Numerical solution of couplestress finite Reynolds equation for squeeze film lubrication of partial porous journal bearings

Abstract

In this paper, the general Reynolds equation of finite porous journal bearing lubricated with couplestress fluid is solved numerically for the assessment of dynamic characteristics of the bearings. The Reynolds type equation governing the steady performance is obtained and solved numerically by finite difference technique. From the numerical results, it is observed that the effect of couple stresses is to increase the load carrying capacity and to lengthen the squeeze film time as compared to the corresponding solid case. The effect of permeability is to reduce the load capacity and to decrease the squeeze film time as compared to the solid case.     

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

Theoretical 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.     

Effect of surface roughness on the couple-stress squeeze film between a sphere and a flat plate

In this paper, a theoretical study of the effect of surface roughness on the hydrodynamic lubrication of couple-stress squeeze film between a sphere and a flat plate is presented on the basis of Christensen's stochastic theory for hydrodynamic lubrication of rough surfaces. The modified Reynolds equation accounting for the couple stresses and the surface roughness is mathematically derived. The modified Reynolds equation is solved for the fluid film pressure and the bearing characteristics, such as the load carrying capacity and the time–height relationship, are obtained. It is found that the surface roughness considerably influences the squeeze film characteristics. The load carrying capacity and squeeze film time are found to increase for an azimuthal roughness pattern as compared to the corresponding smooth case, whereas the reverse trend is observed for a radial roughness pattern.     

Lubrication of a narrow porous journal bearing with a couple stress fluid

This paper describes a theoretical investigation of the rheological effects of couple stress fluids on the performance of narrow porous journal bearings. A most general modified Reynolds equation is derived for narrow porous journal bearings using the Stokes constitutive equations for couple stress fluids. The fluid in the film region and in the porous region has been modelled as a couple stress fluid. The analysis takes into account velocity slip at the porous interface using the Beavers‐Joseph criterion. A closed‐form expression for field pressure is obtained for narrow journal bearings. Eigen‐type expressions for field variations are obtained. The dimensionless load‐carrying capacity, attitude angle, and coefficient of friction are presented for different operating parameters. It is observed that narrow porous journal bearings with couple stress fluids as lubricant show a significant increase in load‐carrying capacity with reduced coefficient of friction as compared to the Newtonian case. The present study predicts the effects of the percolation of polar additives (microstructures) into the porous matrix on the bearing performance.     

A Comparative Study of Static and Dynamic Characteristics of Parabolic and Plane Inclined Slider Bearings Lubricated with MHD Couple Stress Fluids

In this article the couple stress effects on the static and dynamic characteristics of parabolic and plane inclined slider bearings in the presence of an applied magnetic field with squeeze action is analyzed theoretically. The modified magnetohydrodynamic (MHD) couple stress Reynolds equation is derived. The closed-form solution for the film pressure of parabolic and inclined plane slider bearings is obtained and used to study the MHD couple stress static and dynamic characteristics of these bearings. The results predict a higher steady load-carrying capacity, dynamic stiffness, and damping coefficient for the bearings lubricated with MHD couple stress fluid than the corresponding Newtonian case. It is observed that the parabolic-shaped slider bearings provide greater steady load-carrying capacity, dynamic stiffness, and damping coefficients compared to the plane inclined slider bearings.     

Squeeze film characteristics of finite journal bearings: couple stress fluid model

A theoretical study of squeeze film behaviour for a finite journal bearing lubricated with couple stress fluids is presented. On the basis of the microcontinuum theory, the modified Reynolds equation is obtained by using the Stokes equations of motion to account for the couple stress effects due to the lubricant blended with various additives. With the Conjugate Gradient Method of iteration the built-up pressure is calculated, and then applied to predict the squeeze film characteristics of the system. According to the results evaluated, the rheological influence of couple stress fluids is physically apparent. Compared with the case of a Newtonian lubricant, the couple stress effects increase the load-carrying capacity significantly and lengthen the response time of the squeeze film behaviour. On the whole, the presence of couple stresses improves the characteristics of finite journal bearings operating under pure squeeze film motion. The rheological effects of couple stress fluids agree with previous works.     

