Parallel Plane Walls Research Articles

Translation-rotation coupling and the kinematics of non-slip boundary conditions: A rough sphere between two sliding walls.

A non-slip constraint between a particle and a wall is applied at the microscopic level of collision dynamics using the rough sphere model. We analyze the consequences of the translation-rotation coupling of a rough sphere confined between two parallel planar walls and establish that shearing the walls past each other (i) preferentially deposits energy into the rotational degree of freedom and (ii) results in a bounded oscillation of the energy of the confined particle.

Density of condensates of two-component Bose-Einstein condensates restricted between two planar walls

The density of condensates of a binary mixture of Bose gases restricted by two parallel planar walls is investigated within the framework of Cornwal-Jackiw-Tomboulis effective action approach in improved Hartree-Fock approximation. It results that the density of condensates strongly depend not only on the distance between two walls but also on the interspecies interaction strength and they are equal their expectation values of the field operators after adding a term associated with the quantum fluctuations.

Внутридопплеровские резонансы затухания свободной поляризации в ультратонких газовых ячейках

The linear-optical effect of the free polarization decay (FPD) in an ultrathin gas cell, the internal thickness of which is less than or of the order of the wavelength of the exciting monochromatic laser radiation, transmitted orthogonally to the plane-parallel walls of this cell, is theoretically investigated. The considered coherent FPD signal is emitted immediately after an abrupt interruption of such a relatively weak stationary radiation. Sub-Doppler resonances arising at the central frequencies of the studied quantum transitions in the spectra of the intensity and energy of the FPD as a result of the transit-time relaxation of atoms in the gas cell are established and analyzed. A significant dependence of these resonances and the dynamics of the FPD on the ratio of the inner thickness of the ultrathin cell to the wavelength of the exciting monochromatic radiation is shown. The narrow, high-contrast sub-Doppler resonances of the FPD detected and investigated in the linear-optical regime can be used in ultrahigh-resolution atomic (molecular) spectroscopy, as well as references for compact frequency standards.

Features of Free Polarization Decay in Ultrathin Gas Cells

The optical effect of free polarization decay in an ultrathin gas cell the inner thickness of which is less than or of the order of the wavelength of the exciting monochromatic laser pulse transmitted orthogonally to plane-parallel walls of this cell is studied theoretically. A new mechanism of the studied effect is established; it is caused by the specificity of phase mismatch of light-induced atomic dipole moments due to transit-time relaxation of atoms in such a cell. As a result, the dynamics of free polarization decay in the considered situation radically differs from the known case in a usual (macroscopic) gas cell. Nontrivial oscillatory dependences of free polarization decay signals on the ratio of the thickness of such ultrathin cell to the radiation wavelength of the exciting pulse are revealed and analyzed.

Exclusion-Enrichment Effect in Ionic Transistors.

A simple model of a nanofluidic transistor consisting of a uniformly charged central section between a pair of plane parallel walls is considered. The linearized Poisson-Boltzmann equation corresponding to weak surface charge is solved exactly, and the solution is presented as an infinite series. The problem is characterized by three dimensionless parameters, namely the normalized surface charge, the ratio of the channel width to the Debye length, and the length-to-width aspect ratio of the charged section. The first of these parameters is presumed small, but the other two are arbitrary. The dependence of the exclusion-enrichment effect on these three parameters is discussed.

Numerical and perturbation solutions of third-grade fluid in a porous channel: Boundary and thermal slip effects

The steady flow of a third-grade fluid due to pressure gradient is considered between parallel plane walls which are kept at different temperatures. The space between the plane walls is assumed to be a porous medium of constant permeability. The viscosity of the fluid is taken as constant as well as a function of temperature. It is further assumed that the fluid may slip at the wall surfaces. The consequence of this assumption results in non-linear boundary conditions at the plane walls. The temperature field is also supposed to satisfy thermal slip condition at the walls. The governing equations are modelled under these assumptions and the approximate solution is obtained using the perturbation theory. The skin friction coefficient is a decreasing function of slip parameters in the case of temperature-dependent viscosity models while no variation is noted for the case of constant viscosity via boundary slip parameter. The heat transfer rate increases with the boundary slip parameter and decreases with the thermal slip parameter. The validity of the approximated solution is checked by calculating the numerical solution as well. The absolute error is calculated and listed in tabular form in the case of constant and temperature-dependent viscosity via boundary and thermal slip parameters. The influence of various emerging parameters on flow velocity and temperature profile is discussed through graphs.

