Layer Of Viscous Fluid Research Articles

Nonlinear Thermal Instability in a Rotating Viscous Fluid Layer Under Temperature/Gravity Modulation

In the present paper, the effect of time-periodic temperature/gravity modulation on the thermal instability in a rotating viscous fluid layer has been investigated by performing a weakly nonlinear stability analysis. The disturbances are expanded in terms of power series of amplitude of modulation, which has been assumed to be small. The amplitude equation, viz., the Ginzburg–Landau equation, for the stationary mode of convection is obtained and using the same, the effect of temperature/gravity modulation on heat transport has been investigated. The stability of the system is studied and the stream lines are plotted at different slow times as a function of the amplitude of modulation, Rossby number, and Prandtl number. It is found that the temperature/gravity modulation can be used as an external means to augment/diminish heat transport in a rotating system. Further, it is shown that rotation can be effectively used in regulating heat transport.

Flow and heat transfer of couple‐stress permeable fluid sandwiched between viscous fluid layers

AbstractThe fully developed flow and heat transfer in a horizontal channel consisting of couple‐stress permeable fluid sandwiched between viscous fluid layers is investigated analytically. The channel walls are maintained at two different constant temperatures. The transport properties of the fluids in all regions are assumed to be constant. The governing equations are linear ordinary differential equations and hence closed form solutions are obtained. Effects of physical parameters such as viscosity ratio, thermal conductivity ratio, Eckert number, and Prandtl number on the flow are reported. An interesting and new approach is incorporated to analyze the flow for strong, weak, and comparable porosity with couple‐stress parameter. The variation of rate of heat transfer for different values of couple stress parameter and porosity is also discussed. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21007

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→+∞.

Quasipatterns in a parametrically forced horizontal fluid film

We shake harmonically a thin horizontal viscous fluid layer (frequency forcing Ω, only one harmonic), to reproduce the Faraday experiment and using the system derived in Rojas et al. (2010) [34] invariant under horizontal rotations. When the physical parameters are suitably chosen, there is a critical value of the amplitude of the forcing such that instability occurs with at the same time the mode oscillating at frequency Ω/2, and the mode with frequency Ω. Moreover, at criticality the corresponding wave lengths kc and kc′ are such that if we define the family of 2q equally spaced (horizontal) wave vectors kj on the circle of radius kc, then kj+kl=kn′, with |kj|=|kl|=kc,|kn′|=kc′.It results under the above conditions that 0 is an eigenvalue of the linearized operator in a space of time-periodic functions (frequency Ω/2) having a spatially quasiperiodic pattern if q≥4. Restricting our study to solutions invariant under rotations of angle2π/q, gives a kernel of dimension 4.In the spirit of Rucklidge and Silber (2009) [29] we derive formally amplitude equations for perturbations possessing this symmetry. Then we give simple necessary conditions on coefficients, for obtaining the bifurcation of (formally) stable time-periodic (frequency Ω/2) quasipatterns. In particular, we obtain a solution such that a time shift by half the period, is equivalent to a rotation of angleπ/qof the pattern.

An Equivalent Plate Model for Sandwiched Panels With Thermoviscous Fluid Core

Abstract A method has been developed to predict the damping behavior of an equivalent plate formed from sandwich panels with a thermoviscous fluid core, which is based on an asymptotic approach. The technique works on the break-up of the basic physical quantities such as pressure, temperature, and particle velocities as constants and linear functions of the z coordinate, which signifies the perpendicular direction from the plate mid-surface. The asymptotic modeling is performed in a very thin and highly thermoviscous fluid layer on the full set of linearized Navier Stokes equations. The use of highly viscous fluid layers such as Rhodorsil® 47 oils, yields high damping loss factors, which are of the order of 10−1. The effects of the panel thickness and the fluid core thickness are also studied. It is shown that the damping loss factor of the equivalent plate is independent of the frequency. Experimental validation provides a good agreement with the theoretical results.

Dynamics of a viscous layer flowing radially over an inviscid ocean

AbstractWe present a theoretical and experimental study of a viscous fluid layer spreading over a deep layer of denser, inviscid fluid. Specifically, we study an axisymmetric flow produced by a vertical line source. Close to the source, the flow is controlled viscously, with a balance between radial compressive stresses and hoop stresses. Further out, the flow is driven by gradients in the buoyancy force and is resisted by viscous extensional and hoop stresses. An understanding of these different fluid-mechanical relationships is developed by asymptotic analyses for early times and for the near and far fields at late times. Confirmation of the late-time, far-field behaviour is obtained from a series of laboratory experiments in which golden syrup was injected into denser solutions of potassium carbonate. We use our mathematical solutions to discuss a physical mechanism by which horizontal viscous stresses in a spreading ice shelf, such as those in West Antarctica, can buttress the grounded ice sheet that supplies it.

