Inertia Forces in Lubricating Films

1955 ◽  
Vol 22 (3) ◽  
pp. 363-364
Author(s):  
R. S. Brand
Keyword(s):  
Reynolds Number ◽  
Hydrodynamic Lubrication ◽  
Equations Of Motion ◽  
Viscous Forces ◽  
Usual Theory ◽  
Inertia Forces

Abstract The usual theory of hydrodynamic lubrication is based on the assumption that the inertia forces in the equations of motion are negligible compared to the viscous forces. This paper shows that if the Reynolds number ρRω/μ is of the same order as R/h, then the inertia forces will be of the same order as the viscous forces.

Turbulent Lubrication—Its Genesis and Role in Modern Design

1974 ◽  
Vol 96 (1) ◽  
pp. 2-6 ◽  
Author(s):  
D. F. Wilcock
Keyword(s):  
Reynolds Number ◽  
Point Of View ◽  
Thrust Bearings ◽  
Viscous Forces ◽  
Taylor Vortices ◽  
Low Viscosity ◽  
Starting Point ◽  
Process Fluid ◽  
Inertia Forces ◽  
Lubricant Viscosity

Turbulence, a phenomenon well known in fluid flow, was first reported in journal bearings and thrust bearings in 1949. The observations were of higher torques and greater temperature rises than were expected from lower speed data. The transition from laminar behavior occurred at a Reynolds’ number corresponding to the predicted occurrence of Taylor vortices. This was the starting point for efforts to understand the phenomenon and to establish rules of behavior useful for predicting turbulent bearing performance. From an engineering point-of-view, good results in design have been achieved by treating turbulence as an increase in lubricant viscosity, the percent of increase being a function of the ratio of inertia forces to viscous forces, the Reynolds’ number. The effective result is greater film thickness and larger power losses in turbulent lubrication than would be anticipated from laminar theory. Where will the designer of the future encounter turbulence, and how will he treat its effects? Large turbogenerators have already reached a size where turbulent operation is experienced. The gradually increasing use of process-fluid-lubricated machinery, often involving low viscosity fluids such as water, liquid metal, and liquified gases, offers the designer fresh opportunities to understand and take advantage of turbulence in both hydrodynamic and hydrostatic designs.

The Effect of Fluid Inertia in Squeeze Film Damper Bearings: A Heuristic and Physical Description

1983 ◽  
Author(s):  
John A. Tichy
Keyword(s):  
Hydrodynamic Lubrication ◽  
Turbulence Models ◽  
Squeeze Film ◽  
Squeezing Flow ◽  
Fluid Inertia ◽  
Inertia Effects ◽  
Practical Applications ◽  
Viscous Forces ◽  
Squeeze Film Dampers ◽  
Inertia Forces

Fluid inertia forces are comparable to viscous forces in squeeze film dampers in the range of many practical applications. This statement appears to contradict the commonly held view in hydrodynamic lubrication that inertia effects are small. Upon closer inspection, the latter is true for predominantly sliding (rather than squeezing) flow bearings. The basic equations of hydrodynamic lubrication flow are developed, including the inertia terms. The appropriate orders of magnitude of the viscous and inertia terms are evaluated and compared, for journal bearings and for squeeze film dampers. Exact equations for various limiting cases are presented: low eccentricity, high and low Reynolds number. The asymptotic behavior is surprisingly similar in all cases. Due to inertia, the damper force may shift 90° forward from its purely viscous location. Inertia forces are evaluated for typical damper conditions. The effect of turbulence in squeeze film dampers is also discussed. On physical grounds it is argued that the transition occurs at much higher Reynolds numbers than the usual lubrication turbulence models predict.

scholarly journals Measurements of Squeeze Film Bearing Forces to Demonstrate the Effect of Fluid Inertia

1984 ◽  
Author(s):  
John A. Tichy
Keyword(s):  
Dynamic Systems ◽  
High Speed ◽  
Hydrodynamic Lubrication ◽  
Reynolds Numbers ◽  
Lubrication Theory ◽  
Squeeze Film ◽  
Fluid Inertia ◽  
Viscous Forces ◽  
Rotor Dynamic ◽  
Inertia Forces

Squeeze film dampers are commonly applied to high speed rotating machinery, such as aircraft engines, to reduce vibration problems. The theory of hydrodynamic lubrication has been used for the design and modeling of dampers in rotor dynamic systems despite typical modified Reynolds numbers in applications between ten and fifty. Lubrication theory is strictly valid for Reynolds numbers much less than one, which means that fluid viscous forces are much greater than inertia forces. Theoretical papers which account for fluid inertia in squeeze films have predicted large discrepancies from lubrication theory, but these results have not found wide acceptance by workers in the gas turbine industry. Recently, experimental results on the behavior of rotor dynamic systems have been reported which strongly support the existence of large fluid inertia forces. In the present paper direct measurements of damper forces are presented for the first time. Reynolds numbers up to ten are obtained at eccentricity ratios 0.2 and 0.5. Lubrication theory underpredicts the measured forces by up to a factor of two (100% error). Qualitative agreement is found with predictions of earlier improved theories which include fluid inertia forces.

