Plane-parallel Flow Research Articles

Stability of plane parallel flow revisited for particle-fluid suspensions

Abstract An alternative model is proposed for hydrodynamic stability of plane parallel flow of an incompressible Newtonian fluids with suspended solid particles. For heavy particle-laden dusty gases with negligible particle volume fraction, the effective complex-form mean velocity in the modified Orr-Sommerfeld equation derived by the present model is showed to be essentially identical to the well-known Saffman's classical results. In the limit cases of small or large Stokes number of particles, simple formulas are derived for the effective Reynolds number ratio of the particle-laden suspension to the clear fluid without particles under otherwise identical conditions. The derived formula for particles of finite particle-to-fluid density ratio and small Stokes number is verified by comparing predicted results with known data, although a comparison of the derived formula with known results for particles of finite density ratio and large Stokes number cannot be made here due to the lack of available data. It is hoped that the present work could offer a conceptually novel and relatively simplified model for hydrodynamics of solid particle-fluid suspensions.

Diagnostics of the technical condition of wells drilled through freshwater horizons using active thermometry

Relevance. Recently, ESG (Environment, Social, Governance) principles have been given an important role in the agenda of oil and gas companies – this is an actively developing global trend in the modern business environment. One of its key aspects is a responsible attitude to the environment, increasing the priority of solving environmental problems arising from the extraction of hydrocarbons. A significant environmental problem is, in particular, pollution of freshwater horizons during oil production due to their poor-quality insulation during well cementing. At the same time, in the space of the well behind the casing, there is a flow of borehole fluid to freshwater horizons, which has a negative impact on the environmental situation in this region. In this regard, monitoring the technical condition of wells crossing freshwater horizons is an important and urgent task. Object. An operational well drilled through freshwater horizon, in which geophysical studies are carried out by the method of active thermometry – local heating of the casing section is carried out and the dynamics of the temperature field in it is studied. Aim. To identify qualitative patterns and obtain quantitative estimates of the effect of the fluid flow in the space behind the casing (column space) on the temperature field in the casing. Methods. Numerical mathematical modeling of thermal and hydrodynamic processes in a well using the software package Ansys Fluent (ANSYS Academic Research CFD license, agreement with Bashkir State University dated 06/15/2020). Results. The authors have studied the influence of a plane-parallel horizontal fluid flow of the freshwater horizon on the azimuthal temperature distribution of the casing string during induction heating. It is shown that in the case of leakiness of the cement ring in the interval of the freshwater horizon, an inhomogeneity of the azimuthal temperature distribution of the column is observed, reaching a value of few more K. The authors carried out the simulation of induction heating in the well with liquid column flow to the interval of the freshwater horizon. It is shown that the sign indicating the overflow is the occurrence of a temperature disturbance area vertically above the induction heating interval with a length of more than 1 m. In general, the results of the performed studies show the informativeness of the active thermometry method in relation to the diagnosis of the technical condition of wells drilled through freshwater horizons.

Relativistic radiation transfer for plane–parallel flows with the radiative equilibrium

Relativistic radiation transfer for plane–parallel flows with the radiative equilibrium

Singular layer physics informed neural network method for plane parallel flows

We construct in this article the semi-analytic Physics Informed Neural Networks (PINNs), called singular layer PINNs (or sl-PINNs), that are suitable to predict the stiff solutions of plane-parallel flows at a small viscosity. Recalling the boundary layer analysis, we first find the corrector for the problem which describes the singular behavior of the viscous flow inside boundary layers. Then, using the components of the corrector and its curl, we build our new sl-PINN predictions for the velocity and the vorticity by either embedding the explicit expression of the corrector (or its curl) in the structure of PINNs or by training the implicit parts of the corrector (or its curl) together with the PINN predictions. Numerical experiments confirm that our new sl-PINNs produce stable and accurate predicted solutions for the plane-parallel flows at a small viscosity.

Numerical Analysis of the Spectrum of Plane–Parallel Flows

Numerical Analysis of the Spectrum of Plane–Parallel Flows

The nonlinear stability of plane parallel shear flows with respect to tilted perturbations

The nonlinear stability of plane parallel shear flows with respect to tilted perturbations

Electroreduction of divanillin to polyvanillin in an electrochemical flow reactor

The electrochemical conversion of biobased intermediates offers an attractive and sustainable process for the production of green chemicals. One promising synthesis route is the production of the total vanillin-based polymer polyvanillin, which can be produced by electrochemical pinacolization of divanillin (5–5´bisvanillyl). Divanillin can be easily enzymatically generated from vanillin, a renewable intermediate accessible from lignin on an industrial scale. This study investigates systematically the electrochemical production of polyvanillin in a divided plane parallel flow reactor in recirculation mode. Several analytic methods, such as online UV–VIS spectroscopy, size exclusion chromatography (SEC), 2D-NMR (HSQC, 13C/1H), TGA and DSC were used to monitor the reaction progress and to characterize the reaction products under different galvanostatic reaction conditions revealing new insights into the reaction mechanism and structural features of the polymer. Further, by using an electrochemical engineering-based approach determining the limiting current densities, we readily achieved high current densities over 50 mA cm−2 for the polyvanillin synthesis and reached averaged molecular weights up to Mw = 4100 g mol−1 and Mn = 2700 g mol−1. The cathodic polymerization to polyvanillin offers an innovative approach for the electrochemical production of biobased polymers presented on flow cell level.Graphical

