Stokes Micro-continuum Theory Research Articles

Elasto-hydrodynamic lubrication analysis of a porous misaligned crankshaft bearing operating with nanolubricants

In this paper, the combined effects of the characteristic size and concentration of inorganic fullerene-like tungsten disulphide nanoparticles (IF-WS2 NPs) or molybdenum disulphide nanoparticles (IF-MoS2 NPs) on the nonlinear dynamic behaviour of a gasoline engine crankshaft bearing subject to an arbitrary force torsor (effective applied force and moment vector) are theoretically and numerically investigated using the V. K. Stokes micro-continuum theory. These NPs are the most common additives for lubrication purposes due to their excellent tribological characteristics along with their effect on reducing friction and wear. It is assumed that the journal (crankshaft) currently made of a forged steel is rigid and the main bearing consists of a thin poroelastic liner made of low elastic modulus materials like Babbitt metals fixed in a stiff housing as defined by ASTM B23-00. The Krieger-Dougherty law is included in the proposed EHD model to account for the viscosity variation with respect to the volume fraction of nanoparticles dispersed in the base lubricant. On the other hand, the characteristic size of nanomaterials is introduced by a new material entity, denoted l, which is responsible for a couple-stress property. The Reynolds equation is derived in transient conditions and modified to account for the size of nanoparticles and the bearing-liner permeability property. For an arbitrary force torsor, the hydrodynamic pressure distribution, the squeeze film velocities, and the misalignment angular velocities are determined simultaneously by solving the discretized Reynolds equation and the equilibrium equations with the damped Newton-Raphson iterative method at each crank angle step. The crankshaft center trajectories in three sections of the main journal axis as well as the misalignment angles are deduced from the squeeze film velocities and the misalignment angular velocities by means of a Runge-Kutta scheme. According to the obtained results, the combined effects of the size and concentration of fullerene-like nanoparticles on the dynamic behavior of a compliant dynamically loaded crankshaft bearing operating with dynamic misalignment are significant and cannot be overlooked.

On the behavior of a ferrofluid-lubricated herringbone-grooved hybrid slot-entry bearing

The behavior of a slot-entry hybrid herringbone-grooved journal-bearing system lubricated with a ferrofluid lubricant has been numerically studied. The modified Reynolds equation of a ferrofluid bearing model based on the Stokes micro-continuum theory has been numerically solved by a finite-element method. A MATLAB code based on the Gauss–Seidel iteration scheme has been solved to numerically simulate the bearing performance. The simulated results reveal that the use of a ferrofluid lubricant provides enhanced values of lubricant film thickness, fluid film stiffness/damping coefficient, and better stability threshold speed.

Homotopy Analysis of Circular Plates Squeeze Film Bearings Lubricated with Couple Stress Fluids: Piezo-Viscous Model

The piezo-viscous effect is crucial in fluid flows under high-pressure applications such as fluid film lubrication, microfluidics, and geophysics. We have investigated the combined influences of piezo-viscous dependence and non-Newtonian couple stresses on the performance of circular plate squeeze film bearings using Stokes Micro-Continuum theory of couple stress fluids together with the exponential variation of viscosity with pressure. A closed-form solution for film pressure has been obtained using the homotopy analysis method. The numerical results for pressure and load capacity with different values of the viscosity-pressure parameter have been calculated and compared with iso-viscous couple stress and Newtonian lubricants. An enhanced pressure and load capacity are observed in the analysis. The response time for the bearing (plate approach time) has also been calculated and a significant increase is observed.

Non-Newtonian couple-stress squeeze film behaviour between oscillating anisotropic porous circular discs with sealed boundary