On the Non-Linear Dynamics of a Traveling Cable With Small Sag

1995 ◽  
Author(s):  
Christoph G. Reuter ◽  
Peter Hagedorn
Keyword(s):  
Equations Of Motion ◽  
Steady State Solution ◽  
Extended Model ◽  
Transport Velocity ◽  
Non Linear ◽  
Mechanical Models ◽  
Linear Vibrations ◽  
Technical Systems ◽  
Inertia Forces ◽  
Made In

Abstract Traveling cables or threadlines appear in a number of technical applications such as textile machinery, V-belts, ski lifts, funiculars and also in simple models of traveling webs in paper machinery. The mechanical models used so far, most often neglect the effect of sag due to the weight of the cable, although it is well known that in some cases it may be quite important. In this paper, the authors develop a particularly simple model for translating cables using the assumption that the longitudinal inertia forces are negligible in comparison to the transversal inertia forces if the sag of the cable is sufficiently small. This assumption has already been made in a study of linear vibrations of stationary cables in 1970 by Irvine & Caughey. This lead to surprising results which have also been verified experimentally in the laboratory. The extended model presented in this paper includes gyroscopic and nonlinear terms in the equations of motion, related to the cable transport velocity and geometric nonlinearities. As a particular case (zero longitudinal speed and linear theory) the model of Irvine & Caughey is again contained in the present analysis. The linear and non-linear vibrations about a steady state solution are studied. The results show some interesting features which may also be relevant to technical systems if the transport speed is sufficiently high.

scholarly journals Geometrically Similar Rectangular Passive Micromixers and the Scaling Validity on Mixing Efficiency and Pressure Drops

2019 ◽  
Vol 69 (1) ◽  
pp. 69-84
Author(s):  
Veldurthi Naresh ◽  
D. Bodas ◽  
Chandel Sunil ◽  
Bhave Tejashree
Keyword(s):  
Reynolds Number ◽  
Low Reynolds Number ◽  
Scale Factor ◽  
Mixing Efficiency ◽  
Scaling Effect ◽  
Pressure Drops ◽  
Viscous Forces ◽  
Linear Dimensions ◽  
Simulation Results ◽  
Low Reynolds

AbstractIn the present work, two geometrically similar passive geometries with dumbbell shape were designed to perturb the dominating viscous forces in the low Reynolds number (Re) flows of the fluids. The geometries were designated as PDM-I and PDM-II, in which all the linear dimensions were related by a constant scale factor of two. Mixing efficiencies and pressure drops of the species at various Reynolds number (Re) were calculated to estimate the scaling effect validations. Finally, the geometrically similar PDM geometries were fabricated in Polydimethylsiloxane (PDMS) polymer to evaluate the scaling effect on the mixing efficiencies of the dyes and validated with the simulation results of species mixing.

Flow about a fluid sphere at low to moderate Reynolds numbers

1987 ◽  
Vol 177 ◽  
pp. 1-18 ◽  
Author(s):  
D. L. R. Oliver ◽  
J. N. Chung
Keyword(s):  
Reynolds Number ◽  
Equations Of Motion ◽  
Reynolds Numbers ◽  
Solid Sphere ◽  
Fluid Sphere ◽  
State Equations ◽  
Drag Coefficients ◽  
Low Reynolds Number Flows ◽  
Analytical Series ◽  
Reynolds Number Flows

The steady-state equations of motion are solved for a fluid sphere translating in a quiescent medium. A semi-analytical series truncation method is employed in conjunction with a cubic finite-element scheme. The range of Reynolds numbers investigated is from 0.5 to 50. The range of viscosity ratios is from 0 (gas bubble) to 107 (solid sphere). The flow structure and the drag coefficients agree closely with the limited available experimental measurements and also compare favourably with published finite-difference solutions. The strength of the internal circulation was found to increase with increasing Reynolds number. The flow patterns and the drag coefficient show little variation with the interior Reynolds number. Based on the numerical results, predictive equations for drag coefficients are recommended for both moderate- and low-Reynolds-number flows.

