Stratified Viscous Fluid Research Articles

Initial-boundary value problem for the equations of dynamics of a rotating viscous stratified fluid

We study the problem of small motions of a viscous stratified fluid partially filling a container that uniformly rotates around an axis co-directed by gravity. The problem is studied on the basis of an approach related to the application of the so-called operator matrix theory. To this end, we introduce Hilbert spaces and some their subspaces, as well as auxiliary boundary value problems. The original initial-boundary value problem is reduced to the Cauchy problem for a first-order differential equation in some Hilbert space. After a detailed study of the properties of the operator coefficients corresponding to the resulting system of equations, we prove a theorem on the solvability of the Cauchy problem. On this basis, we find sufficient conditions for the existence of a solution of the original initial-boundary value problem describing the evolution of the hydro-system.

Periodic Flows in a Viscous Stratified Fluid in a Homogeneous Gravitational Field

The density of a fluid or gas, which depends on the temperature, pressure and concentration of dissolved substances or suspended particles, changes under the influence of a large number of physical factors. We assume that an undisturbed liquid is heterogeneous. The propagation of periodic flows in viscous, uniformly stratified fluids is considered. The analysis is based on a system of fundamental equations for the transfer of energy, momentum and matter in periodic flows. Taking into account the compatibility condition, dispersion relations are constructed for two-dimensional internal, acoustic and surface linear periodic flows with a positive definite frequency and complex wave number in a compressible viscous fluid exponentially stratified by density. The temperature conductivity and diffusion effects are neglected. The obtained regularly perturbed solutions of the dispersion equations describe the conventional weakly damped waves. The families of singular solutions, specific for every kind of periodic flow, characterize the before unknown thin ligaments that accompany each type of wave. In limited cases, the constructed regular solutions transform into well-known expressions for a viscous homogeneous and an ideal fluid. Singular solutions are degenerated in a viscous homogeneous fluid or disappear in an ideal fluid. The developing method of the fundamental equation system analysis is directed to describe the dynamics and spatial structure of periodic flows in heterogeneous fluids in linear and non-linear approximations.

Formation of Three-Dimensional Internal Waves behind a Body in Motion in a Stratified Viscous Fluid

Formation of Three-Dimensional Internal Waves behind a Body in Motion in a Stratified Viscous Fluid

Formation of Three-Dimensional Internal Waves behind a Body in Motion in a Stratified Viscous Fluid

A uniform motion of a disk in horizontal direction along its axis of symmetry in a stratified viscous fluid at rest is studied. The disk generates three-dimensional internal gravity waves occupying the entire volume between the disk and the location of its start. The waves are observed using two-color, beta-plus visualization of the vortex flow structure calculated within the framework of the system of Navier–Stokes equations in the Boussinesq approximation. The results of the study complete considerably the earlier-published mechanism of the formation of half-waves above the axis of symmetry of the disk, where emphasis was placed on the periodic process of generation of deformed vortex rings above the location of the disk start. Their generation is due to gravitation and shear instabilities, when the left semi-ring is transformed into a half-wave of depressions or crests, while the right one vanishes with time. In this paper it is established that the left parts of the right odd semi-rings are transformed into the axial parts of the crest half-waves.

Marangoni effect on Rayleigh-Taylor instability in viscous stratified fluid layers

Marangoni effect on Rayleigh-Taylor instability in viscous stratified fluid layers

Features of selecting boundary conditions when describing flows of stratified fluids

The article discusses the selection of boundary conditions at the boundary between adjacent layers in a stratified viscous incompressible fluid. It is shown that the "continuity condition + differentiability condition" pair traditionally used in many disciplines gives physically unjustified properties of the resulting exact solution for the velocity field. Although the condition for the differentiability of velocities is close in mathematical form to the stress continuity condition (by virtue of Newton’s law), in terms of physics, taking these conditions into account gives fundamentally different properties of the exact solution of the Navier-Stokes system of equations. It is shown that the consideration of the "the velocity field continuity condition + the stress field continuity condition" pair is more adequate to the physics of the process, which is consistent with the hypothesis of continuity.

