Integration of linear and some model non-linear equations of motion of incompressible fluids

We suggest a new exact method that allows one to construct solutions to a wide class of linear and some model non-linear hydrodynamic-type systems. The method is based on splitting a system into a few simpler equations; two different representations of solutions (non-symmetric and symmetric) are given. We derive formulas that connect solutions to linear three-dimensional stationary and non-stationary systems (corresponding to different models of incompressible fluids in the absence of mass forces) with solutions to two independent equations, one of which being the Laplace equation and the other following from the equation of motion for any velocity component at zero pressure. To illustrate the potentials of the method, we consider the Stokes equations, describing slow flows of viscous incompressible fluids, as well as linearized equations corresponding to Maxwell's and some other viscoelastic models. We also suggest and analyze a differential-difference fluid model with a constant relaxation time. We give examples of integrable non-linear hydrodynamic-type systems. The results obtained can be suitable for the integration of linear hydrodynamic equations and for testing numerical methods designed to solve non-linear equations of continuum mechanics.     

Lagrangian evaluations of viscous models for velocity gradient dynamics in non-stationary turbulence

Simple autonomous dynamical models of velocity gradients are found to be useful in understanding the essential physics of non-linear turbulent processes. Such models can also be employed as closure models for the Lagrangian PDF methods of turbulence computations. The pressure Hessian and the viscous processes incumbent in the exact velocity gradient evolution equation are non-local in nature. Several models have been proposed for these processes. In this work, we focus specifically on two models meant for the incumbent viscous process: the linear Lagrangian diffusion model (LLDM) and the recent fluid deformation closure model (RFDM). Performance of both the models have indeed been examined earlier, but most evaluations have been restricted to statistical stationary flow fields. In this work, we subject these models to further scrutiny. Our evaluation procedure (i) uses direct numerical simulation data of decaying isotropic (non-stationary) turbulence, (ii) follows identified fluid particles (the so-called Lagrangian evolution), (iii) uses both compressible and nearly incompressible flow fields. In nearly incompressible regime, the RFD model is found to be satisfactory, while the LLDM model overestimates viscous effects at late times. In the compressible regime, both the models show inadequacies. For compressible flows, we propose an alternative modelling strategy which shows improvement over both LLD and RFD models.     

A residual‐based Allen–Cahn phase field model for the mixture of incompressible fluid flows

The hydrodynamics of fluid mixtures is receiving more and more attention in many science and engineering applications. Within the techniques for dealing with front displacements and moving boundaries between different density and/or viscosity fluids, phase fields are a class of models in which a diffusive transition region is taken into account instead of a steep interface. Although these models have a physical motivation, they require the definition of extra parameters. In order to make it less parameter dependent, the classic Allen–Cahn phase field model is modified, exploring its similarities with residual‐based discontinuity‐capturing schemes, making the phase field equation dependent on its own residual. We solve the coupling between incompressible viscous fluid flow and the phase field advective–diffusive–reactive transport to simulate the main processes in interface tension and/or buoyancy driven problems. For the solution of the Navier–Stokes and transport equations, we use a stabilized finite element formulation. The implementation has been performed using the libMesh finite element library, written in C++ , which provides support for adaptive mesh refinement and coarsening. A chemical convection benchmark problem is used to validate the proposed model, and then we solve two bubble interaction problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Global Weak Solutions¶for the Two-Dimensional Motion¶of Several Rigid Bodies¶in an Incompressible Viscous Fluid

We consider the two-dimensional motion of several non-homogeneous rigid bodies immersed in an incompressible non-homogeneous viscous fluid. The fluid, and the rigid bodies are contained in a fixed open bounded set of ?². The motion of the fluid is governed by the Navier-Stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies. The time variation of the fluid domain (due to the motion of the rigid bodies) is not known a priori, so we deal with a free boundary value problem. The main novelty here is thedemonstration of the global existence of weak solutions for this problem. More precisely, the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary. We give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero.     

Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels et al. for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier–Stokes system with a modified convective term coupled to a Cahn–Hilliard system. The density of the mixture depends on an order parameter.     