Nonlinear model of a granulated medium containing viscous fluid layers and gas cavities

We analyze nonlinear oscillations and waves in a simple model of a granular medium containing inclusions in the form of fluid layers and gas cavities. We show that in such a medium, the velocity of one of the wave modes is low; therefore, the nonlinearity is high and the effects of interaction are more strongly expressed than usual.

The 3D circulation and cyclone-anticyclone asymmetry in the shallow layers of viscous rotating fluid

A modified von Kármán problem that describes steady vortex flow in a rotating thin viscous fluid layer is solved. An analysis of the effect of bottom friction on the behavior of cyclonic and anticyclonic vortices at arbitrary values of the Rossby number is presented. Several anticyclonic flow patterns are examined. An approximate analytical solution obtained for steady flow is compared with numerical computations of a time-dependent problem. Experimental results on cyclonic and anticyclonic vortices in multiple-vortex quasi-turbulent flow are presented, and their interpretation based on the solution of the numerical model is given.

Analysis of Multi-Layer Immiscible Fluid Flow in a Microchannel

The development of microfluidics platforms in recent years has led to an increase in the number of applications involving the flow of multiple immiscible layers of viscous electrolyte fluids. In this study, numerical results as well as analytic equations for velocity and shear stress profiles were derived for N layers with known viscosities, assuming steady laminar flow in a microchannel driven by pressure and/or electro-static (Coulomb) forces. Numerical simulation results, using a commercial software package, match analytical results for fully-developed flow. Entrance flow effects with centered fluid-layer shrinking were studied as well. Specifically, cases with larger viscosities in the inner layers show a very good agreement with experimental correlations for the dimensionless entrance length as a function of inlet Reynolds number. However, significant deviations may occur for multilayer flows with smaller viscosities in the inner layers. A correlation was deduced for the two-layer electroosmotic flow and the pressure driven flow, both being more complex when compared with single-layer flows. The impact of using power-law fluids on resulting velocity profiles has also been explored and compared to Newtonian fluid flows. The present model readily allows for an exploration of the impact of design choices on velocity profiles, shear stress, and channel distribution in multilayer microchannel flows as a function of layered viscosity distribution and type of driving force.

Flow of a viscous fluid down a slope

A thin layer of viscous fluid flowing down a slope under gravity is considered. The stability of this flow is studied in the framework of a shallow water model. A number of equations of shallow water motion are derived with consideration of the viscous friction. The corresponding Cauchy problem is solved analytically. It is shown that the uniform flow may be unstable with respect to small initial perturbations. A criterion of instability is proposed.

Investigation of the interaction between a viscous incompressible fluid layer and walls of a channel formed by coaxial vibrating discs

The interaction between a viscous incompressible fluid layer and walls of a channel formed by two concentric discs moving perpendicularly to their planes due to vibration of the base on which the channel is mounted is investigated. The case of two absolutely rigid discs with elastic suspension and the case in which one of the discs is an elastic plate with the rigid restrain on the edges are considered. The velocity and pressure distributions over the fluid and the laws of motion of the walls and their frequency characteristics which make it possible to determine the resonance vibration frequencies of the mechanical system considered are found.

Gravitational stability of vertically moving system: Viscous layer and inviscid half-space

The evolution of small initial perturbations is investigated in a system consisting of a heavy layer of a Newtonian viscous fluid covering a half-space of an ideal fluid with another density. The entire system can move as a rigid entity in the vertical direction according to some prescribed law. By using the apparatus of linearization of the equations and the boundary conditions, the characteristic equation is derived and the high-viscosity limit, in which the two parameters determining the stability are the thinning and the overload, is treated analytically.

Effect of gravity modulation on the onset of thermal convection in rotating fluid and porous layer

The combined effect of gravity modulation and rotation on the onset of thermal convection in a horizontal fluid layer and a fluid-saturated porous layer is studied analytically using linear stability theory. The regular perturbation method based on small amplitude of modulation is employed to compute the critical value of Rayleigh number and wavenumber. We considered in the present paper only synchronous mode. The correction critical Rayleigh number is calculated as a function of Taylor number, Prandtl number, Darcy number, frequency of the modulation, normalized porosity, and viscosity ratio. It is shown that in general the gravity modulation produces a stabilizing effect in case of viscous fluid layer and both destabilizing and stabilizing effects in case of Brinkman porous layer while it produces a destabilizing effect in case of Darcy porous layer. The low frequency gravity modulation is found to have a significant effect on the stability of the system. It is shown that the onset of convection can be advanced or delayed by proper tuning of various governing parameters. The results of Darcy limit and viscous flow limit are derived as the degenerate cases of Brinkman model.