Diffusive instabilities of baroclinic lenticular vortices

We consider a model of a circular lenticular vortex immersed into a deep and vertically stratified viscous fluid in the presence of gravity and rotation. The vortex is assumed to be baroclinic with a Gaussian profile of angular velocity both in the radial and axial directions. Assuming the base state to be in cyclogeostrophic balance, we derive linearized equations of motion and seek for their solution in a geometric optics approximation to find amplitude transport equations that yield a comprehensive dispersion relation. Applying the algebraic Bilharz criterion to the latter, we establish that the stability conditions are reduced to three inequalities that define the stability domain in the space of parameters. The main destabilization mechanism is either monotonic or oscillatory axisymmetric instability depending on the Schmidt number (Sc), vortex Rossby number, and the difference between radial and axial density gradients as well as the difference between epicyclic and vertical oscillation frequencies. We discover that the boundaries of the regions of monotonic and oscillatory axisymmetric instabilities meet at a codimension-2 point, forming a singularity of the neutral stability curve. We give an exhaustive classification of the geometry of the stability boundary, depending on the values of the Schmidt number. Although we demonstrate that the centrifugally stable (unstable) Gaussian lens can be destabilized (stabilized) by the differential diffusion of mass and momentum and that destabilization can happen even in the limit of vanishing diffusion, we also describe explicitly a set of parameters in which the Gaussian lens is stable for all Sc > 0.

The problem of normal oscillations of a viscous stratified fluid with an elastic membrane

Normal oscillations of a viscous stratified fluid partially filling an arbitrary vessel and bounded above by an elastic horizontal membrane are studied. In this case, we consider a scalar model problem that reflects the main features of the vector spatial problem. The characteristic equation for the eigenvalues of the model problem is obtained, the structure of the spectrum and the asymptotics of the branches of the eigenvalues are studied. Assumptions are made about the structure of the oscillation spectrum of a viscous stratified fluid bounded by an elastic membrane for an arbitrary vessel. It is proved that the spectrum of the problem is discrete, located in the right complex half-plane symmetrically with respect to the real axis, and has a single limit point $+\infty$. Moreover, the spectrum is localized in a certain way in the right half-plane, the location zone depends on the dynamic viscosity of the fluid.

A new upper bound for the largest growth rate of linear Rayleigh\u2013Taylor instability

We investigate the effect of (interface) surface tensor on the linear Rayleigh–Taylor (RT) instability in stratified incompressible viscous fluids. The existence of linear RT instability solutions with largest growth rate Λ is proved under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold vartheta _{mathrm{c}}) by the modified variational method of PDEs. Moreover, we find a new upper bound for Λ. In particular, we directly observe from the upper bound that Λ decreasingly converges to zero as ϑ goes from zero to the threshold vartheta _{mathrm{c}}.

Simplified solution to the problem plane-parallel motion partial control of an autonomous rigid body in incompressible stratified viscous fluid

The report presents a mathematical model of the motion control of an autonomous solid body moving in incompressible stratified viscous fluid and analytical and numerical analysis of this model. It is assumed that the body does not have its own propulsion system, but is equipped with controlled rudders - wings of finite span. The body is moved by the influence of the buoyancy force and wings lift. Partial movement control is achieved by the angle of attack of the wing change (for ensuring access to the given point by this solid body). This body motion is considered to be plane-parallel motion.

Instability of the abstract Rayleigh–Taylor problem and applications

Based on a bootstrap instability method, we prove the existence of unstable strong solutions in the sense of [Formula: see text]-norm to an abstract Rayleigh–Taylor (RT) problem arising from stratified viscous fluids in Lagrangian coordinates. In the proof we develop a method to modify the initial data of the linearized abstract RT problem by exploiting the existence theory of a unique solution to the stratified (steady) Stokes problem and an iterative technique, such that the obtained modified initial data satisfy the necessary compatibility conditions on boundary of the original (nonlinear) abstract RT problem. Applying an inverse transform of Lagrangian coordinates to the obtained unstable solutions and taking then proper values of the parameters, we can further obtain unstable solutions of the RT problem in viscoelastic, magnetohydrodynamics (MHD) flows with zero resistivity and pure viscous flows (with/without interface intension) in Eulerian coordinates.

