Convective Inertia Research Articles

The dynamics and rheology of a dilute suspension of periodically forced neutrally buoyant spherical particles in a quiescent Newtonian fluid at low Reynolds numbers

We study the effects of both convective and unsteady inertia on the dynamics and rheology of a dilute suspension of periodically forced neutrally buoyant spherical particles, at low Reynolds numbers, in a quiescent Newtonian fluid. The inclusion of inertia results in additional terms in the equation governing the dynamics of the particle that represent a fading memory of the entire history of the particle motion. The inclusion of convective inertia in the low Reynolds number limit makes the memory term nonlinear. Several tests were performed to show that the results presented in this paper are physically reasonable and correct. A perturbation analysis of the problem yields strong evidence that the results of our simulations are correct. It is observed that there is a preferred direction in this system which manifests itself in the properties of the solution. This preferred direction is identified as the direction of the initial motion of the particle. We present here results on the behavior of various parameters with respect to Reynolds numbers and the amplitude of the periodic force. These include phase-space plots between particle displacement and particle velocity and the variation of a rheological parameter, namely a ‘normal stress’ with respect to Reynolds number and the amplitude of the periodic force. We believe that our results may be technologically important since the rheological parameter depends strongly on controllable parameters such as the Reynolds number and the amplitude of the periodic force. Further, this system is one of the simplest systems whose rheology shows non-Newtonian behavior, such as the presence of a normal stress. In addition, this system represents a physically realizable system for experimentally testing the frameworks developed to calculate the collective behavior of systems of oscillators with memory.

A Review of Tilting Pad Bearing Theory

A theoretical basis for static and dynamic operation of tilting pad journal bearings (TPJBs) has evolved over the last 50 years. Originally demonstrated by Lund using the pad assembly method and a classic Reynolds equation solution, the current state of the art includes full thermoelastohydrodynamic solutions of the generalized Reynolds equation that include fluid convective inertia effects, pad motions; and thermal and mechanical deformations of the pads and shaft. The development of TPJB theory is reviewed, emphasizing dynamic modeling. The paper begins with the early analyses of fixed geometry bearings and continues to modern analyses that include pad motion and stiffness and damping effects. The development of thermohydrodynamic, thermoelastohydrodynamic, and bulk-flow analyses is reviewed. The theories of TPJB dynamics, including synchronous and nonsynchronous models, are reviewed. A discussion of temporal inertia effects in tilting pad bearing is considered. Future trends are discussed, and a path for experimental verification is proposed.

Dynamics and ‘normal stress’ evaluation of dilute suspensions of periodically forced prolate spheroids in a quiescent Newtonian fluid at low Reynolds numbers

The problem of determining the force acting on a particle in a fluid where the motion of the fluid and the particle is given has been considered in some detail in the literature. In this work, we propose an example of a new class of problems where, the fluid is quiescent and the effect of an external periodic force on the motion of the particle is determined at low non-zero Reynolds numbers. We present an analysis of the dynamics of dilute suspensions of periodically forced prolate spheroids in a quiescent Newtonian fluid at low Reynolds numbers including the effects of both convective and unsteady inertia. The inclusion of both forms of inertia leads to a nonlinear integro — differential equation which is solved numerically for the velocity and displacement of the individual particle. We show that a ‘normal stress’ like parameter can be evaluated using standard techniques of Batchelor. Hence this system allows for an experimentally accessible measurable macroscopic parameter, analogous to the ‘normal stress’, which can be related to the dynamics of individual particles. We note that this ‘normal stress’ arises from the internal fluctuations induced by the periodic force. In addition, a preliminary analysis leading to a possible application of separating particles by shape is presented. We feel that our results show possibilities of being technologically important since the ‘normal stress’ depends strongly on the controllable parameters and our results may lead to insights in the development of active dampeners and smart fluids. Since we see complex behaviour even in this simple system, it is expected that the macroscopic behaviour of such suspensions may be much more complex in more complex flows.

Forced Convection Flow of Power-Law Fluids Over a Flat Plate Embedded in a Darcy-Brinkman Porous Medium

The steady forced convection flow of a power-law fluid over a horizontal plate embedded in a saturated Darcy-Brinkman porous medium is considered. The flow is driven by a constant pressure gradient. In addition to the convective inertia, also the “porous Forchheimer inertia” effects are taken into account. The pertinent boundary value problem is investigated analytically, as well as numerically by a finite difference method. It is found that far away from the leading edge, the velocity boundary layer always approaches an asymptotic state with identically vanishing transverse component. This holds for pseudoplastic (0 1) fluids as well. The asymptotic solution is given for several particular values of the power-law index n in an exact analytical form. The main flow characteristics of physical and engineering interest are discussed in the paper in some detail.