The thrust of this paper is to investigate theoretically the non-Newtonian couple stress squeeze film behaviour between oscillating circular discs based on V. K. Stokes micro-continuum theory. The lubricant squeezed out between parallel porous and rigid facings is supposed to be a concentrated suspension which consists of small particles dispersed in a Newtonian base fluid (solvent). The effective viscosity of the suspension is determined by using the Krieger-Dougherty viscosity model for a given volume fraction of particles in the base fluid. For low frequency and amplitude of sinusoidal squeezing where cavitation as well as turbulence are unlikely, the governing equations including the modified Reynolds equation coupled with the modified Darcy's equation are derived and solved numerically using the finite difference method and a sub-relaxed iterative procedure. The slip velocity at the porous-fluid interface is directly evaluated by means of the modified Darcy's law considering laminar and isothermal squeezing flow. For a given volume fraction, the couple stress effects on the squeeze film characteristics are analyzed through the dimensionless couple stress parameterℓ˜considering sealed and unsealed boundary of the porous disc. The obtained relevant results reveal that the use of couple stress suspending fluids as lubricants and the effect of sealing the boundary of the porous matrix improves substantially the squeeze film behaviour by increasing the squeeze film force. On the other hand, side leakage flow calculated in the sealed case remains constant in comparison to that of open end (unsealed) porous disc for all values of couple stress parameter and volume fraction of particle.

Squeeze Film Analysis of Three-layered Parallel Plate and Partial Journal Bearing Lubricated with Couple Stress Fluids for Skeletal Joint Applications

Abstract Squeeze film lubrication in surfaces approaching each other with a normal velocity, play an important role in skeletal joints. The objective of this study is to analyse the squeeze film load capacity characteristics of three-layered parallel plate and partial journal bearing lubricated with couple stress fluids for skeletal joint applications using one-dimensional analysis. The nondimensional squeeze film pressure is derived based on modified Reynolds equation. The couple stress fluids take into account of lubricant properties with additives are analyzed based on Stokes micro-continuum theory. The Brinkman model is utilized to model the flow in the porous region. In this work, bearing performance improvement under the influence of lubricants with additives are evaluated using surface (or porous) adsorbent layer on stationary plate (or bearing), couple stress fluid film core region, and surface adsorbent layer on moving plate (or journal). The nondimensional squeeze film load capacity increases with (i) increase in couple stress effects, (ii) increase in dynamic viscosity ratio of surface to core layer, and (iii) decrease in permeability of porous layer. The surface (or porous) adsorbent layer and couple stress fluid film core layer improve the squeeze film bearing characteristics for skeletal joints.

Analysis of couple‐stress effects in gas foil bearings using the V. K. Stokes micro‐continuum theory

AbstractThis paper presents an original theoretical investigation of the steady‐state and dynamic characteristics of foil journal bearings lubricated with contaminated air taking into account couple stresses because of the presence of pollutant substances. The substance can be solid particles (eg dust, ash, pollen, and smoke). In aerodynamic lubrication theory, a fluid with couple stresses so‐called couple stress or polar fluid may be considered as one characterised by the 2 physical constants μ and η, which are the absolute viscosity assumed to be independent of the particle volume fraction and the new material constant responsible for couple stress or polar property. The effects of couple stresses on the oil‐lubricated sliding bearings are usually studied by defining the parameter or its normalised form . The couple‐stress parameter l, which has the dimension of length, can be thought of as a fluid property depending on the size of the particle contained in the base fluid (solvent).To determine the aerodynamic pressure and the power loss, the governing modified Reynolds' equation and the viscous dissipation term appearing on the right‐hand side of the modified energy equation are derived by using the Vijay Kumar Stokes micro‐continuum theory.The system of partial differential equations resulting from analytical perturbation of the transient modified Reynolds' equation is solved for the steady‐state and dynamic pressures by means of the finite difference method considering both the static and dynamic deformations of the bump foil. Using the complex variable technique, the analytical perturbation process leads to 2 first‐order uncoupled partial differential equations instead of 4 coupled equations as it is usually found in the technical literature.On the other hand, the 2 Cartesian coordinates of the equilibrium position for a given applied static load are iteratively determined by solving the nonlinear equilibrium equations of the journal bearing system with the improved Newton‐Raphson method.According to the obtained results, the effects of couple stresses on the steady‐state and dynamic behaviour of foil self‐acting gas bearings are significant and cannot be overlooked. The results also show that the dynamic deformation of the bump foil should be included in calculating the dynamic performance characteristics of foil journal bearings.