On an Initial-Boundary Value Problem Which Arises in the Dynamics of a Viscous Stratified Fluid

On an Initial-Boundary Value Problem Which Arises in the Dynamics of a Viscous Stratified Fluid

Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows

The paper presents a numerical study of the efficiency of the newly proposed far-field boundary simulations of wall-bounded, stably stratified flows. The comparison of numerical solutions obtained on large and truncated computational domain demonstrates how the solution is affected by the adopted far-field conditions. The mathematical model is based on Boussinesq approximation for stably stratified viscous variable density incompressible fluid. The three-dimensional numerical simulations of the steady flow over an isolated hill were performed using a high-resolution compact finite difference code, with artificial compressibility method used for pressure computation. The mutual comparison of the full domain reference solution and the truncated domain solution is provided and the influence of the newly proposed far-field boundary condition is discussed.

Об одной начально-краевой задаче, возникающей в динамике вязкой стратифицированной жидкости

An initial-boundary value problem is considered, which describes the linear vibrations of a viscous stratified fluid in a bounded vessel with elastic membrane. We find sufficient existence conditions for a strong (with respect to the time variable) solution of the initial-boundary value problem describing the evolution of the specified hydrodynamics system.

Evolution of the Diffusion-Induced Flow over a Disk Submerged in a Stratified Viscous Fluid

The paper presents the results of the mathematically simulated evolution of a 3D diffusion-induced flow over a disk (with diameter d and thickness H = 0.76 d) immersed in a linearly density-stratified incompressible viscous fluid (described by a system of the Navier–Stokes equations in the Boussinesq approximation). The disk rests at the level of the neutral buoyancy (coinciding with its symmetry axis z) and disturbs the homogeneity of the background diffusion flux in the fluid forming a complex system of slow currents (internal gravitational waves). Over time, two thin horizontal convection cells are formed at the upper and lower parts of the disk stretching parallel to the z axis and adjacent to the base cell with thickness d/2. This work is the first to analyze in detail the fundamental mechanism for the formation of each new half-wave near the vertical axis x (passing through the center of the disk) during half the buoyancy period of the fluid Tb. This mechanism is based on gravitational instability. The emergence of this instability is first detected at 0.473Tb at a height of 3.9 d above the center of the disk. The same mechanism is also implemented over the place where the body moves in the horizontal direction. The 3D vortex structure of the flow is visualized by constructing the isosurfaces of the imaginary part of the complex-conjugate eigenvalues of the velocity gradient tensor. In mathematical simulation, we employed a numerical method SMIF that has proved itself over three decades with an explicit hybrid finite-difference scheme for the approximation of the convective terms of the equations (second-order approximation, monotonicity).

Process of the Formation of Internal Waves Initiated by the Start of Motion of a Body in a Stratified Viscous Fluid

The mechanism of formation of three-dimensional internal gravity waves initiated by the start of motion of a disk of given diameter and finite thickness in the horizontal direction along the disk symmetry axis from right to left at a given uniform velocity in an incompressible viscous linearly density-stratified fluid is first considered in detail. The consideration is carried out on the basis of numerical solution of the system of Navier—Stokes equations in the Boussinesq approximation and visualization of the three-dimensional vortex structure of the flow calculated. The obtained fields of the velocity vectors and pressure perturbations possess horizontal and vertical symmetry planes passing through the disk symmetry axis. The process of formation of flow in the upper half-space caused by the shear and gravitational instabilities is described. In this process, two horizontal vortex filaments are initially formed between the back disk face and the place of pulsed start and then transformed into legs of the hairpin vortex loop whose head is located to right of the start point. Thereafter, vortex rings are periodically formed above the start point during half the buoyancy period of fluid. The left-hand halves of the rings are transformed into half-waves occupying space between the disk and the start point.