Особенности затухания свободной поляризации в ультратонких газовых ячейках

Optical effect of the free polarization decay is studied theoretically in the ultrathin gas cell whose inner thickness is less or order of the wavelength of exciting monochromatic laser pulse transmitted orthogonally to plane-parallel walls of the cell. New mechanism of the investigated effect is established which is caused by specifics of phase mismatch of light induced atomic dipole moments because of transit-time relaxation in such a cell. As a result, dynamics of the free polarization decay in the considered situation radically different from the known case in a usual (macroscopic) gas cell. Nontrivial oscillatory dependences of signals of the free polarization decay on the ratio of thickness of such a cell to the radiation wavelength of the exciting pulse have been discovered and analyzed.

Poiseuille flow and thermal transpiration of a rarefied gas between plane parallel walls with nonuniform surface properties: integral equation approach

Poiseuille flow and thermal transpiration of a rarefied gas between plane parallel walls with nonuniform surface properties are studied on the basis of kinetic theory. Specifically, one of the walls is a diffuse reflection boundary, and the other wall is a Maxwell-type boundary whose accommodation coefficient has a periodic distribution in the direction perpendicular to the flow. The behavior of the gas is studied on the basis of the linearized Bhatnagar–Gross–Krook–Welander model of the Boltzmann equation. The extended integral equation for the macroscopic flow velocity is derived for the analysis at large Knudsen numbers, and is numerically solved. Features in the dependence of the mass flow rate on the distribution of the accommodation coefficient observed in the previous study (Doi 2015 ASME J. Fluids. Eng. 137 101103) were found to hold over a quite wide range of the Knudsen number up to 1000. A successive approximation method of the solution for a large Knudsen number is numerically tested using the integral equation.

Poiseuille, thermal transpiration and Couette flows of a rarefied gas between plane parallel walls with nonuniform surface properties in the transverse direction and their reciprocity relations

Slow flows of a rarefied gas between two plane parallel walls with nonuniform surface properties are studied based on kinetic theory. It is assumed that one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary whose accommodation coefficient varies periodically in the direction perpendicular to the flow. The time-independent Poiseuille, thermal transpiration and Couette flows are considered. The flow behavior is numerically studied based on the linearized Bhatnagar–Gross–Krook–Welander model of the Boltzmann equation. The flow field, the mass and heat flow rates in the gas, and the tangential force acting on the wall surface are studied over a wide range of the gas rarefaction degree and the parameters characterizing the distribution of the accommodation coefficient. The locally convex velocity distribution is observed in Couette flow of a highly rarefied gas, similarly to Poiseuille flow and thermal transpiration. The reciprocity relations are numerically confirmed over a wide range of the flow parameters.

Reciprocity relations in flows of a rarefied gas between plane parallel walls with nonuniform surface properties

Flows of a rarefied gas between plane parallel walls with nonuniform surface properties are studied based on kinetic theory. It is assumed that one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary whose accommodation coefficient varies periodically in the longitudinal direction. Four fundamental flows are studied, namely, Poiseuille flow, thermal transpiration, Couette flow, and the heat transfer problem. These flow problems are numerically studied based on the linearized Bhatnagar–Gross–Krook–Welander model of the Boltzmann equation over a wide range of the mean free path and the parameters characterizing the distribution of the accommodation coefficient. The flow fields, the mass and heat flow rates through a cross section or the wall surfaces, and the tangential force acting on the wall surfaces are studied. Due to the nonuniform surface properties, a longitudinal motion of the gas induces a local heat transfer through the wall surface in Poiseuille, thermal transpiration, and Couette flows; a temperature difference between the walls induces a motion of a gas and a local tangential stress on the walls in the heat transfer problem. However, the heat flow rate through the wall surface in the first three flow problems and the tangential force acting on the wall surface in the heat transfer problem vanish if integrated over one period. No net mass flow is induced in the heat transfer problem. Six reciprocity relations among the flow rates in the aforementioned four flows are numerically confirmed. Among the six relations, both hand sides of three relations vanish. The background of this phenomenon is discussed based on the flow field of the gas.

Transient motion of and heat transfer in a rarefied gas between plane parallel walls with different surface properties

Transient motion of and heat transfer in a rarefied gas between plane parallel walls with different surface properties are studied based on kinetic theory. It is assumed that one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary, and the transient behavior of the gas caused by a sudden heating of one of the walls is studied. The linearized Boltzmann equation for a hard-sphere molecular gas is numerically studied using the modified hybrid scheme of the characteristic coordinate and finite difference methods, to correctly describe the discontinuities in the velocity distribution function. The transient motion of the gas from an early time stage to the final time-independent state is studied over a wide range of the mean free path and the accommodation coefficient of the boundary. Between the two transient flows caused by the heating of the respective walls, the values of the heat flow on the heated wall are different, whereas those on the unheated wall coincide identically. This property, which is a consequence of the symmetric relation of the linearized Boltzmann equation, is numerically confirmed over a wide range of the mean free path. The long time behavior of the heat flow on the walls is quite similar to that of the shear stress in the Couette flow problem, whereas a distinct wavy behavior is observed in an early time stage.

