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In order to investigate the gas squeeze film characteristics with mixed-modal disk excitation, an actuator with piezo-electric components is designed. Experiments show that different modal shapes of the disk are excited along with the changes of the actuator excitation frequency. The amplitude of the modal shape can reach the same order of magnitude as the squeeze film thickness, so the modal effects on the squeeze film characteristics cannot be ignored. In this paper, the simulating and fitting of the exciting disk modal shapes were finished by ANSYS and MATLAB. The actual film thickness equation was amended by the modal shape fitting curve. The gas film characteristic can be obtained by solving the Reynolds equation which is coupled with film thickness and motion equation. In this study, finite element simulation and differential numerical calculation results provided a good guidance to the piezoelectric actuator design.

The asymptotic behavior of a Stokes flow with Fourier boundary condition on one part on the boundary and Tresca free boundary friction condition on the other, when one dimension of the fluid domain tends to zero is studied. The strong convergence of the velocity is proved, a specific Reynolds equation is obtained, and the uniqueness of the limit velocity and pressure distributions is established.

We provide a rigorous derivation of the compressible Reynolds system as a singular limit of the compressible (barotropic) Navier–Stokes system on a thin domain. In particular, the existence of solutions to the Navier–Stokes system with non-homogeneous boundary conditions is shown that may be of independent interest. Our approach is based on new a priori bounds available for the pressure law of hard sphere type. Finally, uniqueness for the limit problem is established in the one-dimensional case.

We present a space–time (ST) computational flow analysis method with built-in Reynolds-equation limit. The method enables solution of lubrication fluid dynamics problems with a computational cost comparable to that of the Reynolds-equation model for the comparable solution quality, but with the computational flexibility to go beyond the limitations of the Reynolds-equation model. The key components of the method are the ST Variational Multiscale (ST-VMS) method, ST Isogeometric Analysis (ST-IGA), and the ST Slip Interface (ST-SI) method. The VMS feature of the ST-VMS serves as a numerical stabilization method with a good track record, the moving-mesh feature of the ST framework enables high-resolution flow computation near the moving fluid–solid interfaces, and the higher-order accuracy of the ST framework strengthens both features. The ST-IGA enables more accurate representation of the solid-surface geometries and increased accuracy in the flow solution in general. With the ST-IGA, even with just one quadratic NURBS element across the gap of the lubrication fluid dynamics problem, we reach a solution quality comparable to that of the Reynolds-equation model. The ST-SI enables moving-mesh computation when the spinning solid surface is noncircular. The mesh covering the solid surface spins with it, retaining the high-resolution representation of the flow near the surface, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. We present detailed 2D test computations to show how the method performs compared to the Reynolds-equation model, compared to finite element discretization, at different circumferential and normal mesh refinement levels, when there is an SI in the mesh, and when the no-slip boundary conditions are weakly-enforced.

Micro/nanoscale lubrication must take into account the fractal profile of the shaft and bearing surfaces. A new fractal rheological model is proposed to describe the properties of the non-Newtonian fluid, and a fractal variational principle is established by the semi-inverse method, and finally the Lagrange multipliers can be found in the obtained variational formulation. This work provides a new fractal approach to nano/micro lubrication.

In this paper, problem of squeeze film damping in dual axis torsion microactuators is modeled and closed form expressions are provided for damping torques around tilting axes of the actuator. The Reynolds equation which governs the pressure distribution underneath the actuator is linearized. The resulting equation is then solved analytically. The obtained pressure distribution is used to calculate the normalized damping torques around tilting axes of the actuator. Dependence of the damping torques on the design parameters of the dual axis torsion actuator is studied. It is observed that with proper selection of the actuator's aspect ratio, damping torque along one of the tilting directions can be eliminated. It is shown that when the tilting angles of the actuator are small, squeeze film damping would act like a linear viscous damping. The results of this paper can be used for accurate dynamical modeling and control of torsion dual axis microactuators.

Recently, wavelet theory has become a well recognized promising tool in science and engineering field; especially, wavelets are successfully used in fast algorithms for easy execution. In this paper, we developed wavelet lifting scheme using orthogonal and biorthogonal wavelets for the numerical solution of dynamic Reynolds equation for micropolar fluid lubrication. The numerical results gained through proposed scheme are compared with the exact solution to expose the accuracy with speed of convergence in lesser computational time as compared with the existing methods. The examples are given to demonstrate the applicability and attractiveness of proposed method.