Cross-diffusion effects on an exponentially stretching sheet in a doubly stratified viscous fluid

This article analyses the influence of double stratification and cross-diffusion effects on the viscous fluid flow over a porous sheet stretching exponentially by applying the velocity slip at the boundary. The numerical solutions to the governing equations are evaluated using successive linearisation procedure and the Chebyshev collocation method. The results obtained are compared with the existing results in the literature for special cases. The influence of the pertinent parameters on the physical quantities are displayed through graphs.

On the qualitative theory of the rotating Boussinesq and quasi-geostrophic equations

The author studies the rotating Boussinesq equations describing the motion of a viscous incompressible stratified fluid in a rotating system which is relevant, e.g., a Lagrangian and Eulerian analysis of a geophysical fluid flow. This article focuses on the initial boundary value problem to the rotating Boussinesq equations. The equations are derived from conservation laws in continuum physics followed by the formulation of the problem as initial value problem on Hilbert spaces. By using Faedo-Galerkin approximation and semigroup technique, existence and uniqueness of solutions are proved. Additionally, the manuscript outlines how Lyapunov functions can be used to assess energy stability criteria. The author also addresses singular problems for which the equation has parabolic structure (rotating Boussinesq equations for the atmosphere and ocean) and the singular limit is hyperbolic (quasi-geostrophic equations for the atmosphere and ocean) in the asymptotic limit of small Rossby number. In particular, this approach gives as a corollary a constructive proof of the well-posedness of the problem of quasi-geostrophic potential vorticity equations governing modons or Rossby solitons.

Entropy Generation Analysis in Nonlinear Convection Flow of Thermally Stratified Fluid in Saturated Porous Medium With Convective Boundary Condition

This article emphasizes the significance of entropy generation analysis and nonlinear temperature density relation on thermally stratified viscous fluid flow over a vertical plate embedded in a porous medium with a thermal dispersion effect. In addition, the convective surface boundary condition is taken into an account. By using the suitable transformations, the governing flow equations in dimensional form are converted into set of nondimensional partial differential equations. Then the local similarity and nonsimilarity procedures are applied to transform the set of nondimensional partial differential equations into set of ordinary differential equations and then the resulting system of equations are solved by Chebyshev spectral collocation method along with the successive linearization. The effect of pertinent parameters, namely, Biot number, mixed convection parameter, and thermal dispersion on velocity, temperature, entropy generation rate, and heat transfer rate are displayed graphically and the salient features are explored in detail.

Two-layer electrified pressure-driven flow in topographically structured channels

The flow of two stratified viscous immiscible perfect dielectric fluids in a channel with topographically structured walls is investigated. The flow is driven by a streamwise pressure gradient and an electric field across the channel gap. This problem is explored in detail by deriving and studying a nonlinear evolution equation for the interface valid for large-amplitude long waves in the Stokes flow regime. For flat walls, the electrified flow is long-wave unstable with a critical cutoff wavenumber that increases linearly with the magnitude of the applied voltage. In the nonlinear regime, it is found that the presence of pressure-driven flow prevents electrostatically induced interface touchdown that has been observed previously – time-modulated nonlinear travelling waves emerge instead. When topography is present, linearly stable uniform flows become non-uniform spatially periodic steady states; a small-amplitude asymptotic theory is carried out and compared with computations. In the linearly unstable regime, intricate nonlinear structures emerge that depend, among other things, on the magnitude of the wall corrugations. For a low-amplitude sinusoidal boundary, time-modulated travelling waves are observed that are similar to those found for flat walls but are influenced by the geometry of the wall and slide over it without touching. The flow over a high-amplitude sinusoidal pattern is also examined in detail and it is found that for sufficiently large voltages the interface evolves to large-amplitude waves that span the channel and are subharmonic relative to the wall. A type of ‘walking’ motion emerges that causes the lower fluid to wash through the troughs and create strong vortices over the peaks of the lower boundary. Non-uniform steady states induced by the topography are calculated numerically for moderate and large values of the flow rate, and their stability is analysed using Floquet theory. The effect of large flow rates is also considered asymptotically to find solutions that compare very well with numerical computations.