Combined piezoresponse force microscopy and Raman scattering investigation of domain boundaries in BiFeO3 ceramics

Domain structure of BiFeO3 ceramics has been investigated using a combination of piezoresponse force microscopy (PFM) and Raman scattering techniques. Both methods demonstrate the presence of ferroelastic domains, separated by almost parallel planar domain walls, as well as the presence of large homogeneous single ferroelastic domain regions. In addition to highly resolved domain pattern obtained by PFM, small frequency shifts of the Raman-active modes give us complementary information about the angle φ between the surface normal and the rhombohedral axis of the BiFeO3 crystal for any measured position at the surface of the sample.

Flow generated by oscillatory uniform heating of a rarefied gas in a channel

Kinetic theory provides a rigorous foundation to explore the unsteady (oscillatory) flow of a dilute gas, which is often generated by nanomechanical devices. Recently, formal asymptotic analyses of unsteady (oscillatory) flows at small Knudsen numbers have been derived from the linearised Boltzmann–Bhatnagar–Gross–Krook (Boltzmann–BGK) equation, in both the low- and high-frequency limits (Nassios & Sader, J. Fluid Mech., vol. 708, 2012, pp. 197–249 and vol. 729, 2013, pp. 1–46; Takata & Hattori, J. Stat. Phys., vol. 147, 2012, pp. 1182–1215). These asymptotic theories predict that unsteadiness can couple strongly with heat transport to dramatically modify the overall gas flow. Here, we study the gas flow generated between two parallel plane walls whose temperatures vary sinusoidally in time. Predictions of the asymptotic theories are compared to direct numerical solutions, which are valid for all Knudsen numbers and normalised frequencies. Excellent agreement is observed, providing the first numerical validation of the asymptotic theories. The asymptotic analyses also provide critical insight into the physical mechanisms underlying these flow phenomena, establishing that mass conservation (not momentum or energy) drives the flows – this explains the identical results obtained using different previous theoretical treatments of these linear thermal flows. This study highlights the unique gas flows that can be generated under oscillatory non-isothermal conditions and the importance of both numerical and asymptotic analyses in explaining the underlying mechanisms.

Transient Couette flow of a rarefied gas between plane parallel walls with different surface properties

Transient Couette flow of a rarefied gas between plane parallel walls with different surface properties induced by a sudden start-up of one of the walls is studied based on kinetic theory. The linearized Boltzmann equation for a hard sphere molecular gas is analyzed under the assumptions that one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary. The initial and boundary value problem is solved numerically by using a modified hybrid scheme of characteristic coordinate and finite difference methods, to describe the discontinuities in the velocity distribution function correctly. The time evolution of the flow and the approach to the final time-independent state are studied over a wide range of the mean free paths and the accommodation coefficient of the boundary. In the transient process, the shear force acting on the moving wall depends on which wall moves. In contrast, the shear force acting on the wall at rest is independent of which wall moves; this property is a consequence of the symmetric relation of the Boltzmann equation [S. Takata, “Symmetry of the unsteady linearized Boltzmann equation in a fixed bounded domain,” J. Stat. Phys. 140, 985 (2010)]. The speed of approach to the time-independent state is fastest at an intermediate value of the mean free path. The behavior of the gas in the final time-independent state, including the heat flow in the isothermal gas, is also discussed.

Novel nano bearings constructed by physical adsorption.

The paper proposes a novel nano bearing formed by the physical adsorption of the confined fluid to the solid wall. The bearing is formed between two parallel smooth solid plane walls sliding against one another, where conventional hydrodynamic lubrication theory predicted no lubricating effect. In this bearing, the stationary solid wall is divided into two subzones which respectively have different interaction strengths with the lubricating fluid. It leads to different physical adsorption and slip properties of the lubricating fluid at the stationary solid wall respectively in these two subzones. It was found that a significant load-carrying capacity of the bearing can be generated for low lubricating film thicknesses, because of the strong physical adsorption and non-continuum effects of the lubricating film.

Solvent effects on lasing characteristics for Rh B laser dye

We demonstrate pulsed, photopumped multimode laser emission in the visible spectral range from rhodamine B dye dissolved in various solvents. The laser emission is characterized by a well-defined, low threshold pump power at which the emission spectral intensity dramatically increases and collapsed into several dominant laser modes with reduced mode spacing and spectral width. The modes were found to originate from the subcavities formed by the plane-parallel walls of the cuvette containing the gain medium. The cavity lasing spectral structure and the numbers of longitudinal modes were easily controlled by changing the solvents. A shift in the emission spectra has been also observed by changing the solvents will allow a limited range of tuning of laser emission wavelength. We also determined the gain coefficient and stimulated emission cross-section for the Rh B dye dissolved liquid laser system. A detailed discussion of the solvent effect in the lasing characteristics of Rh B in different solution is explained along with the computational data.