Displacement of yield-stress fluids in a fracture

We consider a displacement of several yield-stress fluids in a Hele-Shaw cell. The topic is relevant to the development of a model for the flow of multiple phases inside a narrow fracture with application to hydraulically fracturing a hydrocarbon-bearing underground formation. Existing models for fracturing flows include only pure power-law models without yield stress, and the present work is aimed at filling this gap. The fluids are assumed to be immiscible and incompressible. We consider fluid advection in a plane channel in the presence of density gradients. Gravity is taken into account, so that there can be slumping and gravitational convection. We use the lubrication approximation so that governing equations are reduced to a 2D width-averaged system formed by the quasi-linear elliptic equation for pressure and transport equations for volume concentrations of fluids. The numerical solution is obtained using a finite-difference method. The pressure equation is solved using an iterative algorithm and the Multigrid method, while the transport equations are solved using a second-order TVD flux-limiting scheme with the superbee limiter. This numerical model is validated against three different sets of experiments: (i) gravitational slumping of fluids in a closed Hele-Shaw cell, (ii) viscous fingering of fluids with a high viscosity contrast due to the Saffman–Taylor (S–T) instability in a Hele-Shaw cell at microgravity conditions, (iii) displacement of Bingham fluids in a Hele-Shaw cell with the development of fingers due to the S–T instability. Good agreement is observed between simulations and laboratory data. The model is then used to investigate the joint effect of fingering and slumping. Numerical simulations show that the slumping rate of yield-stress fluid is significantly less pronounced than that of a Newtonian fluid with the same density and viscosity. If a low-viscosity Newtonian fluid is injected after a yield-stress one, the S–T instability at the interface leads to the development of fingers. As a result, fingers penetrating into a fluid with a finite yield stress locally decrease the pressure gradient and unyielded zones develop as a consequence.     

Numerical study of PPE source term errors in the incompressible SPH models

In a recent work by Gui et al. 13 , an incompressible SPH model was presented that employs a mixed pressure Poisson equation (PPE) source term combining both the density‐invariant and velocity divergence‐free formulations. The present work intends to apply the model to a wider range of fluid impact situations in order to quantify the numerical errors associated with different formulations of the PPE source term in incompressible SPH (ISPH) models. The good agreement achieved between the model predictions and the documented data is taken as a further demonstration that the mixed source term formulation can accurately predict the fluid impact pressures and forces, both in the magnitude and in the spatial and temporal patterns. Furthermore, an in‐depth numerical analysis using either the pure density‐invariant or velocity divergence‐free formulation has revealed that the pure density‐invariant formulation can lead to relatively large divergence errors while the velocity divergence‐free formulation may cause relatively large density errors. As compared with these two approaches, the mixed source term formulation performs much better having the minimum total errors in all test cases. Although some recent studies found that the weakly compressible SPH models perform somewhat better than the incompressible SPH models in certain fluid impact problems, we have shown that this could be largely caused by the particular formulation of PPE source term in the previous ISPH models and a better formulation of the source term can significantly improve the accuracy of ISPH models. Copyright © 2014 John Wiley & Sons, Ltd.     

Hydrodynamic action of the fluid slowly flowing around a spheroidal particle covered by a viscous film

A steady problem of a slow axisymmetric flow of a viscous incompressible fluid around an oblate spheroid covered by a viscous film is solved analytically with the use of the Stokes approximation. Surface tension on the interface between the fluids is taken into account. Expressions for velocity components and stream functions are presented. A formula for determining the force action of the incoming flow onto the oblate spheroid is derived.     

Structure of Averaged Flow Driven by a Vibrating Body with a Large-Curvature Edge

The averaged viscous incompressible fluid flow driven by a vibrating body with a large-curvature edge is investigated experimentally and numerically. The case of an axisymmetric body immersed in fluid and performing translational vibrations along its axis is considered. Experiments carried out on fluids of various viscosity over a wide vibration frequency and amplitude range and direct numerical calculations based on the complete time-dependent equations of viscous fluid dynamics show that the global structure of the averaged flow significantly depends on the relation between the curvature radius of the body edge and the viscous skin-layer thickness. Different averaged flow regimes are detected and the flow restructuring process is investigated as a function of the vibration amplitude and frequency.     