The present paper is aimed to formulate a numerical model for calculating the time-dependent velocity field and the space and time distribution of sand concentration inside the bottom boundary layer in the surf zone. The numerical results are compared with recent laboratory data. The 2DV (two-dimensional vertical) numerical module for the velocity field is based on the Reynolds averaged form of the Navier-Stokes equations while the sand transport module uses the turbulent convection-diffusion equation. The equi-phase mean values for the horizontal and vertical velocities within the bottom boundary layer (the flow field) and the equi-phase mean values for the sand concentration are obtained. A transformation of the coordinate system is performed in order to enable the easy implementation of the numerical modelling. Details of the numerical modeling together with discussions on the boundary conditions are also presented. The comparison of numerical and laboratory results reveals relatively good agreement within certain limitations of the present model. The laboratory data, which are used to compare the hydrodynamic numerical model, have been obtained in the surf zone while the ones for verifying the sand concentration numerical model are obtained as a result of pure oscillatory movement.

Practical two-dimensional numerical models investigating the internal characteristics of plunging breakers on a uniform slope are proposed under the assumption that the energy dissipation associated with the turbulence and vortex motions after wave breaking can be evaluated directly by the Reynolds stress terms. In these models, the Reynolds stress terms are expressed using Prandtl's mixing length model. The energy dissipation is evaluated using internal kinematic quantities (vorticity and shear distortion rate) because laboratory experiments have revealed that the energy dissipation of plunging breakers demonstrate a strong correlation to internal kinematic quantities. A new concept termed "vortex radius" is defined as the circulation divided by the local horizontal velocity, and the representative scale of length is given by this vortex radius. Even though the representative scale of length is evaluated using the zero-order turbulence model, the scale changes spatially in accordance with flow conditions. The qualitative and quantitative accuracies of the models are examined by comparing turbulence properties with experimental results for plunging breakers on a uniform slope. Furthermore, the internal characteristics of plunging breakers are numerically investigated.

We show that a certain stochastic perturbation of the flow of perfect incompressible fluid under some special external force on the flat n-dimensional torus yields a solution of Navier–Stokes equation without external force in the tangent space at unit of volume preserving diffeomorphism group. If that external force is absent, the equation turns into the one of Reynolds type. For the flow of diffuse matter this construction yields the Burgers equation.

Asymptotic analysis of an incompressible Stokes fluid in a dynamic regime in a three-dimensional thin domain ${\mathrm{\Omega }}^{𝜀}$ with mixed boundary conditions and Tresca friction law is studied in this paper. The problem statement and variational formulation of the problem are reformulated in a fixed domain. In which case, the estimates on velocity and pressure are proved. These estimates will be useful in order to give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.

Synovial fluid functions as a biological lubricant and lubricates articular cartilage to reduce friction and wear. Lubricin and hyaluronic acid are the primary components of synovial fluid responsible for its lubricating properties. The loss of properties in synovial fluid and articular cartilage due to aging and activities can restrict mobility in synovial joints, resulting in the degradation of articular cartilage and ultimately leading to pathological synovial joints, which is a major cause of disability. Thus, research on synovial joints remains crucial, and despite previous investigations on synovial joint lubrication, there are several issues related to squeeze film lubrication that require further attention. The Newtonian model of squeeze film lubrication in synovial joints needs to be extensively studied. In this study, lubrication and other related properties of synovial fluid are studied theoretically. In this paper, we have studied the flow of synovial fluid in the diseased synovial joint. Further, we have studied the effect of the viscosity of the synovial fluid, the permeability of articular cartilage, thickness of articular cartilage, and fluid film thickness on the characteristic of the squeeze film formed between the articular cartilages of the diseased human knee joint. The flow of synovial fluid is modeled by considering it as a viscous, incompressible and Newtonian fluid. We have derived the modified Reynolds equation using the principle of hydrodynamic lubrication and continuum mechanics theory and solved it by applying the suitable boundary conditions according to the physical considerations. Subsequently, we obtained the expression for pressure distribution in fluid film, load-bearing capacity, and squeeze time and have done the theoretical analysis on these properties for different parameters. Pressure increases with squeeze velocity and viscosity of the synovial fluid and decreases with permeability and fluid film thickness. Human knee joint becomes diseased due to excessive pressure, and the mobility of the knee joint decreases. The load capacity increases with viscosity and squeeze velocity and decreases with permeability resulting in the reduction of the load-carrying capacity of the knee joint in diseased conditions. Moreover, the squeeze time also increased in the diseased state of the joint.

Based on the analytical solution of Reynolds equation for the plane slider of finite width, the effects of the length ratio, ratio of length to width and rotation angle on the average pressure are analyzed and optimized. The results show that there exists optimal ratio of length to width to maximize the average pressure while the area or width of the plane slider is constant. While the ratio of length to width is greater than 2, there exists optimal length ratio to maximize the average pressure. The hydrodynamic pressure obtains the maximum while the directions of the height gradient and movement of surface texture are the same.