Analysis of couple-stresses and piezo-viscous effects in a layered connecting-rod bearing

In this work, the combined effects of couple-stresses and piezo-viscosity on the dynamic behavior of a compression ignition engine big-end connecting-rod bearing with elastic layer are investigated using the V. K. Stokes micro-continuum theory. It is assumed that the journal (crankpin) is rigid and the big-end bearing consists of a thin compressible elastic liner fixed in an infinitely stiff housing. The governing Reynolds' equation and the viscous dissipation term appearing on the RHS of energy equation are modified using the V. K. Stokes micro-continuum theory. The non-Newtonian effect is introduced by a new material constant η, which is responsible for couple-stress property, and the piezo-viscosity effect by the pressure–viscosity coefficient α appearing in the well-known Barus' law. In the proposed model, the nonlinear transient modified Reynolds equation is discretized by the finite difference method, and the resulting system of algebraic equations is solved by means of the subrelaxed successive substitutions method to obtain the fluid-film pressure field as well as the film thickness distribution. The crankpin center trajectories for a given load diagram are determined iteratively by solving the nonlinear equilibrium equations of the journal bearing system with the improved and damped Newton–Raphson method for each time step or crankshaft rotation angle. According to the obtained results, the effects of couple-stresses and piezo-viscosity on the nonlinear dynamic behavior of dynamically loaded bearings with either stiff or compliant liners are significant and cannot be overlooked.

Derivation of Non-Newtonian Magnetic Fluid Lubricated Rough Centrosymmetric Squeeze Film Reynolds Equation and its Application

AbstractBased upon the Shliomis ferromagnetic fluid model and the Stokes microcontinuum theory incorporating with the Christensen stochastic model, a modified Reynolds equation of centrosymmetric squeeze films has been derived in this paper. The Reynolds equation includes the combined effects of non-Newtonian rheology, magnetic fluids with applied magnetic fields, rotational inertia forces, and surface roughness. To guide the use of the derived equation, the squeeze film of rotational rough-surface circular disks lubricated with non-Newtonian magnetic fluids is illustrated. According to the results obtained, the effects of rotation inertia decrease the load capacity and the squeeze film time of smooth circular disks. By the use of non-Newtonian magnetic fluids with applied magnetic fields, the rotational circular disks predict better squeeze film performances. When the influences of circumferential roughness patterns are considered, the non-Newtonian magnetic-fluid lubricated rotational rough disks with applied magnetic fields provide further higher values of the load capacity and the squeeze film time as compared to those of the smooth case.

Study of annular recess hydrostatic tilted thrust pad bearing under the influence of couple stress lubricant behaviour

This paper concerns with the theoretical investigation to predict the combined influence of couple stress lubricant and tilt between bearing pad surfaces, on the performance of externally pressurised annular recess thrust pad bearing. Stokes micro-continuum theory is used to derive modified Reynolds equation accounting for the presence of couple stress additives in lubricant. A MATLAB code has been developed to solve the governing equation using FEM technique. Influence of annular recess location and tilt between pad surfaces are investigated on the performance of hydrostatic thrust pad bearing. It has been observed that the use of couple stress lubricant significantly improves the performance of annular recessed hydrostatic thrust pad bearing. Presence of tilt between bearing surfaces proved to be beneficial in terms of dynamic characteristics of bearing system.

Combined influence of couple stress lubricant, recess geometry and method of compensation on the performance of hydrostatic circular thrust pad bearing

This study concerns with the theoretical investigation to show the influence of recess geometry and method of compensation on the performance of recessed hydrostatic circular thrust pad bearing operating with couple stress lubricant. Modified Reynolds equation for couple stress lubricant has been derived using Stokes microcontinuum theory. Stokes theory describes the behavior of fluids consisting of long chain polymer additives having sizes comparable to that of the fluid film thickness. Finite element method technique and commercially available software MATLAB 2013 have been used to solve the modified Reynolds equation and, thus, performance characteristics of bearing have been computed. The performance of capillary- and orifice-compensated hydrostatic thrust pad bearings performance have been compared vis-à-vis to different geometric shapes of recess. Each recess shape has identical ratio of bearing pad area to recess area. It has been observed that the couple stress additives have significant positive influence on the static and dynamic performance of the compensated hydrostatic thrust pad bearing provided that restrictors operates with lower values of the restrictor deign parameter[Formula: see text]. Limiting values of 10 and 5 for restrictor design parameter [Formula: see text] have been noticed for capillary and orifice restrictor; beyond these values of [Formula: see text], use of couple stress lubricant on the performance of hydrostatic thrust pad bearing become insignificant.

