Layer Of Viscous Incompressible Fluid Research Articles

Thermal convection with a Cattaneo heat flux model

The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one.

Traveling Wave Solutions to the Free Boundary Incompressible Navier‐Stokes Equations

AbstractIn this paper we study a finite‐depth layer of viscous incompressible fluid in dimension , modeled by the Navier‐Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity‐capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time‐independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension and without surface tension in dimension , for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well‐known, to the best of our knowledge this is the first construction of viscous traveling wave solutions.Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal‐stress to normal‐Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Vanishing Viscosity and Surface Tension Limits of Incompressible Viscous Surface Waves

Consider the dynamics of a layer of viscous incompressible fluid under the influence of gravity. The upper boundary is a free boundary with the effect of surface tension taken into account, and the...

Change of the properties of turbulent flows in a spherical layer under the action of modulation of rotation velocity

Isothermal turbulent flows in a layer of viscous incompressible fluid confined between counter-rotating spherical boundaries have been experimentally studied. The influence of modulation of the velocity of inner sphere rotation on the characteristics of turbulence is considered. Measurements of flow velocity were carried out by laser Doppler anemometer. A local minimum is found in the dependence of the magnitude of turbulent pulsations on the amplitude of velocity modulation. It is established that, in the absence of modulation, the slopes of velocity spectra are close to–11/5. Modulation amplitude enhancing can lead to the formation of velocity spectra typical to two-dimensional turbulence, with slopes of–5/3 at low frequencies and–3 at high frequencies.

Triply resonant penetrative convection

A thermal convection model is considered that consists of a layer of viscous incompressible fluid contained between two horizontal planes. Gravity is acting vertically downward, and the fluid has a density maximum in the active temperature range. A heat source/sink that varies with vertical height is imposed. It is shown that in this situation there are three possible (different) sub-layers that may induce convective overturning instability. The possibility of resonance between the motion in these layers is investigated. A region is discovered where a very sharp increase in Rayleigh number is observed. In addition to a linearized instability analysis, two global (unconditional) nonlinear stability thresholds are derived.

Positive solutions to a nonlinear fourth-order partial differential equation

A class of fourth-order partial differential equations with Dirichlet boundary conditions which has important applications in phase transformation theory and in the description of the motion of a very thin layer of viscous incompressible fluid is studied. By transforming the higher order problem into a second-order elliptic system and using a fixed point method, the existence and uniqueness of steady state solutions are obtained. Moreover, by using a semi-discrete method, the existence of strong solutions of the evolution equation is obtained if the initial function is in the neighborhood of a positive steady state solution. Finally, the solution of the evolution equation converges to its steady state solution as the time t→+∞.

Study of stability and secondary regimes of thermocapillary flow in a liquid layer under localized heating

The problem of thermocapillary convection in a thin horizontal layer of viscous incompressible fluid with a deformable free boundary under the action of space-inhomogeneous localized temperature field is considered. The system of nonlinear differential equations for temperature, surface deformation and vorticity amplitudes in the long-wave approximation is solved numerically. The Galerkin method with polynomial basic functions is used to investigate steady states and their stability to infinitesimal two-dimensional disturbances in the cases of planar and axisymmetrical heat fluxes. The dependences of a disturbance decrement on the wave number are obtained for different steady states and parameter values. The forms of eigenfunctions for the most dangerous disturbances are presented. The nonlinear behavior of the disturbances of localized one-dimensional equilibrium states is investigated by the pseudo-spectral method in the two-dimensional formulation. It is shown that there is a domain of parameters in which these states are steady. Beyond its limits, diverse variants of the nonlinear development of disturbances can be realized, which, depending on the parameter values and disturbance magnitude, could lead to formation of the localized structures of another symmetry, global cellular structures of different symmetry and oscillatory regimes of thermocapillary convection.

On the linear evolution of the surface of a viscous incompressible fluid

We consider a layer of viscous incompressible fluid on a plate which can be vibrated vertically and we derive an equation which is non local in time for the linear evolution of the deformation of the surface in contact with the atmosphere. The equation is used to study the decay of a deformation of the surface. We characterize an interesting regime of high viscosity where the equation becomes local and we use it to study Faraday's instability. A new scenario is found for low frequency of the forcing. Key words : viscous fluid, Faraday's instability, lubrication regime.

2-D elastodynamic scattering from a semi-infinite cracklike flaw with interfacial forces

The canonical problems of 2-D elastodynamic scattering of longitudinal (L), ‘vertically’ polarized transverse (TV), and horizontally polarized transverse (TH) waves from a semi-infinite cracklike flaw are studied. The boundary conditions on the flaw are of a type which incorporate restoring forces as well as energy dissipation, thus enabling the modelling of a number of flaw types apart from the open crack or the closed friction-free crack. These types include a thin layer of viscous incompressible fluid and a thin inclusion of a material of Maxwell or Kelvin type. It is also argued that the boundary conditions, though linear, can be adapted to serve as an approximate model, capturing at least the aspect of energy dissipation, for a closed crack under a background pressure with friction between the faces of the crack. Both the uncoupled TH case, and the more complicated L-TV case, where mode coupling enters, are solved by means of the Wiener-Hopf technique. Both the geometrical elastodynamic and the edge-diffracted body waves are computed, the latter as high frequency approximations expressed in terms of diffraction coefficients. Similarly the diffraction coefficients for body-to-surface and surface-to-body waves are computed, providing the canonical solutions needed for a GTD computation of the 2-D scattering of L or TV waves from e.g. a finite cracklike flaw of the type considered.

Unsteady thermocapillary convection in a nonuniformly heated fluid layer

In many technological processes, thin extended layers of nonuniformly heated fluid are used [1–3]. If they are sufficiently thin, thermocapillary forces have a decisive influence on the occurrence and development of motion of the fluid [4–6]. Investigation of convective motion in such a layer is of great interest for estimating the intensity of heat and mass transfer in technological processes. This paper is a study of unsteady thermocapillary motion in a layer of viscous incompressible fluid with free surface in which a thermal inhomogeneity is created at the initial time. Approximate expressions are obtained for the fields of the velocity, temperature, and pressure in the fluid, and also for the shape of the free surface.

Influence of rotation on the damping of surface waves in a cylindrical layer of viscous fluid

A study is made of the oscillations of the free surface of a cylindrical layer of viscous incompressible fluid attracted by gravity to a solid, uniformly rotating cylinder. Logarithmic decay rates are found for the damping of the surface gravitational waves when large Reynolds numbers are assumed. It is shown that rotation introduces an asymmetry into the damping of the waves traveling in and against the direction of rotation of the fluid.

Separation and the Taylor-column problem for a hemisphere

A layer of viscous incompressible fluid is confined between two horizontal plates which rotate rapidly in their own plane with a constant angular velocity. A hemisphere has its plane face joined to the lower plate and when a uniform flow is forced past such an obstacle, a Taylor column bounded by thin detached vertical shear layers forms. The linear theory for this problem, wherein the Rossby number ε is set equal to zero on the assumption that the flow is slow, is examined in detail. The nonlinear modifications of the shear layers are then investigated for the case when ε ∼ E½, where E is the Ekman number. In particular, it is shown that provided that the Rossby number is large enough separation occurs in the free shear layers. The extension of the theory to flow past arbitrary spheroids is indicated.