function exists almost everywhere in the domain for the Hopf class solutions. The proposed regularization proves there exist more regular weak solutions to the initial problem that do not impose smallness restrictions on the input data. The theorem of uniqueness is proven for the 2D case.

Simulation of flows with breaking internal waves generated by an obstacle

Numerical study of a stably stratified flow with obstacle of different shapes is performed by means of the Navier–Stokes and scalar equations for incompressible fluid with the Boussinesq approximation of buoyancy effects. The results of computations agree well with the concurrent experiment study with the towed body in the water tank with pycnocline.

Large data analysis for Kolmogorov’s two-equation model of turbulence

Kolmogorov seems to have been the first to recognize that a two-equation model of turbulence might be used as the basis of turbulent flow prediction. Nowadays, a whole hierarchy of phenomenological two-equation models of turbulence is in place. The structure of their governing equations is similar to the Navier–Stokes equations for incompressible fluids, the difference is that the viscosity is not constant but depends on two scalar quantities that measure the effect of turbulence: the average of the kinetic energy of velocity fluctuations (i.e. the turbulent energy) and the measure related to the length scales of turbulence. For these two scalar quantities two additional evolutionary convection–diffusion equations are added to the generalized Navier–Stokes system. Although Kolmogorov’s model has so far been almost unnoticed, it exhibits interesting features. First of all, in contrast to other two-equation models of turbulence, there is no source term in the equation for the frequency. Consequently, nonhomogeneous Dirichlet boundary conditions for the quantities measuring the effect of turbulence are assigned to a part of the boundary. Second, the structure of the governing equations is such that one can find an “equivalent” reformulation of the equation for turbulent energy that eliminates the presence of the energy dissipation acting as the source in the original equation for turbulent energy and which is merely an L1 quantity. Third, the material coefficients such as the viscosity and turbulent diffusivities may degenerate, and thus the a priori control of the derivatives of the quantities involved is unclear.We establish long-time and large-data existence of a suitable weak solution to three-dimensional internal unsteady flows described by Kolmogorov’s two-equation model of turbulence. The governing system of equations is completed by initial and boundary conditions; concerning the velocity we consider generalized stick–slip boundary conditions. The fact that the admissible class of boundary conditions includes various types of slipping mechanisms on the boundary makes the result robust from the point of view of possible applications.

Математическое моделирование вихревых структур при электроконвекции в канале ячейки электродиализатора на модельных мембранах с двумя проводящими участками

The article deals with mathematical modeling of the mechanism of electroconvection in electromembrane systems. The simulation is carried out by solving two-dimensional Navier-Stokes equations for incompressible fluid with boundary adhesion conditions and a given volume force distribution in accordance with the Rubinstein theory. It is shown that the volume force induced by current flow that can generate paired electroconvective vortices (electroosmosis of the second kind). It is shown that the most important parameters affecting the electroconvection are the limit current, the size of the inhomogeneities and the value of the space charge.

A relation between membrane permeability and flow rate at low Reynolds number in circular pipe

A relation between membrane permeability to a pure solvent and flow rate is derived in a circular pipe driven by a constant pressure difference across the pipe length. Membrane is assumed to be non-deformable and zero thickness, and the permeate flux of pure solvent due to pressure discontinuity across the flat membrane is coupled with the governing equations for incompressible Newtonian fluid in the Stokes regime. The permeation flow rate (normalised by that of the Hagen-Poiseuille flow for the no-membrane case or with a membrane of infinite permeability) is represented as a function of a non-dimensional permeability including the aspect ratio of the pipe geometry. The relation is established through comparison with a fully-validated numerical simulation result: the numerical discretisation is based on our original discrete-forcing immersed boundary method, which guarantees (i) conservations of mass and momentum even in the immediate vicinity of the membrane surface and (ii) consistency between incompressible velocity and pressure fields. Inverse analysis of the above formula yields the permeability as function of the flow rate through the pipe, comprising three equations covering the entire permeability ranges. The established permeability formulae are expected to be useful for identifying effective permeabilities of membrane to single-component fluid or pure solvent in practical applications.