Short and long waves over a muddy seabed

The available experimental results have shown that in time-periodic motion the rheology of fluid mud displays complex viscoelastic behaviour. Based on the measured rheology of fluid mud from two field sites, we study the interaction of water waves and fluid mud by a two-layered model in which the water above is assumed to be inviscid and the mud below is viscoelastic. As the fluid-mud layer in shallow seas is usually much thinner than the water layer above, the sharp contrast of scales enables an approximate analytical theory for the interaction between fluid mud and small-amplitude waves with a narrow frequency band. It is shown that at the leading order and within a short distance of a few wavelengths, wave pressure from above forces mud motion below. Over a much longer distance, waves are modified by the accumulative dissipation in mud. At the next order, infragravity waves owing to convective inertia (or radiation stresses) are affected indirectly by mud motion through the slow modulation of the short waves. Quantitative predictions are made for mud samples of several concentrations and from two different field sites.

Convective inertia effects on the squeeze film between rectangular parallel plates

This paper investigates the effects of convective inertia on the lubricating characteristics of the squeeze film between parallel rectangular plates. Relaxing a simplifying assumption of the conventional lubrication theory, the paper retains some terms of the convective inertia terms of the Navier-Stokes equations, leading to a derivation of a modified Reynolds equation. The introduction of the convective inertia terms induces two inversely related dimensionless parameters, m and R (the Reynolds number), into the characteristics of the flow behavior. The dimensionless normal (or squeezing) velocity increases monotonically from the lower plate to the upper plate for given value of R. At every point of the flow space, the squeezing velocity decreases with increase in R due to an increase in the fluid's resistance to the motion of the upper plate and an increase in the rate of loss of momentum per unit volume of fluid flow. The dimensionless pressure and load capacity decrease with increase in the Reynolds number R due to an increase the kinetic energy of the fluid.

Dynamic and Static Characteristics of Wavy Annular Seals in Turbulent Flow

This paper presents results of a theoretical analysis to determine the leakage flow and dynamic characteristics of circumferential wave geometry annular seals using turbulent lubrication theory. Turbulent lubrication equations based on an eddy viscosity model have been used to develop the turbulent Reynolds equation. Convective fluid inertia effects have been incorporated using the perturbation approach in turbulent lubrication. To determine the dynamic characteristics i.e. stiffness, damping and whirl ratio, perturbation theory for small amplitude vibration of the journal center has been adopted. In general, it has been found that circumferential wave geometry improves the performance of the annular seal at higher speeds and increases in the L/D and taper ratios.

The Dynamics of Periodically Forced Prolate Spheroids in a Quiescent Newtonian Fluid with Weak Inertia

The effects of convective and unsteady inertia on the dynamics of periodically forced neutrally buoyant prolate spheroids in a quiescent Newtonian fluid medium, at low Reynolds numbers have been modeled. The resulting nonlinear equations have been solved using appropriate numerical methods. Several tests including a perturbation analysis are performed to validate results. A preferred direction, which is identified as the initial direction of motion, is observed that manifests itself in the properties of the solution. Results of the behavior of various parameters with respect to the Reynolds number, aspect ratio of the spheroid and the amplitude of the periodic force are presented. The results are technologically important as they may lead to insights in the development of active dampeners and smart fluids.

Polar wander of the Earth associated with the Quaternary glacial cycle on a convecting mantle

SUMMARY Observed true polar wander (TPW) at the present-day, with the speed of ∼1° Myr−1 towards Hudson Bay, has been generally examined based on the glacial isostatic adjustment (GIA) for the Quaternary glacial cycle. The observation is, however, affected by the present-day melting events of polar ice sheets and mountain glaciers and convective processes in the mantle. In this study, I examine TPW due to GIA and convective processes by developing the Liouville equation including these two processes, referred to as generalized Liouville equation here. The generalized Liouville equation makes it possible to evaluate the effects of convective related non-forcing inertia elements (I11, I22, I33 and I12) and forcing elements (I13 and I23) on the polar wander for the Quaternary ice age phase, and clearly indicates that the excess flattening of the Earth, inferred from the non-hydrostatic geoid, stabilizes the polar motion as discussed by Mitrovica et al. (2005). Also, numerical experiments for examining the effects of time-dependent convective processes on TPW indicate that reasonable convection scenarios with a time dependence in the inertia tensor of ∼1031 kg m2 Myr−1 (Ricard et al. 1993) insignificantly contribute to the observed present-day TPW; moreover, the predicted TPW with magnitude of convective related forcing rates, dI13/dt and dI23/dt, larger than ∼5 × 1031 kg m2 Myr−1 is significantly different from the observation. These results may provide a framework for separating the contributions of GIA and convection to the observed present-day polar wander.