Influence of couple stress lubricant on the performance of textured two-lobe slot-entry hybrid journal bearing system

This paper concerns with theoretical investigation to predict the combined influence of textured surface and couple stress lubricant behavior on the performance of circular/two-lobe slot-entry hybrid journal bearing. The flow of incompressible couple stress lubricant through the clearance of a textured journal bearing system is governed by the modified Reynolds equation. The modified Reynolds equation has been derived on the basis of Stokes micro-continuum theory and is solved using a finite element method technique for computing the bearing performance characteristics parameters. The static and dynamic characteristics of circular/two-lobe slot-entry hybrid journal bearing have been computed for textured and non-textured bearings with influence of couple stress lubricant. The present study indicates that the influence of couple stress lubricant behavior is significantly more on the values of fluid film stiffness coefficients [Formula: see text], critical mass of journal [Formula: see text], and stability threshold speed margin [Formula: see text] for textured surface than that of non-textured surface in case of a slot-entry hybrid journal bearing system.

Squeeze film characteristics of conical bearings with combined effects of piezo-viscous dependency and non-Newtonian couple stresses

In this paper, the analysis of squeeze film characteristics of conical bearings with combined effects of piezo-viscous dependency and couple stress fluid is presented. On the basis of the Stokes microcontinuum theory of couple stress fluid model and Barus experimental research, a modified Reynolds equation is derived, the standard perturbation technique is used to solve the highly non-linear Reynolds equation and approximate analytical solution is obtained for the squeeze film pressure, load carrying capacity and squeeze film time. According to the results obtained, the effect of viscosity pressure dependency on the squeeze film lubrication of conical bearings with couple stress fluids is to improve the load carrying capacity significantly and lengthen the squeeze film time as compared to iso-viscous Newtonian case.

Effect of Piezoviscous Dependency and Couple Stress on Squeeze Film Lubrication Between Parallel Stepped Plates

ABSTRACTTheoretical analysis of the effect of piezoviscous dependency and couple stress on squeeze film lubrication between parallel stepped plates is presented in this article. According to the Stokes microcontinuum theory of couple stress fluids, the modified Reynolds equation is derived by considering viscosity variation along the film thickness. The standard perturbation technique is used to solve the nonlinear Reynolds equation and an approximate analytical solution for the film pressure is obtained. It is found that the effect of couple stresses and pressure-dependent viscosity variation increases the load-carrying capacity and lengthens the squeeze film time.

Nonlinear Dynamic Response of an Unbalanced Flexible Rotor Supported by Elastic Bearings Lubricated with Piezo-Viscous Polar Fluids

On the basis of the V. K. Stokes micro-continuum theory, the effects of couple stresses on the nonlinear dynamic response of the unbalanced Jeffcott’s flexible rotor supported by layered hydrodynamic journal bearings is presented in this paper. A nonlinear transient modified Reynolds’ equation is derived and discretized by the finite element method to obtain the fluid-film pressure field as well as the film thickness by means of the implicit Euler method. The nonlinear orbits of the rotor center are determined by solving the nonlinear differential equations of motion with the explicit Euler’s scheme taking into account the flexibility of rotor. According to the obtained results, the combined effects of couple stresses due to the presence of polymer additives in lubricant and the pressure dependent viscosity on the nonlinear dynamic response of the rotor-bearing system are significant and cannot be ignored or overlooked. As expected, these effects are more noticeable for polymers characterized by higher length molecular chains.

Couple stress fluids in porous circular squeeze films

The squeeze film flow characteristics of a couple stress fluid between porous circular disks is investigated. The modified Reynold’s equation is derived using Stokes microcontinuum theory and is solved analytically. The expressions for the squeeze film pressure and the squeeze film force are obtained in terms of Fourier-Bessel series. Numerical results are presented for the sinusoidal motion of the upper porous disk. Further, the equation for the gap width between the disks is obtained by considering a constant force squeezing state and is solved numerically. The effects of couple stresses and the porous facing on the squeeze film behaviour are analysed through the squeeze film pressure, force and the film thickness as a function of response time.