Flow of truncated power-law fluid between parallel walls for hydraulic fracturing applications

An essential closure of hydraulic fracturing models is the solution of the momentum equation for flow between plane parallel walls. Newtonian or simple power-law rheology is usually assumed. In real treatments, fracturing fluid often has more complicated rheology, such as Carreau. An earlier introduced modification to the power-law model enables a fair approximation to Carreau rheology. Unlike Carreau, it also enables a closed-form solution for the flow rate between plane parallel walls. The computational cost is, however, considerably smaller than with Carreau. Closed-form solution for the flow rate versus pressure gradient is obtained which is useful in hydraulic fracturing simulations. Compared to simple power-law model, the truncated power-law model improves accuracy of flow computations in small-aperture and large-aperture parts of the fracture, thereby improving the overall accuracy of hydraulic fracturing simulation.

Density functional study of condensation in capped capillaries

We study liquid adsorption in narrow rectangular capped capillaries formed by capping two parallel planar walls (a slit pore) with a third wall orthogonal to the two planar walls. The most important transition in confined fluids is arguably condensation, where the pore becomes filled with the liquid phase which is metastable in the bulk. Depending on the temperature T, the condensation in capped capillaries can be first-order (at ) or continuous (at ), where is the capillary wetting temperature. At , the capping wall can adsorb mesoscopic amounts of metastable under-condensed liquid. The onset of condensation is then manifested by the continuous unbinding of the interface between the liquid adsorbed on the capping wall and the gas filling the rest of the capillary volume. In wide capped capillaries there may be a remnant of wedge filling transition, which is manifested by the adsorption of liquid drops in the corners. Our classical statistical mechanical treatment predicts a possibility of three-phase coexistence between gas, corner drops and liquid slabs adsorbed on the capping wall. In sufficiently wide capillaries we find that thick prewetting films of finite length may be nucleated at the capping wall below the boundary of the prewetting transition. Prewetting then proceeds in a continuous manner manifested by the unbinding interface between the thick and thin films adsorbed on the side walls. Our analysis is based on a detailed numerical investigation of the density functional theory for the fluid equilibria for a number of illustrative case studies.

Structure and stability of steady porous medium convection at large Rayleigh number

A systematic investigation of unstable steady-state solutions of the Darcy–Oberbeck–Boussinesq equations at large values of the Rayleigh number $\mathit{Ra}$ is performed to gain insight into two-dimensional porous medium convection in domains of varying aspect ratio $L$. The steady convective states are shown to transport less heat than the statistically steady ‘turbulent’ flow realised at the same parameter values: the Nusselt number $\mathit{Nu}\sim \mathit{Ra}$ for turbulent porous medium convection, while $\mathit{Nu}\sim \mathit{Ra}^{0.6}$ for the maximum heat-transporting steady solutions. A key finding is that the lateral scale of the heat-flux-maximising solutions shrinks roughly as $L\sim \mathit{Ra}^{-0.5}$, reminiscent of the decrease of the mean inter-plume spacing observed in turbulent porous medium convection as the thermal forcing is increased. A spatial Floquet analysis is performed to investigate the linear stability of the fully nonlinear steady convective states, extending a recent study by Hewitt et al. (J. Fluid Mech., vol. 737, 2013, pp. 205–231) by treating a base convective state, and secondary stability modes, that satisfy appropriate boundary conditions along plane parallel walls. As in that study, a bulk instability mode is found for sufficiently small-aspect-ratio base states. However, the growth rate of this bulk mode is shown to be significantly reduced by the presence of the walls. Beyond a certain critical $\mathit{Ra}$-dependent aspect ratio, the base state is most strongly unstable to a secondary mode that is localised near the heated and cooled walls. Direct numerical simulations, strategically initialised to investigate the fully nonlinear evolution of the most dangerous secondary instability modes, suggest that the (long time) mean inter-plume spacing in statistically steady porous medium convection results from a balance between the competing effects of these two types of instability.

When is natural convection completely passive?

Momentum and energy equations for vertical flow with viscous dissipation are derived and shown to require that the cross‐section mean density is taken as the reference density for calculation of buoyancy forces under the Boussinesq approximation. Solutions are obtained for flow between parallel plane walls, with and without the pressure work as an explicit term in the energy equation. Both walls are at the same temperature, so there is no thermal forcing, but solutions are obtained for all admissible values of dynamic pressure gradient. The passive convection condition, whereby the flow is driven entirely by buoyancy forces resulting from heat generated by the flow's own viscous dissipation, is found on one branch of the dual solutions. However, while theoretically possible, passive convection is not physically realisable with any real fluid.