Computer Modeling for the Complex Response Analysis of Nonstandard Structural Dynamics Problems

Over the past several decades, two intriguing classes of problems, having a wide range of applications in engineering, have been of interest to many researchers: (1) coupled dynamics of a distributed parameter system traversed by one or more moving oscillators; and (2) transient dynamic analysis of axially moving media (and associated phenomena of parametric resonances). Bridge vehicle interaction falls into the first class of problems, and the analysis of flexible appendages deployed from a satellite or a spacecraft is typical of the second class. Mathematically, these two problems are dual to each other, and they often are highly nonlinear in nature and typically involve large overall motion in space with complex effects of convective inertia terms in their governing equations of dynamic equilibrium. The “nonstandard” analytical nature of these problems stems from the fact that we are dealing with one or more of the following peculiarities: (1) variable problem domain; (2) varying spatial distribution o...

Three-dimensional elastohydrodynamics of a thin plate oscillating above a wall

We consider deflections of a thin rectangular elastic plate that is submerged within a Newtonian fluid. The plate is clamped along one edge and supported horizontally over a plane horizontal wall. We consider both external driving, where the clamped edge is vibrated vertically at high frequencies, and thermal driving, where the plate fluctuates under Brownian motion. In both cases, the amplitude of oscillation is assumed sufficiently small that the resulting flow has little convective inertia, although the oscillation frequency is sufficiently high to generate substantial unsteady inertia in the flow, a common scenario in many nano- and microdevices. We exploit the plate's thinness to develop an integral-equation representation for the three-dimensional flow (a so-called thin-plate theory) which offers considerable computational savings over a full boundary-integral formulation. Limiting cases of high oscillation frequencies and small wall-plate separation distances are studied separately, leading to further simplified descriptions for the hydrodynamics. We validate these reduced integral representations against full boundary-integral computations, and identify the parameter ranges over which these simplified formulations are valid. Addressing the full flow-structure interaction, we also examine the limits of simpler two-dimensional hydrodynamic models. We compare the responses of a narrow plate under two- and three-dimensional hydrodynamic loading, and report differences in the frequency response curves that occur when the plate operates in water, in contrast to the excellent agreement observed in air.

Nonisothermal Transient Flow in Natural Gas Pipeline

Abstract The fully implicit finite-difference method is used to solve the continuity, momentum, and energy equations for flow within a gas pipeline. This methodology (1) incorporates the convective inertia term in the conservation of momentum equation, (2) treats the compressibility factor as a function of temperature and pressure, and (3) considers the friction factor as a function of the Reynolds number and pipe roughness. The fully implicit method representation of the equations offers the advantage of guaranteed stability for a large time step, which is very useful for gas pipeline industry. The results show that the effect of treating the gas in a nonisothermal manner is extremely necessary for pipeline flow calculation accuracies, especially for rapid transient process. It also indicates that the convective inertia term plays an important role in the gas flow analysis and cannot be neglected from the calculation.

Euler equations in geophysics and astrophysics

The use of Euler equations in Geophysics and Astrophysics is reviewed. Recent developments and new applications are emphasized. Examples are buoyancy columns in rotating fluids, possible preference for axisymmetric inertial convection at low Prandtl numbers, resonance properties of precessing spheroidal fluid filled cavities, and the possible absence of turbulence in rotating shear flows in the limit of high Reynolds numbers.

Combined effects of non-Newtonian couple stresses and fluid inertia on the squeeze film characteristics between a long cylinder and an infinite plate

On the basis of the Stokes micro-continuum theory together with the averaged inertia principle, the combined effects of non-Newtonian couple stresses and convective fluid inertia forces on the squeeze film motion between a long cylinder and an infinite plate are presented. A closed-form solution has been derived for squeeze film characteristics including the film pressure, the load capacity and the response time. Comparing with the Newtonian-lubricant non-inertia case, the combined effects of couple stresses and convective inertia forces provide an increase in the film pressure, the load capacity and the response time. In addition, the quantitative effects of couple stresses and convective inertia forces are more pronounced for cylinder–plate system operating at a larger couple stress parameter and film Reynolds number, as well as a smaller squeeze film height. To guide the use of the present study, a numerical example is also illustrated for engineers when considering both the effects of non-Newtonian couple stresses and fluid convective inertia forces.