Couple Stress Squeeze Films with VPD in a Curved Circular Geometry

The problem of couple stress squeeze films considering viscosity pressure dependence (VPD) has been analysed in a curved circular geometry. Using Stokes microcontinuum theory and the Barus formula, the Reynolds type equation has been derived. The approximate analytical expressions for the squeeze film pressure and load carrying capacity are obtained using a perturbation technique. The numerical solutions for the squeeze film pressure and load carrying capacity are presented for the sinusoidal motion of the upper curved disk, assuming an exponential form for the curvature. The effects of curvature, the non-Newtonian couple stresses, and VPD and their combined effects are investigated through the squeeze film pressure and the load carrying capacity of the squeeze film.

MAGNETO-HYDRODYNAMIC NON-NEWTONIAN CURVED CIRCULAR SQUEEZE FILMS

The squeeze film performances between curved circular plates lubricated with an electrically conducting non-Newtonian fluid in the presence of external magnetic fields are investigated in this paper. Based upon the magneto-hydrodynamic flow theory together with the Stokes microcontinuum theory, the magneto-hydrodynamic non-Newtonian Reynolds equation is derived and applied to predict the curved circular squeeze film behaviors. Comparing with the hydrodynamic Newtonian case, the squeeze film characteristics for curved circular plates are improved by the use of an electrically conducting non-Newtonian fluid in the presence of external magnetic fields. Numerical values of the load capacity and the approaching time are provided in Tables for engineering applications.

Effect of surface roughness on the hydrodynamic lubrication of porous inclined slider bearing considering slip velocity and squeeze velocity with couple stress fluids

This paper presents the theoretical study of the effect of surface roughness on the hydrodynamic lubrication of porous inclined slider bearing considering slip velocity and squeeze velocity. A generalized form of surface roughness characterized by a stochastic random variable with non-zero mean, variance and skewness is assumed to define the bearing surface topography.The modified Reynolds equation governing fluid film pressure is derived on the basis of stokes microcontinuum theory of couple stress fluids by using slip boundary conditions at the fluid-porous interface. The stochastic form of Reynolds equation is solved and closed form expressions are obtained for pressure, load capacity, friction on the slider and coefficient of friction to examine possible effects on the system. The numerical computations results show that the negatively skewed surface roughness pattern decreases the load capacity and increases the coefficient of friction by considering the slip velocity. For the other case adverse effects were found for the positively skewed surface roughness patter which is very well in conformity with the existing results in the literature. Computed values of load capacity, frictional force, and coefficient of friction are displayed in graphical form. Keywords : couple stress fluid, surface roughness, porous inclined slider bearing, slip velocity.

Behaviour of Couple Stress Fluids in Porous Annular Squeeze Films

The laminar squeeze flow of an incompressible couple stress fluid between porous annular disks is studied using hydrodynamic lubrication theory. The modified Reynolds’ equation is derived using Stokes microcontinuum theory and is solved analytically. Analytical expressions for the squeeze film pressure and the load carrying capacity are obtained in terms of Fourier-Bessel series. Numerical results are obtained for the sinusoidal motion of the upper disk. The effect of couple stresses and that of porous facing on the squeeze film behaviour are analysed through the squeeze film pressure and the load carrying capacity. Further, the equation for the gap width between the disks is obtained from the inverse problem.

Analysis of Journal Bearing with Double-Layer Porous Lubricant Film: Influence of Surface Porous Layer Configuration

This article presents an analysis of a long journal bearing with a double-layer porous lubricant film using couple stress and Newtonian fluids. The porous layer with infinite permeability analyzed in this study simulates the surface layer. The Brinkman model was utilized to model the flow in the porous region. The effects of couple stresses were analyzed based on Stokes microcontinuum theory. A double-layer porous lubricant film configuration, with a low-permeability porous layer on top of a high-permeability bearing adherent porous layer, improved the journal bearing performance characteristics. A surface porous layered lubricant film configuration increased the load-carrying capacity and reduced the coefficient of friction in a journal bearing.