Exact, Free-Surface Equatorial Flows with General Stratification in Spherical Coordinates

This paper is concerned with the construction of a new exact solution to the geophysical fluid dynamics governing equations for inviscid and incompressible fluid in the equatorial region. This solution represents a steady purely-azimuthal flow with a free-surface. The novel aspect of the solution we derive is that the flow it prescribes accommodates a general fluid stratification: the density may vary both with depth, and with latitude. The solution is presented in the terms of spherical coordinates, hence at no stage do we invoke approximations by way of simplifying the geometry in the governing equations. Following the construction of our explicit solution, we employ functional analytic considerations to prove that the pressure at the free-surface defines implicitly the shape of the free-surface distortion in a unique way, exhibiting also the expected monotonicity properties. Finally, using a short-wavelength stability analysis we prove that certain flows defined by our exact solution are stable for a specific choice of the density distribution.

Multi-bubble motion behavior of electric field based on phase field model* *Project supported by the National Natural Science Foundation of China (Grant Nos. 51661020, 11504149, and 11364024), the Postdoctoral Science Foundation of China (Grant No. 2014M560371) and the Funds for Distinguished Young Scientists of Lanzhou University of Technology, China (Grant No. J201304).

By coupling the phase field model with the continuity equation of incompressible fluid, Navier–Stokes equation, electric field equation, and other governing equations, a multi-field coupling model for multi-bubble coalescence in a viscous fluids is established. The phase field method is used to capture the two-phase interface. The motion and coalescence of a pair of coaxial bubbles under an external uniform electric field and the effects of different electric field strengths on the interaction and coalescence of rising bubbles are studied. The results show that the uniform electric field accelerates the collision and coalescence process of double bubbles in the fluid, and increases the rising velocity of the coalesced bubble. The electric field with an intensity of E = 2 kV/mm is reduced about 2 times compared with that without electric field in coalescence time. When the electric field strength is strong (E ≥ 1 kV/mm), the coalesced bubble will rupture before it rises to the top of the calculation area, and the time of the bubble rupturing also decreases with the increase of the electric field strength. The phase field method is compared with the simulation results of Lattice Boltzmann Method (LBM), and the shape of bubble obtained by the two methods is in good agreement, which verifies the correctness of the calculation model.

Formation of waves, vortices and ligaments in 2D stratified flows around obstacles

Wave and vortex fine structure of unsteady stratified flows were studied numerically based on the system of fundamental equations for non-homogeneous incompressible fluid with no-slip and no-flux boundary conditions, and experimentally using high-resolution Schlieren visualization techniques. Different flow regimes, including creeping, wave and vortex ones, which are widespread in the ocean and atmosphere, were investigated in the single problem formulation, starting from the slowest diffusion-induced flows up to the fastest and unsteady ones. A classification of flow components is proposed, which contains waves, vortices, and ligaments, which manifest themselves in the form of extended interfaces with their transverse scales being determined by the dissipative properties of a fluid and the flow dynamics. Due to interrupting the molecular flux of a stratifying agent on a motionless obstacle, fine structural diffusion-induced flows are formed around the obstacle in the form of a multilevel set of vortex cells which are transformed near the edges into horizontally extended streaky structures. A moving body in a stratified fluid produces upstream disturbances, groups of attached internal waves, leading-edge vortices, and a vortex street in the wake, which coexist in the stratified flow as an organic whole and are separated with ligaments. In the unsteady vortex flow regime, the shape of the bluff body essentially affects the flow structure, determining whether it will consist of a quasi-steady chain of vortices with a quite regular vortex shedding structure, or leading-edge vortices and vortex street would evolve into a much more complex and multi-scale vortex evolution. The numerical and experimental visualizations of diffusion-induced flows, internal waves, vortex streets, and fine structural ligaments, are in a qualitative agreement with each other.

Steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles

We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. We first develop a new approach to establish the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets/entropy waves by nonlinear arguments. This is the first result on the global existence of solutions of the multidimensional steady compressible full Euler equations with free boundaries, which are not necessarily small perturbations of piecewise constant background solutions. The subsonic–sonic limit of the solutions is also shown. Finally, through the incompressible limit, we establish the existence and uniqueness of incompressible Euler flows in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solutions with vortex sheets. The methods and techniques developed here will be useful for solving other problems involving similar difficulties.

Compressible structures in incompressible hydrodynamics and their role in turbulence onset

The formation of the coherent vortical structures in the form of thin pancakes is studied for three-dimensional flows at the high Reynolds regime when, in the leading order, the development of such structures can be described within the 3D Euler equations for ideal incompressible fluids. Numerically and analytically on the base of the vortex line representation we show that compression of such structures and respectively increase of their amplitudes are possible due to the compressibility of the continuously distributed vortex lines. It is demonstrated that this growth can be considered as analog of breaking for the divergence-free vorticity field. At high amplitudes this process has a self-similar behavior connected the maximal vorticity and the pancake width by the Kolmogorov type relation ωmax ∝ l-2/3. The role of such structures in the Kolmogorov spectrum formation is also discussed.