Unsteady flows of an inhomogeneous incompressible viscous fluid

This paper deals with a theoretical analysis of the transfer of reactive impurities by open and filtration flows of an incompressible viscous fluid. The first section of the paper studies the model of an inhomogeneous incompressible viscous fluid, which is widely used in meteorology and oceanology, with additional allowance for the drag of the magnetic field or porous medium. Another object of research in this paper is the model of filtration of an inhomogeneous incompressible fluid in porous media proposed by V. N. Monakhov (1977) (Section 2). In both models, hydrodynamic flows determine the motion of the mixture as a whole and the temperature and concentration distributions of the components of an inhomogeneous fluid are described by a common nonlinear system of equations of diffusive heat and mass transfer.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 44–51, March–April, 2005.     

Conjugate heat exchange and hydrodynamics for a viscous incompressible fluid moving in a rectangular cavity

Numerical simulation was performed of the motion of a viscous incompressible nonisothermal fluid in an open rectangular cavity under conditions of forced convection and conjugate heat exchange. The effect of the jet dynamic parameter (Reynolds number) and fluid flow conditions on the character of motion and heat exchange of viscous incompressible nonisothermal fluids in rectangular cavities is studied. A hydrodynamic pattern of viscous flow in an open cavity under forced convection conditions (in the conjugate and nonconjugate formulations of the problem) is obtained. The effect of parameters of the model on the character of motion is studied. Temperature profiles for the solid and fluid phases are obtained. The effect of parameters of the model on the character of temperature distribution in both phases is studied.     

Front‐tracking by the level‐set and the volume penalization methods in a two‐phase microfluidic network

Two‐phase immiscible fluids in a two‐dimensional micro‐channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow, while the Oldroyd‐B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro‐channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level‐set method is used, and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two‐phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one. Copyright © 2015 John Wiley & Sons, Ltd.     

Experimental studies of model porous media fluid dynamics

The three-dimensional, steady flow velocity components of a viscous, incompressible, Newtonian fluid in model porous media were measured. The model porous geometries were constructed from 3 mm glass rods. A laser Doppler anemometer was used to measure two of the velocity components and the third was calculated by integrating the continuity equation. The effects of viscous drag, inertial flow fields and eddy losses in the model were studied. The results showed that the measured flow was laminar and stable such that micromixing of the fluid was absent. Inertial flow effects were absent due to high viscous drag coefficients.     

Separation of species of a binary fluid mixture confined between two concentric rotating circular cylinders in presence of a strong radial magnetic field

The effect of a radial magnetic field on separation of a binary mixture of incompressible viscous thermally and electrically conducting fluids confined between two concentric rotating circular cylinders with different angular velocity is examined. The equations governing the motion, temperature and concentration in cylindrical polar coordinate are solved analytically. The solution obtained in closed form for concentration distribution is plotted against the radial distances from the surface of the inner circular cylinder for various values of non-dimensional parameters. It is found that the non-dimensional parameters viz. the Hartmann number, thermal diffusion number, baro diffusion number, rotational Reynolds number, the product of Prandtl number and Eckert number, magnetic Prandtl number and the ratio of the angular velocities of inner and outer cylinders affects the species separation of rarer and lighter component significantly. The problem discussed here derives its application in the basic fluid dynamics separation processes to separate the rarer component of the different isotopes of heavier molecules where electromagnetic method of separation does not work.     