Combined effects of non-Newtonian rheology and fluid inertia forces on the non-linear transient behaviour in circular squeeze films

On the basis of the momentum integral method and the microcontinuum theory, the combined effects of total fluid inertia forces and non-Newtonian couple stresses on the non-linear squeeze film characteristics between parallel circular discs are presented. The squeeze-film lubrication equation is derived from the Stokes couple-stress motion equation retaining total fluid inertia terms to take into account the effects of non-Newtonian couple stresses resulting from the base oil blended with various types of additives, and the effects of local fluid inertia and convective fluid inertia. The non-linear motion equation for the squeezing disc is then numerically solved by using fourth-order Runge-Kutta method. Comparing with the Newtonian-lubricant inertialess case, the combined effects of fluid inertia forces and non-Newtonian couple stresses provide a significant increment in the response time of the squeeze film. The operating life of squeeze film discs are therefore lengthened, especially for larger couple stress parameters, Reynolds numbers and body inertia parameters. A design example is provided to facilitate the engineers assessing the extent to design the non-linear squeezing film discs considering the combined effects of total fluid inertia forces and non-Newtonian couple stresses.

Design and performance of an experiment for the investigation of open capillary channel flows

In this paper we report on the set-up and the performance of an experiment for the investigation of flow-rate limitations in open capillary channels under low-gravity conditions (microgravity). The channels consist of two parallel plates bounded by free liquid surfaces along the open sides. In the case of steady flow the capillary pressure of the free surface balances the differential pressure between the liquid and the surrounding constant-pressure gas phase. A maximum flow rate is achieved when the adjusted volumetric flow rate exceeds a certain limit leading to a collapse of the free surfaces. The flow is convective (inertia) dominated, since the viscous forces are negligibly small compared to the convective forces. In order to investigate this type of flow an experiment aboard the sounding rocket TEXUS-41 was performed. The aim of the investigation was to achieve the profiles of the free liquid surfaces and to determine the maximum flow rate of the steady flow. For this purpose a new approach to the critical flow condition by enlarging the channel length was applied. The paper is focussed on the technical details of the experiment and gives a review of the set-up, the preparation of the flight procedures and the performance. Additionally the typical appearance of the flow indicated by the surface profiles is presented as a basis for a separate continuative discussion of the experimental results.

Asymptotic theory of inertial convection in a rotating cylinder

Inertial convection in a fluid contained in a rotating cylinder heated uniformly from below is investigated on the basis of the assumption that convection at leading order can be represented by a single or several inertial wave modes which propagate either in the prograde or retrograde direction. Buoyancy forces appear at the next order to drive inertial convection against the effect of viscous damping. Asymptotic expressions for inertial convection for four different combinations of the sidewall boundary condition are derived for a cylinder of arbitrary aspect ratio. New convection patterns in rotating cylinders are revealed by the asymptotic analysis. A fully numerical solution of the same problem is also carried out, demonstrating a quantitative agreement between the asymptotic and numerical analysis.

Optical altimetry: a new method for observing rotating fluids with applications to Rossby and inertial waves on a polar beta-plane

The entire free-surface elevation field of a rotating fluid in the laboratory can be imaged and analysed, by using it as a parabolic Newtonian telescope mirror. This ‘optical altimetry’ readily achieves a precision of better than 1 μm of surface elevation. The surface topography corresponds to the pressure field just beneath the surface. It is the streamfunction for the geostrophic hydrostatic circulation, which can be resolved to better than 0.1 mm s−1. Still and animated images thus produced, of the entire surface elevation field, are of value in themselves, and using a projected image (a speckle pattern), have the promise of providing quantitative slope and height field data recovered by PIV (particle imaging velocimetry) techniques. With homogeneous fluid, geostrophic flow is the same at all depths. Yet of equal interest are sheared stratified rotating flows where the surface pressure is associated with inertial waves, convection, and other motions, geostrophic or ageostrophic.Although the technique is designed for experiments in which Coriolis effects are strong, it is possible to use reflective imaging for flows at such high Rossby number that Coriolis effects are negligible, and hence this becomes a tool of more general interest in non-rotating fluid dynamics (for example, illuminating surface gravity waves).Examples are given, involving (i) the Taylor–Proudman effect with very slow flows over topography; (ii) quasi-geostrophic and inertial-wave flows over a mountain (f-plane); (iii) inertial waves generated by oscillatory forcing; (iv) Kelvin waves (v) free oscillatory Rossby waves on a polar β-plane; and (vi) stationary waves, blocking, jets and wakes with β-plane zonal flow past a mountain. Movies are available with the online version of the paper.