A model of particulate matter dispersion from unfiltered air conditioner indoor

The air conditioner is used not only for indoor temperature control but also as a particulate filtering system. However, bad maintenance of the air conditioner would lead to the damage of the filtering system. The result is a bad indoor air quality due to particulate matters depositing from outdoor. By the fact that a room with an air conditioner system is isolated, the risk of the human to be exposed is high. In this research, we built a model to simulate the particulate dispersion from air conditioner without active filtering system to find the effective range of the exposure. The research is purposely to model particulate matters, especially for PM2.5 and PM0.1. The model was built in the Navier-Stokes equation for incompressible fluids by using a Lagrangian approach in the grid system. The model simulated the dispersion of particulate matters in an 8 × 8 × 3 m room with a single air conditioner. The temperature of the room was settled at 24°C without any external particulate source. The speed of the airflow was set at 3 m/s while the concentration of the PM was set by change the air conditioner temperature setting. In the simulation, we found the increase of the effective range of particulate matters in the function of time. In the initial condition, the particulate matters are dispersed into the ceiling with the highest concentration. In the less than 5 minutes, our simulation shows the particulates dispersing in the whole room in the varied concentration. In the minute 50th, the concentration is similar to the air conditioner exposure. Our measurement in real condition showed a similar condition by a maximum different of 15% than the simulation both in the dispersion time and 2.5% in the concentration. These results were achieved after we compared the simulation in t time concentration with the real particulate dispersion in the identical room dimension with the simulation. In conclusion, the model can be performed to simulate an indoor PM dispersion from non-filtering air conditioner well.

Global existence and asymptotic stability of the fractional chemotaxis–fluid system in [formula omitted

In this paper, we consider a model arising from biology, consisting of fractional chemotaxis equations coupled to viscous incompressible fluid equations throughout transport and external force in R3. We establish the existence and uniqueness of global solution with small initial data by combining the local existence and a priori estimates as well as continuation argument. Moreover, the decay rates of the solutions and their higher-order spatial derivatives are established towards the equilibrium by introducing the negative Sobolev space Ḣ−s(0⩽s<32).

Error Estimates for Spectral Semi-Galerkin Approximations of Incompressible Asymmetric Fluids with Variable Density

We consider spectral semi-Galerkin approximations for global strong solutions of the equations for variable density asymmetric incompressible fluids in a bounded domain $$\Omega $$ of $$\mathbb {R}^3$$ . We prove an optimal uniform in time error estimate in the $${\varvec{H}}^{1}$$ norm for approximations of both the linear and angular velocity of particles of the fluid. We also derive an error bound for approximations of the density in some Lebesgue spaces $$L^{r}(\Omega )$$ .

Analysis of landslides employing a space–time single-phase level-set method

Landslide dynamics are characterized by a phase transition from a solid like behavior to a fluid one. Since the material is moving through space special discretization techniques are required. Four-dimensional space–time finite elements for fluid–structure interactions are applied together with the single-phase level-set method to investigate the dynamics of 3D landslides interacting with flexible walls. The wall is modeled as a geometric nonlinear solid with elastic–viscoplastic material behavior, whereas the liquefied soil is described with the Navier–Stokes equations for incompressible Bingham fluids. Between fluid and solid advanced coupling conditions are taken into account incorporating friction. In order to solve the governing equations of the multi-field problem, the weighted-residuals method is applied, which is discretized by time-discontinuous space–time elements. Within the monolithic solution procedure for the coupled problem, the kinematics of both solid and fluid are described using velocities as primary variables. A pseudo-structure adapts the coordinates of the fluid mesh to the deformations of adjacent structures. The single-phase level-set method describes the motion of the free surface of the sliding material. To accurately transport the free surface a pde-based extrapolation is used for the velocities. Various effects in 3D landslide dynamics are investigated.

Weak solutions with improved regularity for the nonhomogeneous asymmetric fluids equations with vacuum

We prove the existence of local and global in time weak solutions for the equations of nonhomogeneous incompressible asymmetric fluids in a bounded domain Ω ⊊ R 3 , when the initial vacuum is allowed, i.e., the initial density ρ 0 is not necessarily strictly positive. Our solutions here are more regular than previous results and so satisfy the initial conditions in a stronger sense. The global in time solutions are obtained under smallness assumptions on the initial data and forcing terms.