A simple SPH algorithm for multi‐fluid flow with high density ratios

In this paper, we describe an SPH algorithm for multi‐fluid flow, which is efficient, simple and robust. We derive the inviscid equations of motion from a Lagrangian together with the constraint provided by the continuity equation. The viscous flow equations then follow by adding a viscous term. Rigid boundaries are simulated using boundary force particles in a manner similar to the immersed boundary method. Each fluid is approximated as weakly compressible with a speed of sound sufficiently large to guarantee that the relative density variations are typically 1%. When the SPH force interaction is between two particles of different fluids, we increase the pressure terms. This simple procedure stabilizes the interface between the fluids. The equations of motion are integrated using a time stepping rule based on a second‐order symplectic integrator. When linear and angular momentum should be conserved exactly, they are conserved to within round‐off errors. We test the algorithm by simulating a variety of problems involving fluids with a density ratio in the range 1–1000. The first of these is a free surface problem with no rigid boundaries. It involves the flow of an elliptical distribution with one fluid inside the other. We show that the simulations converge as the particle spacing decreases, and the results are in good agreement with the exact inviscid, incompressible theory. The second test is similar to the first but involves the nonlinear oscillation of the fluids. As in the first test, the agreement with theory is very good, and the method converges. The third test is the simulation of waves at the interface between two fluids. The method is shown to converge, and the agreement with theory is satisfactory. The fourth test is the Rayleigh–Taylor instability for a configuration considered by other authors. Key parameters are shown to converge, and the agreement with other authors is good. The fifth and final test is how well the SPH method simulates gravity currents with density ratios in the range 2–30. The results of these simulations are in very good agreement with those of other authors and in satisfactory agreement with experimental results.Copyright © 2012 John Wiley & Sons, Ltd.     

Stability criteria of an immersed annulus swirling jet in a zero gravity environment

The present work deals with the stability properties of an immersed annulus swirling jet in a zero gravity environment. The considered system is composed of two streaming coaxial fluid cylinders, embedded in a third streaming fluid, where the intermediate one (annulus) has a uniform swirl speed. The fluids are immiscible, inviscid, and incompressible. The linear stability criteria of the model are discussed analytically and stability diagrams are obtained. We conclude that the radii ratio, the Weber number and the swirl number play a significant role in determining the dynamics of the developing interfacial patterns.     

Strong Solutions for a Phase Field Navier–Stokes Vesicle–Fluid Interaction Model

In this paper we study a mathematical model for the dynamics of vesicle membranes in a 3D incompressible viscous fluid. The system is in the Eulerian formulation, involving the coupling of the incompressible Navier–Stokes system with a phase field equation. This equation models the vesicle deformations under external flow fields. We prove the local in time existence and uniqueness of strong solutions. Moreover, we show that, given T > 0, for initial data which are small (in terms of T), these solutions are defined on [0, T] (almost global existence).     

Semi-inverse solutions of a non-newtonian fluid

Solutions for the equations of motion of an incompressible second grade fluid are obtained by employing semi-inverse methods in which we assume certain geometrical or kinematical properties of the fields. Specifically the problems studied in viscous fluids by Jeffery, Hamel and Görtler and Wieghardt, etc. are considered in a second grade fluid and the results for stream lines, velocities and pressure distribution are compared in the two cases.     

Effect of Density Dependent Viscosities on Multiphasic Incompressible Fluid Models

In this paper, we look at the influence of the choice of the Reynolds tensor on the derivation of some multiphasic incompressible fluid models, called Kazhikhov–Smagulov type models. We show that a compatibility condition between the viscous tensor and the diffusive term allows us to obtain similar models without assuming a small diffusive term as it was done for instance by A. Kazhikhov and Sh. Smagulov. We begin with two examples: The first one concerning pollution and the last one concerning a model of combustion at low Mach number. We give the compatibility condition that provides a class of models of the Kazhikhov–Smagulov type. We prove that these models are globally well posed without assumptions between the density and the diffusion terms.     

Finite element methods for a class of continuum models for immiscible flows with moving contact lines

In this paper, we present a finite element method for two‐phase incompressible flows with moving contact lines. We use a sharp interface Navier–Stokes model for the bulk phase fluid dynamics. Surface tension forces, including Marangoni forces and viscous interfacial effects, are modeled. For describing the moving contact lines, we consider a class of continuum models that contains several special cases known from the literature. For the whole model, describing bulk fluid dynamics, surface tension forces, and contact line forces, we derive a variational formulation and a corresponding energy estimate. For handling the evolving interface numerically, the level‐set technique is applied. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space. We apply a Nitsche technique to weakly impose the Navier slip conditions on the solid wall. A unified approach for discretization of the (different types of) surface tension forces and contact line forces is introduced. Results of numerical experiments are presented, which illustrate the performance of the solver. Copyright © 2016 John Wiley & Sons, Ltd.