A Multiscale Method Modeling Surface Texture Effects

In this paper a multiscale method is presented that includes surface texture in a mixed lubrication journal bearing model. Recent publications have shown that the pressure generating effect of surface texture in bearings that operate in full film conditions may be the result of micro-cavitation and/or convective inertia. To include inertia effects, the Navier–Stokes equations have to be used instead of the Reynolds equation. It has been shown in earlier work (de Kraker et al., 2006, Tribol. Trans., in press) that the coupled two-dimensional (2D) Reynolds and 3D structure deformation problem with partial contact resulting from the soft EHL journal bearing model is not easy to solve due to the strong nonlinear coupling, especially for soft surfaces. Therefore, replacing the 2D Reynolds equation by the 3D Navier–Stokes equations in this coupled problem will need an enormous amount of computing power that is not readily available nowadays. In this paper, the development of a micro–macro multiscale method is described. The local (micro) flow effects for a single surface pocket are analyzed using the Navier–Stokes equations and compared to the Reynolds solution for a similar smooth piece of surface. It is shown how flow factors can be derived and added to the macroscopic smooth flow problem, that is modeled by the 2D Reynolds equation. The flow factors are a function of the operating conditions such as the ratio between the film height and the pocket dimensions, the surface velocity, and the pressure gradient over a surface texture unit cell. To account for an additional pressure buildup in the texture cell due to inertia effects, a pressure gain is introduced at macroscopic level. The method also allows for microcavitation. Microcavitation occurs when the pressure variation due to surface texture is larger than the average pressure level at that particular bearing location. In contrast with the work of Patir and Cheng (1978, J. Lubrication Technol., 78, pp. 1–10), where the microlevel is solved by the Reynolds equation, and the Navier–Stokes equations are used at the microlevel. Depending on the texture geometry and film height, the Reynolds equation may become invalid. A second pocket effect occurs when the pocket is located in the moving surface. In mixed lubrication, fluid can become trapped inside a pocket and squeezed out when the pocket is running into an area with higher contact load. To include this effect, an additional source term that represents the average fluid inflow due to the deformation of the surface around the pocket is added to the Reynolds equation at macrolevel. The additional inflow is computed at microlevel by numerical solution of the surface deformation for a single pocket that is subject to a contact load. The pocket volume is a function of the contact pressure. It must be emphasized that before ready-to-use results can be presented, a large number of simulations to determine the flow factors and pressure gain as a function of the texture parameters and operating conditions have yet to be done. Before conclusions can be drawn, regarding the dominanant mechanism(s), the flow factors and pressure gain have to be added to the macrobearing model. In this paper, only a limited number of preliminary illustrative simulation results, calculating the flow factors for a single 2D texture geometry, are shown to give insight into the method.

Heat Transfer in a Rotating Twin-Pass Trapezoidal-Sectioned Passage Roughened by Skewed Ribs on Two Opposite Walls

An experimental study of heat transfer in a radially rotating twin-pass trapezoidal-sectioned duct with two opposite walls roughened by 45° staggered ribs was performed. Two channel orientations of 0° and 45° from the direction of rotation were tested. At each Reynolds number of 5000, 7500, 10000, 12500, and 15000, local Nusselt numbers along the centerlines of two rib-roughened surfaces with five different heating levels were acquired at rotating numbers of 0, 0.1, 0.3, 0.5, 0.7, and 1. A selection of experimental results illustrates the isolated and interactive influences of convective inertial, Coriolis, and rotating buoyancy forces on local and centerline-averaged heat transfers. The isolated Coriolis force-effect improves heat transfer over two unstable surfaces of the rotating twin-pass channel. The rotating buoyancy effect undermines local heat transfer, but its influence is alleviated when the rotating number increases. At rotating number of 0.7 and 1, the rotating buoyancy force acting with counter-flow manner considerably impairs local heat transfer in the end-region of the first passage with radially outward flow. With the rotating numbers in the range of 0.1 to 1, the heat transfer differences between the two channels with orientations of 0° and 45° are in the range of 5–26%. As a strategic aim of the present study, heat transfer correlations are derived to evaluate the centerline-averaged Nusselt numbers over two rib-roughened surfaces that permit the individual and interactive influences of convective inertia, Coriolis force, and rotating buoyancy to be quantified. As the full-field spatial heat transfer variations in the present rotating channel are not measured, the local heat transfer results generated by the present study are limited to the locations measured.