Effect of surface roughness on magneto‐hydrodynamic couple‐stress squeeze film lubrication between circular stepped plates

In this paper, a theoretical analysis of the problem of magneto‐hydrodynamic couple‐stress squeeze film lubrication between rough circular stepped plates is presented. The modified averaged Reynolds equation is derived for the two types of one‐dimensional roughness structures, namely the radial roughness pattern and the azimuthal roughness pattern. The closed‐form expressions are obtained for the mean squeeze film pressure, load‐carrying capacity and squeeze film time. The results are presented for different operating parameters. It is observed that the effect of azimuthal (radial) roughness pattern on the bearing surface is to increase (decrease) the mean load‐carrying capacity and squeeze film time. The applied magnetic field increases the load‐carrying capacity and lengthens the squeezing time. Copyright © 2012 John Wiley & Sons, Ltd.     

Squeeze film characteristics of long partial journal bearings lubricated with couple stress fluids

On the basis of microcontinuum theory, a theoretical analysis of hydrodynamic squeeze film behaviour for long partial journal bearings lubricated by fluids with couple stresses is presented. To take into account the couple stress effects due to the lubricant containing additives or suspended particles, the modified Reynolds equation governing the film pressure is derived by using the Stokes constitutive equations. Various bearing characteristics are then calculated. According to the results obtained, the influence of couple stress effects on the performance of the system is physically apparent and not negligible. Compared with the Newtonian lubricant case, the couple stress effects provide an enhancement in the load-carrying capacity and lengthen the response time of the squeeze film action. On the whole, the presence of couple stresses signifies an improvement in the squeeze film characteristics of the system.     

Combined effects of surface roughness and couple stresses on static and dynamic behaviour of squeeze film lubrication of porous journal bearings

Abstract

The combined effects of surface roughness and the couple stresses on the static and dynamic characteristics of squeeze film lubrication in porous journal bearings with no journal rotation are theoretically studied. The Stokes couple stress fluid model is considered to model the lubricants with additives. The surface roughness on the porous journal bearing is mathematically modelled by a random variable with non-zero mean, variance and skewness. The generalised stochastic Reynolds type equation is derived for the problem under consideration. The applied load is considered as a sinusoidal function of time to simulate the bearings operating under cyclic loads. The closed form expressions for the bearing characteristics are obtained for the short porous journal bearings. It is observed that the negatively skewed surface roughness and couple stresses improve the performance of the porous journal bearings as compared to the smooth journal bearings with Newtonian lubricants. However, the presence of positively skewed surface roughness on the bearing surface affects its performance.     

Micropolar fluid squeeze film lubrication of finite porous journal bearing

In this paper, the effect of micropolar fluid on the static and dynamic characteristics of squeeze film lubrication in finite porous journal bearings is studied. The finite modified Reynolds equation is solved numerically using the finite difference technique and the squeeze film characteristics are obtained. According to the results obtained, the micropolar fluid effect significantly increases the squeeze film pressure and the load-carrying capacity as compared to the corresponding Newtonian case. Under cyclic load, the effect of micropolar fluid is to reduce the velocity of the journal centre. Effect of porous matrix is to reduce the film pressure, load-carrying capacity and to increase the journal centre velocity.     

Hydrodynamic analysis of rough curved pivoted porous slider bearings with couple stress fluid

Abstract

The purpose of this paper is to study the effect of surface roughness on the performance of curved pivoted porous slider bearings lubricated with couple stress fluid. The modified B–J slip boundary condition is utilised at the porous/fluid film interface to derive the Reynolds type equation for the problem under consideration. To mathematically model the surface roughness due to non-uniform rubbing of bearing surfaces, a stochastic random variable with non-zero mean, variance and skewness is considered. The closed form solution is obtained for the averaged Reynolds equation, and the compact expressions for the mean fluid film pressure mean load carrying capacity, frictional force and the centre of pressure are obtained. The numerical computations of the results show the improved performance due to the couplestresses and the presence of negatively skewed surface roughness. However, the presence of porous facing and positively skewed surface roughness affects the performance of the pivoted porous slider bearing.     