On the non-self-adjoint and multiscale character of passive scalar mixing under laminar advection

Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter $q=2{\rm i}k\,{\textit {Pe}}$ , where $k$ is the non-dimensional wavenumber of the tracer in the streamwise direction, and ${\textit {Pe}}$ is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.

On exact solutions of the equation of continuity

The continuum of exact solutions of the two-dimensional continuity equation for stationary, planeparallel fluid flow is found in the article. Only specific solutions of this equation corresponding to a vortex-free fluid flow are described in the classical literature. At the same time, components of an arbitrary analytical function of a complex variable are used, playing the role of a velocity potential and a current function. In this paper we give a generalization of this method, which also gives vortex-free solutions to the continuity equation.

Plane-Parallel Laminar Flow of Viscous Fluid in the Transition Zone of the Inlet Section

A study was conducted to analyze how hydrodynamic parameters change in the entrance region of plane-parallel flow under stationary flow conditions, with an initial arbitrary distribution of velocities in the entrance section. This study was based on boundary layer equations, and a boundary problem was formed under the conditions of plane-parallel flow. The boundary conditions were chosen to reflect the pattern of arbitrary velocity distribution in the entrance section. A general solution of the approximating Navier-Stokes equations is provided, which depends on the initial conditions and the Reynolds number. The boundary conditions are established based on the nature of the motion, and the boundary value problem is described. A method for integrating the boundary value problem has been developed, and regularities for the change in velocities along the length of the inlet section have been obtained for both constant and parabolic velocity distributions in the entrance sections. Analytical solutions have been derived to obtain patterns of velocity and pressure changes in any given flow direction. Through computer analysis, velocity change patterns in various sections along the inlet transition area have been constructed, allowing for the determination of fluid velocity at any point on the section and an estimation of the length of the transition area. These proposed solutions provide a framework for accurately constructing individual units of hydromechanical equipment.

Singular Nonlinear Problems for Phase Trajectories of Some Self-Similar Solutions of Boundary Layer Equations: Correct Formulation, Analysis, and Calculations

We study a singular initial value problem for a nonlinear non-autonomous ordinary differential equation of the second order, defined on a semi-infinite interval and degenerating in the initial data for the phase variable. The problem arises in the dynamics of a viscous incompressible fluid as an auxiliary problem in the study of self-similar solutions of the boundary layer equations for a stream function with a zero pressure gradient (plane-parallel laminar flow in a mixing layer). It is also of independent mathematical interest. Using the previously obtained results on singular nonlinear Cauchy problems and parametric exponential Lyapunov series, a correct formulation and a complete mathematical analysis of this singular initial value problem are given. Restrictions on the “self-similarity parameter” for the global existence of solutions are formulated, two-sided estimates of solutions, and results of calculations of the phase trajectories of solutions for different values of this parameter are given.

Исследование влияния неоднородности плазмы с учётом тормозного и фоторекомбинационного излучений на поглощение альфвеновской волны

We numerically study the nonlinear absorption of a plane Alfven wave incident on a fixed boundary of a dissipative inhomogeneous plasma, which is caused by dissipative factors, including bremsstrahlung and photorecombination radiation. Density inhomogeneities specified using a Gaussian distribution are considered. The study is based on the equations of two-fluid electromagnetic hydrodynamics with full allowance for the inertia of electrons. The proposed implicit difference scheme for calculating plane-parallel flows of a two-fluid plasma made it possible to reveal a number of important absorption patterns. In particular, it is shown that absorption with allowance for bremsstrahlung and photorecombination radiation for an inhomogeneous plasma leads to blocking of the Alfven wave in a dissipative plasma. The dependences of the depth of penetration of the Alfven wave and the maximum temperatures of electrons and ions on the size of the density inhomogeneity are obtained.

Criteria for integro-differential modeling of plane-parallel flow of viscous incompressible fluid

For a liquid with a nonmonotonic flow curve in the stationary case in the region of the descending branch, setting the velocity at the boundary does not uniquely determine the shear stress, strain rate distribution, and velocity profile that arise since the problem is known to have many unstable solutions. At the same time, the problem of the motion of such fluid under the action of a given pressure difference has no more than three solutions, two of which are stable, and the third is unstable and not reproducible. Which of the two stable solutions is realized depends on the loading history. The problem of determining the velocity profile for a fluid characterized by a nonmonotonic rheological flow curve between parallel planes is considered. The existence of a solution is realized by reducing the problem posed to a singular integral equation of normal type, which is known by the Carleman – Vekua regularization method developed by S.G. Mikhlin and M.M. Smirnov equivalently reduces to the Fredholm integral equation of the second kind, and the solvability of the latter follows from the uniqueness of the solution delivered problem describing of criteria for integro–differential modeling of a plane-parallel flow of a viscous incompressible fluid.