An almost-inviscid geostrophic flow

1962 ◽  
Vol 12 (1) ◽  
pp. 129-134 ◽  
Author(s):  
L. M. Hocking
Keyword(s):  
Reynolds Number ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Axial Direction ◽  
Reynolds Numbers ◽  
Rigid Rotation ◽  
Velocity Variation ◽  
Geostrophic Flow ◽  
The Difference ◽  
Streaming Motion

An almost rigid rotation of a viscous fluid is produced by dividing the containing cylinder into two sections and rotating them at slightly different speeds. The fluid velocity can be separated into two parts, a swirl about the axis and a streaming motion in the axial planes. When the difference in the speeds of rotation of the two sections is small, the equations of motion can be linearized. The solution is found for large Reynolds numbers and provides an illustration of the way in which the conditions of geostrophic flow (no velocity variation in the axial direction and an inability to insist on undistrubed flow at infinity) are approached as the Reynolds number tends to infinity.

scholarly journals Inertia and Couple-Stress Effects in a Curvilinear Thrust Hydrostatic Bearing

2017 ◽  
Vol 22 (3) ◽  
pp. 759-767 ◽  
Author(s):  
A. Walicka ◽  
P. Jurczak ◽  
J. Falicki
Keyword(s):  
Couple Stress ◽  
Equations Of Motion ◽  
Inertia Effects ◽  
Numerical Examples ◽  
Thrust Bearings ◽  
Hydrostatic Bearing ◽  
Stress Effects ◽  
Pressure Distributions ◽  
Radial Thrust ◽  
Inertia Forces

AbstractThe flow of a couple-stress lubricant in a clearance of a curvilinear thrust hydrostatic bearing with impermeable walls is considered. The flow in the bearing clearance is considered with inertia forces. The equations of motion are solved by an averaged inertia method. As a result, the formulae for pressure distributions without and with inertia effects were obtained. Radial thrust bearings and spherical bearings are discussed as numerical examples. It is shown that inertia effects influence the bearing performance considerably.

Two-dimensional sink flow of a stratified fluid contained in a duct

1972 ◽  
Vol 53 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Jorg Imberger
Keyword(s):  
Layer Thickness ◽  
Schmidt Number ◽  
Critical Distance ◽  
Force Balance ◽  
Viscous Force ◽  
Order Unity ◽  
Outer Flow ◽  
Viscous Forces ◽  
Primary Focus ◽  
Inertia Forces

A reservoir is assumed to be filled with water which has a linear variation of density with depth. The geometry of the boundaries is simplified to a parallel walled duct with the line sink at the centre of the fluid. The primary focus is on partitioning the flow into distinct flow regimes and predicting the withdrawal-layer thickness as a function of the distance from the sink; the predictions are verified experimentally.For fluids with a Schmidt number of order unity, the withdrawal layer is shown to be composed of distinct regions in each of which a definite force balance prevails. The outer flow, where inertia forces are neglected, changes from a parallel uniform flow upstream to a symmetric self-similar withdrawal layer near the sink. For distances from the sink smaller than a critical distance, dependent on the flow parameters, inertia forces become of equal importance to buoyancy and viscous forces. The equations valid in this inner region are derived. Using the inner limit of the outer flow as the upstream boundary condition, these inner equations are solved approximately for the withdrawal-layer thickness by an integral method. The inner and outer variations of δ, the withdrawal-layer thickness, are combined to yield a composite solution and it is seen that the inclusion of inertia forces yields layers thicker than those obtained from a strict buoyancy-viscous force balance. In terms of the inner variables the only parameter remaining is the Schmidt number.Laboratory experiments were carried out to verify the theoretical conclusions. The observed withdrawal-layer thicknesses were shown to be closely predicted by the integral solution. Furthermore, the data could be represented in terms of the inner variables by a single curve dependent only on the Schmidt number.

The steady flow due to a rotating sphere at low and moderate Reynolds numbers

1980 ◽  
Vol 101 (2) ◽  
pp. 257-279 ◽  
Author(s):  
S. C. R. Dennis ◽  
S. N. Singh ◽  
D. B. Ingham
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Reynolds Numbers ◽  
Low Reynolds Numbers ◽  
Finite Set ◽  
Flow Variables ◽  
Theoretical Results ◽  
Radial Variable ◽  
Good Agreement

The problem of determining the steady axially symmetrical motion induced by a sphere rotating with constant angular velocity about a diameter in an incompressible viscous fluid which is at rest at large distances from it is considered. The basic independent variables are the polar co-ordinates (r, θ) in a plane through the axis of rotation and with origin at the centre of the sphere. The equations of motion are reduced to three sets of nonlinear second-order ordinary differential equations in the radial variable by expanding the flow variables as series of orthogonal Gegenbauer functions with argument μ = cosθ. Numerical solutions of the finite set of equations obtained by truncating the series after a given number of terms are obtained. The calculations are carried out for Reynolds numbers in the range R = 1 to R = 100, and the results are compared with various other theoretical results and with experimental observations.The torque exerted by the fluid on the sphere is found to be in good agreement with theory at low Reynolds numbers and appears to tend towards the results of steady boundary-layer theory for increasing Reynolds number. There is excellent agreement with experimental results over the range considered. A region of inflow to the sphere near the poles is balanced by a region of outflow near the equator and as the Reynolds number increases the inflow region increases and the region of outflow becomes narrower. The radial velocity increases with Reynolds number at the equator, indicating the formation of a radial jet over the narrowing region of outflow. There is no evidence of any separation of the flow from the surface of the sphere near the equator over the range of Reynolds numbers considered.

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