Lubricant additives effects on the hydrodynamic lubrication of misaligned conical–cylindrical bearings

Conical–cylindrical bearings are used in electrohydraulic servo systems to improve the control accuracy, eliminate the static friction and increase the normal load‐carrying capacity. A non‐Newtonian rheological model to investigate theoretically the effects of lubricant additives on the performance of misaligned conical–cylindrical bearings is proposed in this study. In this model, the non‐Newtonian behaviour resulting from blending the lubricant with polymer additives is simulated by Stokes couple stress fluid model. The formed boundary layer at the bearing surface is described through the use of a hypothetical porous medium layer that adheres to the bearing surface. The Brinkman‐extended Darcy equations are utilised to model the flow in the porous region. A stress jump boundary condition is applied at the porous media/fluid film interface. The misalignment of the cylinder rod is also considered. A modified form of the Reynolds equation is derived and solved numerically using a finite difference scheme. The effects of bearing geometry and non‐Newtonian behaviour of the lubricant on the steady‐state performance characteristics such as pressure distribution, load‐carrying capacity and coefficient of friction are presented and discussed. The results showed that lubricant additives significantly increase the load‐carrying capacity and reduce the coefficient of friction as compared to the Newtonian lubricants. Furthermore, the misalignment of the piston rod has significant effects on the performance of conical–cylindrical bearings. Copyright © 2007 John Wiley & Sons, Ltd.     

Squeeze film characteristics between a sphere and a rough porous flat plate with micropolar fluids

The effects of surface roughness on the squeeze film characteristics between a sphere and flat plate covered with a thin porous layer are investigated in this paper. The sphere and the plate are separated with a non‐Newtonian lubricant of a micropolar fluid. The well‐established Christensen stochastic theory of hydrodynamic lubrication of rough surfaces is used to incorporate the effects of surface roughness into the Reynolds equation. The film pressure distribution is solved and other squeeze film characteristics, such as the load‐carrying capacity,and time–height relationship, are obtained. The results indicate that lubrication by a micropolar fluid will increase the load‐carrying capacity and lengthen the squeeze film time, regardless to the surface rough and porosity of the flat plate. It is also found that excessive permeability of the porous layer causes a significant drop in the squeeze film characteristics and minimises the effect of surface roughness. For the case of limited or no permeability, the azimuthal roughness is found to increase the load‐carrying capacity and squeeze time, whereas the reverse results are obtained for the case of radial roughness.     

Effect of roughness on hydromagnetic squeeze films between porous rectangular plates

The theoretical investigations made in this paper are to study the combined effects of unidirectional surface roughness and magnetic effect on the performance characteristics of a porous squeeze film lubrication between two rectangular plates. The stochastic Reynolds equation accounting for the magnetic effect and randomized surface roughness structure is mathematically derived. The expressions for dimensionless pressure, load carrying capacity and squeeze film time are obtained. Results are computed numerically and it is observed that a roughness effect enhances pressure, load carrying capacity and squeeze film time.     

A study of optimum load-bearing capacity for slider bearings lubricated with couple stress fluids in magnetic field

A theoretical study of slider bearings in general form is presented, considering the lubricant to be an isothermal, incompressible electrically conducting couple stress fluid in the presence of a uniform magnetic field. An expression for a modified Reynolds equation is derived in order to obtain pressure gradient in terms of inlet–outlet (IO) film height ratio (simply IO film ratio) of slider bearings. As a special case, a study of the IO film ratio of four particular types of slider bearings is analysed. For the study of optimum load capacity, three-dimensional geometry has been assumed in order to consider side flow. Values of maximum load capacity for various values of couple stress and magnetic parameters, and the corresponding IO film ratio of the four bearings are obtained numerically, and a comparative study of these values has been presented. A comparative study of optimum load-carrying capacity for finite and infinite slider bearings has also been made. It is observed that both the values of maximum load capacity and the corresponding IO film ratio depend on couple stress and magnetic parameters and the shape of bearings conjointly. The present results are also compared with the similar available data in Newtonian cases.     