Influence of the liquid layer thickness on the stability of a plane-parallel flow in a two-phase system with evaporation

Influence of the liquid layer thickness on the stability of a plane-parallel flow in a two-phase system with evaporation

Development of production decline curves for non-Darcy oil flow in low-permeability reservoirs

The purpose of work is creating a new technique for result interpretation of long-term well monitoring by production decline curve analysis in low-permeability reservoirs. For these reservoirs oil flow does not obey Darcy law, in some cases threshold pressure gradient exists. Standard techniques for well test results interpretation do not allow to determine reservoir parameters reliably. Currently, most of works devoted to the interpretation of well test results for nonlinear flow use the approach of numerical modeling and the method of type-curve matching to determine reservoir parameters. Of course, these methods are powerful tool, but analytical solutions of the considered problems are also of practical interest. Obtained solutions make it possible to determine not only reservoir parameters but also flow law parameters and to diagnose characteristic regimes of non-Darcy flow. That is why, it becomes necessary to develop new approaches for reservoir parameters and flow law determination, which was implemented in the proposed paper. A one-dimensional isothermal problem of single-phase non-Darcy flow is considered in two geometries: plane parallel (flow from fracture) and plane radial (flow from single well). Well operates with constant bottomhole pressure. Flow equation is characterized by two parameters: threshold pressure gradient and coefficient describing the change in flow rate near the threshold gradient value. To obtain analytical solutions, integral equation method was applied. The behavior of flow rate at different times was investigated, the results showed good agreement with numerical simulation. Decline flow rate curves were analyzed when changing parameters of flow equation. Also, the possibility of reservoir parameters determination was investigated.

Friedrichs Inequalities and Sharpened Sufficient Stability Conditions of Plane-Parallel Flows

Friedrichs Inequalities and Sharpened Sufficient Stability Conditions of Plane-Parallel Flows

New Lagrangian view of vorticity evolution in two-dimensional flows of liquid and gas

Purpose of the study is to obtain formulas for such a speed of imaginary particles that the circulation of the speed of a (real) fluid along any circuit consisting of these imaginary particles changes (in the process of motion of imaginary particles) according to a given time law. (Until now, only those speeds of imaginary particles were known, at which the mentioned circulation during the motion remained unchanged). Method. Without implementation of asymptotic, numerical and other approximate methods, a rigorous analysis of the dynamic equation of motion (flow) of any continuous fluid medium, from an ideal liquid to a viscous gas, is carried out. Plane-parallel and nonswirling axisymmetric flows are considered. The concept of motion of imaginary particles is used, based on the K. Zoravsky criterion (which is also called A. A. Fridman’s theorem). Results. Formulas for the speed of imaginary particles are proposed. These formulas include the parameters of the (real) flow, their spatial derivatives and the function of time, which determines the law of the change in time of the (real fluid) velocity circulation along the contours moving together with the imaginary particles. In addition, it turned out that for a given function of time (and, as a consequence, for a given law of change in circulation with respect to time), the speed of imaginary particles is determined ambiguously. As a result, a method is proposed to change the speed and direction of motion of imaginary particles in a certain range (while maintaining the selected law of changes in circulation in time). For a viscous incompressible fluid, formulas are proposed that do not include pressure and its derivatives. Conclusion. A new Lagrangian point of view on the vorticity evolution in two-dimensional flows of fluids of all types is proposed. Formulas are obtained for the velocity of such movement of contours, at which the real fluid velocity circulation along any contour changes according to a given time law. This theoretical result can be used in computational fluid dynamics to limit the number of domains when using a gridless method for calculating flows of a viscous incompressible fluid (the method of viscous vortex domains).

Stability of plane-parallel flow of magnetic fluids under external magnetic fields

Stability of plane-parallel flow of magnetic fluids under external magnetic fields

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.

Construction of Inhomogeneous Velocity Fields Using Expansions in Terms of Eigenfunctions of the Laplace Operator

In this paper we present a method for constructing inhomogeneous velocity fields of an incompressible fluid using expansions in terms of eigenfunctions of the Laplace operator whose weight coefficients are determined from the problem of minimizing the integral of the squared divergence. A number of examples of constructing the velocity fields of plane-parallel and axisymmetric flows are considered. It is shown that the problem of minimizing the integral value of divergence is incorrect and requires regularization. In particular, we apply Tikhonov’s regularization method. The method proposed in this paper can be used to generate different initial conditions in investigating the nonuniqueness of the solution to the Navier – Stokes equations.