Lubrication theory applied to the convergent flows of two stacked liquid layers

With the aim of describing the mountain building process, we have previously applied the lubrication approximation to obtain the evolution equations of the problem of two stacked layers of viscous fluids with different densities and different viscosities. The lubrication approximation is a perturbation method where the small parameter is the aspect ratio (thickness/lenght) of the current. This approximation is widely used to study the slow flow of one layer of a viscous fluid, but it is not well known under which conditions it can be applied in more general settings. Here we analyze in detail the assumptions needed to apply the lubrication theory to study the flow of two stacked viscous fluid layers. We employ the same perturbation method and we found that, besides the usual conditions (low Reynolds number and gentle slope), we must require that the viscosity and density ratios are of the order of unity. These requirements determine the range of validity of the equations of our model of the mountain building.

The onset of convection in a rotating layer of viscous fluid with an imposed magnetic field: the dependence on the prandlt numbers

The onset of Boussinesq convection in a horizontal layer of an electrically conducting incompressible fluid is considered. The layer rotating about a vertical axis is heated from below; a vertical magnetic field is imposed. Rigid electrically insulating boundaries are assumed. The loss of stability of the trivial steady state, which occurs as the Rayleigh numbers increase, can be accompanied by the development of a monotonic or an oscillatory instability, depending on the parameter values of the problem at hand (the Taylor number, the Chandrasekhar number, the kinematic and the magnetic Prandtl numbers). When the instability is monotonic, the emerging convective rolls themselves are also unstable if the Taylor number is sufficiently large (the so-called Kuppers-Lortz instability takes place). In the present work it is studied how the critical value of the Rayleigh number, the type of the trivial steady state instability, and the critical value of the Taylor number for the Kuppers-Lortz instability depend on the kinematic and the magnetic Prandtl numbers. We consider the values of the Prandtl number not exceeding 1, which is typical for the outer core of the Earth.

Frequency shifts in a quartz plate piezoelectric resonator in contact with a viscous fluid under a separated electrode

We study thickness-shear vibration of a rotated Y-cut quartz crystal plate whose one surface is in contact with a viscous fluid layer of finite thickness. The fluid is under an el ectrode separated from the crystal plate and the driving electric field is in the plate thickness direction. The theory of piezo electricity and the theory of Newtonian fluids are used. Both free and forced vibration solutions are obtained. An approximate expression for the frequency shifts in the crystal plate du e to the fluid is presented. The admittance of the structure is also ca lculated. The results are useful for acoustic wave fluid sens ors.

Vortex patterns in quasi-two-dimensional flows of a viscous rotating fluid

A modified von Karman problem that describes steady vortex flow in a rotating thin viscous fluid layer is solved. An analysis of the effect of bottom friction on the behavior of cyclonic and anticyclonic vortices at arbitrary values of the Rossby number is presented. Several anticyclonic flow patterns are examined. An approximate analytical solution obtained for steady flows is compared with numerical computations of a time-dependent problem. Experimental results on cyclonic and anticyclonic vortices in multiple-vortex quasi-turbulent flow are presented, and their interpretation based on the solution of the model problem is given.

Faraday patterns in lubricated thin films

We study the patterns observed in the vicinity of a Faraday instability, in the limit of a very thin layer of viscous fluid. We numerically solve our previous model [N.O. Rojas et al., Phys. Rev. Lett. 104, 187801 (2010)] and compare our predictions to experiments. Our model captures quantitatively the threshold of instability. The direct simulation of our system permits us to predict the patterns observed in experiments.

Fluid-induced frequency shift in a piezoelectric plate driven by lateral electric fields

We study thickness-shear vibration of a piezoelectric crystal plate whose one surface is in contact with a viscous fluid layer of finite thickness. The crystal plate is driven by a lat eral electric field. The theory of piezoelectricity and the t heory of Newtonian fluids are used. Both free and forced vibration sol utions are obtained. An approximate expression for the frequency shift in the crystal plate due to the presence of the fluid is pr esented. The admittance of the structure is also calculated . The results are useful for acoustic wave fluid sensors.

Nonlinear development of two-layer Couette–Poiseuille flow in the presence of surfactant

The two-dimensional nonlinear evolution of the interface between two superposed layers of viscous fluid moving in a channel in the presence of an insoluble surfactant is examined. A pair of coupled weakly nonlinear equations is derived describing the interfacial and surfactant dynamics when one of the two fluid layers is very thin in comparison to the other. In contrast to previous work, the dynamics in the thin film are coupled to the dynamics in the thicker layer through a nonlocal integral term. For asymptotically small Reynolds number, the flow in the thicker layer is governed by the Stokes equations. A linearized analysis confirms the linear instability identified by previous workers and it is proven that the film flow is linearly unstable if the undisturbed surfactant concentration exceeds a threshold value. Numerical simulations of the weakly nonlinear equations reveal the existence of finite amplitude traveling-wave solutions. For order one Reynolds number, the flow in the thicker layer is governed by the linearized Navier–Stokes equations. In this case the weakly nonlinear film dynamics are more complex and include the possibility of periodic traveling-waves and chaotic flow.