The influence of couple stresses in squeeze films

A theoretical study of non-Newtonian flow effects in a squeeze film configuration is carried out with special reference to synovial joints. The material model taken is that of the Stokes' couple stress fluid. It is found that the bearings with couple stresss fluid as lubricant provide significant load supporting capacities which result in longer bearing life. The squeeze film time is found to be considerably longer for couple stress fluid than in the case of Newtonian fluid of the same viscosity. These are the most desirable advantages which render the model close to the natural characteristics of synovial joints.     

Surface roughness effects on curved pivoted slider bearings with couple stress fluid

In this paper the effect of surface roughness on the performance of curved pivoted slider bearings is studied. A more general type of surface roughness is mathematically modelled by a stochastic random variable with nonzero mean, variance and skewness. The averaged modified Reynolds type equation is derived on the basis of Stokes microcontinuum theory for couple stress fluids. The closed‐form expressions for the mean pressure, load‐carrying capacity, frictional force and the centre of pressure are obtained. Numerical computations show that the performance of the slider bearing is improved by the use of lubricants with additives (couple stress fluid) as compared to Newtonian lubricants. Further, it is observed that the negatively skewed surface roughness increases the load‐carrying capacity and frictional force and reduces the coefficient of friction, whereas the positively skewed surface roughness on the bearing surface adversely affects the performance of the pivoted slider bearings. Copyright © 2006 John Wiley & Sons, Ltd.     

Misalignment effects on steady‐state and dynamic behaviour of compliant journal bearings lubricated with couple stress fluids

This work concerns the steady‐state and dynamic analysis of misaligned compliant journal bearings considering the effects of couple stresses arising from the lubricant blended with polymer additives. Based on the Stokes micro‐continuum theory, a modified form of the Reynolds equation is derived. The displacement field at the fluid film–bearing liner interface due to pressure forces is determined using the elastic thin liner model. The effects of the misalignment and the couple stress parameters on static and dynamic performances such as pressure distribution, load‐carrying capacity, power loss, side leakage flow, misalignment moment, critical mass and whirl frequency are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Modified Reynolds Equation for Different Types of Finite Plates with the Combined Effect of MHD and Couple Stresses

We make an effort to analyze the behavior of squeeze film characteristics of different finite plates with couple stress fluid in the presence of a transverse magnetic field. On the basis of the Stokes couple stress fluid model and hydromagnetic flow model, a modified Reynold's equation is derived, which is solved by using appropriate boundary conditions to obtain squeeze film pressure, load-carrying capacity, and squeeze film time. The graphical representation of the results suggests that the different bearing systems register an enhanced performance with couple stresses compared to that of a bearing system working with a conventional lubricant in the presence of a transverse magnetic field. It is observed that the effect of applied magnetic field on the squeeze film lubrication between different finite plates with conducting couple stress fluids is to increase the load-carrying capacity significantly and to delay the time of approach compared to the corresponding nonconducting Newtonian case. It is seen that for all of the finite plates of different shapes, the circular shape gives the maximum load and time.     

Micropolar fluid squeeze film lubrication between rough anisotropic poroelastic rectangular plates: special reference to synovial joint lubrication

Abstract

The effect of anisotropic permeability on micropolar squeeze film lubrication between poroelastic rectangular plates is studied. The non-Newtonian synovial fluid is modelled by Eringen’s micropolar fluid, and the poroelastic nature of cartilage is taken in to account. The stochastic modified Reynolds equation, which incorporates the elastic as well as randomised surface roughness structure of cartilage with micropolar fluid as lubricant, is derived. Modified equations for the mean fluid film pressure, mean load carrying capacity and squeeze film time are obtained using the Christensen’s stochastic theory for the study of roughness effects. The effects of surface roughness, micropolar fluid and anisotropic permeability on the squeeze film characteristics of synovial joint are discussed. It is found that the surface roughness effects are more pronounced for micropolar fluids as compared to the Newtonian fluids, and the anisotropic nature of permeability of cartilage off-squares the plate size for optimum performance.