RESEARCH ON AQUATIC POLLUTION LEVEL OF MALEIA RIVER BY SIMULATION IN COMPUTATIONAL FLUID DYNAMICS

Watercourses that transit rural / urban areas are susceptible to the phenomenon of aquatic pollution due to various types of polluting species. The important issues of river pollution and of polluting species dispersion, require an approach with predictive tools (models for transport of polluting species) which can evaluate the performance of depollution measures/actions to reduce pollution and to take optimal management decisions. Thus, from January 2017 to May 2017, a monitoring investigation was carried out on the Maleia watercourse, polluting species concentrations measured by different physical-chemical methods being considered as input data for the Surface Water Modelling Systems (SMS) software in order to establish the dispersion of pollutants in the aquatic environment. The hydrodynamics of the river sector has been simulated through a module of the Surface Water Modelling Systems using the Reynolds form of Navier-Stokes equation system, along with the continuity equation for incompressible fluids in turbulent motion with free surface. The numerical simulation of advection-diffusion processes at an average depth of the studied river sector, was used for analyzing the space-time evolution of aquatic pollutants. The originality of this paper starts from the desideratum that rivers crossing inhabited areas are subjected to discharge of pollutants which implies the analysis of the quality of the studied water course (Maleia), as well as the illustration of the dispersion of certain pollutant species by estimating the dilution times as well as the determination of the iso-concentration field. Numerical models have been obtained for a sector of the Maleia River, providing the possibility of simulating both common and accidental pollution (related to space-time evolution of transport and dispersion of pollutants). The models obtained allowed the estimation of water quality in each finite element of the studied sector, and not just in one sampling point, as it is usually measured.

A Formal Passage From a System of Boltzmann Equations for Mixtures Towards a Vlasov-Euler System of Compressible Fluids

A formal asymptotics leading from a system of Boltzmann equations for mixtures towards either Vlasov-Navier-Stokes or Vlasov-Stokes equations of incompressible fluids was established by the same authors and Etienne Bernard in: A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory Commun. Math. Sci., 15: 1703–1741 (2017) and A Derivation of the Vlasov-Stokes System for Aerosol Flows from the Kinetic Theory of Binary Gas Mixtures. KRM, 11: 43–69 (2018). With the same starting point but with a different scaling, we establish here a formal asymptotics leading to the Vlasov-Euler system of compressible fluids. Explicit formulas for the coupling terms are obtained in two typical situations: for elastic hard spheres on one hand, and for collisions corresponding to the inelastic interaction with a macroscopic dust speck on the other hand.

Analysis of new forms of orifice plates using computational fluid dynamics

In many technologies, such as process industry or water supply, there is a need to measure fluid flowrates. Orifice plates are the most common instruments for measuring the fluid flowrate through pipelines due to their many advantages. On the other side, their use increases operating costs of industrial plants and pipelines. In this work, three new forms of orifice plates were designed and tested. These new forms and one standard, which served as a reference, were designed by using the SolidWorks software package. The aim of the new designs was energy savings, and consequently reduction of operating costs. Energy savings can be achieved by such a design, which decreases the orifice plate resistance an element of the pipeline. This was achieved by increasing the open part of the orifice plate permitting the fluid flow. CAD models of orifice plates were transferred to STL files that were further used for CFD simulation as well as 3D printing of experimental replicas. According to the proposed algorithm, the new designs were tested by CFD simulation performed in the COMSOL Multiphysics software package, by using a finite-difference method. Equations used were based on the Reynolds form of Navier-Stokes equations (RANS, Reynolds-averaged Navier-Stokes), and the continuity equation for incompressible fluids. Next, as we have proposed in our algorithm of development of new orifice plate designs, experimental orifice plates were made by using 3D printing technology and FDM (Fused Deposition Modeling) procedure and tested at laboratory conditions. The results of laboratory tests were compared with the results of CFD simulation. A considerable amount of energy saving was indicated, which was achieved already by the first of the three new orifice plate forms (V1) as compared to the reference (V0). For the other two proposed forms, the effect of energy savings was considerably lower. By using CFD simulation, data can be obtained based on which a decision can be made whether the new shape of the measuring device should be corrected or is appropriate for further laboratory tests. Based on the presented results it can be concluded that the proposed testing algorithm proved useful in designing new forms of orifice plates.