Non-Newtonian couple stress poroelastic squeeze film

doi:10.1016/j.triboint.2013.03.006

Tribology International

Volume 64, August 2013, Pages 116-127

https://doi.org/10.1016/j.triboint.2013.03.006 Get rights and content

Highlights

•
Numerical analysis of couple stress poroelastic squeeze film between two discs.
•
Small elastic deformation of the porous disc calculated by the thin elastic layer model.
•
Film thickness and friction coefficient increase with increasing the couple stress.
•
These properties are reduced by permeability and elastic deformation.

Abstract

The aim of this paper is to develop a new model of the interaction of a fluid film with a porous medium. The model takes into account the fluid inertia in both the lubricant and the porous matrix. Non-Newtonian behavior of the fluid, viscous effects in the porous matrix, and poroelasticity of the matrix are also considered. The main concerns are modeling and simulation of the squeeze film lubrication between two discs when one has a porous facing. The fluid flow is described using a reduced version of the Navier–Stokes equations in the fluid film, and the Darcy–Brinkman–Forchheimer generalized model in the porous matrix.

The present study focuses on the combined effects of the non-Newtonian fluid lubricant and porous matrix deformation. The non-Newtonian behavior of the lubricant is described by the so-called couple stress model. The porous interface deformation is obtained using the thin elastic layer approach. The partial differential equations established in this study are discretized by finite differences. The resulting algebraic equations are solved using the Gauss–Seidel relaxation method.

The numerical results of the present simulations show that all these effects have a significant influence on the porous squeeze film performance.

Introduction

The porous squeeze film hydrodynamic lubrication problem describes the action of a rigid body in normal motion on a confined lubricating film. This basic configuration has attracted the attention of many researchers because it is often confronted in industry and biomechanics [1], [2], [3].

Modeling of lubrication flow depends on the rheological equation being used. The assumption of a Newtonian fluid may no be longer valid in the case of lubricants with polymer additives to improve viscosity-temperature control in various operating conditions [4], [5]. Likewise, in human articulations synovial fluid contains long chain hyaluronic acid as natural additives [6], [7], [8]. The resulting nonlinear behavior significantly influences the characteristics of the fluid, thus non-Newtonian effects must be taken into account. In addition to this non-Newtonian effect, the elastic deformation of a contiguous porous matrix [9] may be significant. This typical case could correspond, for example, to the case of cartilage that deforms under loading of articular contact [10], [11]. Several prior studies of this problem have been conducted. They are based on a modified Reynolds equation for non-Newtonian fluids, and a mixture theory which models the cartilage as a biphasic continuum, without taking into account the fluid inertial and viscous effects [2], [12], [13].

Accounting for viscous shear effects on the squeeze film characteristics for porous circular discs, Lin [14] introduced the Darcy–Brinkman model simplified due to the thin elastic layer assumption. He found that these viscous effects on the results obtained are significant and thus cannot be neglected.

Recently Nabhani et al. [15] used the full Darcy–Brinkman model to describe the flow within a fixed porous disc in a squeezing situation of a Newtonian fluid film by another mobile disc. These authors have extended this model to account for inertial effects of the fluid using the Darcy–Brinkman–Forchheimer model in the porous disc, and reduced Navier–Stokes equations (RNSP) in the film [16].

In this paper, we write the RNSP equations with an additional term taking into account non-Newtonian effects through the couple stress fluids model. We also propose an approach to investigate the elastic deformation effect of the porous interface on contact performance. This approach is based on the thin elastic layer model. Use of this model is a simple preliminary approximation allowing for consideration of the porous interface deformation.

The flow model in the porous medium is based on the Darcy–Brinkman–Forchheimer equations. The velocity, and consequently the position, of the moving disc are two unknowns of the problem; but it is necessary to add an additional equation to the system of equations to be solved.

The numerical solution of this coupled fluid film—porous medium problem is not achieved easily since it involves solving a highly nonlinear problem. A particular numerical approach is developed to avoid the convergence difficulties encountered during the solution process. These difficulties are partly related to the coupled nonlinear equations. The approach is based on a sequential coupling for separately solving the various equations of the problem, and also establishes an iterative process between the consecutive solutions. Some results of this numerical simulation are presented and discussed. These results show the non-Newtonian fluid effects and the influence of elastic deformation of the porous interface on the performance of lubricated contact.

Section snippets

Governing equations in the fluid film for couple stress fluids

Consider two parallel circular discs of the same radius R, one of which has a poroelastic face, separated by a fluid film lubricant (see Fig. 1). Both discs are considered immersed in the lubricant.

The poroelastic disc is fixed in reference frame (Or, Oz), whose the origin point O is positioned in the middle of the lower impermeable surface. The rigid disc, supporting a constant load W, is animated by a squeezing movement of instantaneous velocity dg/dt along the axis (Oz). The thickness of the

Governing equations in the porous medium

The porous medium is considered homogeneous, isotropic, saturated by a Newtonian fluid having the same dynamic viscosity as in the fluid film. The porous layer deforms elastically under hydrodynamic pressure. Suspended particles in the fluid film are assumed sufficiently large to not penetrate into the porous medium. This is the typical case of articular cartilage, which filters large molecules of hyaluronic acid from the synovial fluid [18], [19], [20].

Equation of motion of the upper disc

External forces acting on the upper disc include the resultant force due to the hydrodynamic pressure of the fluid film, the inertia force of the disc in movement, and its weight. In the system of Fig. 1, the equation of motion along the axial component is written: $m \frac{d^{2} g}{d t^{2}} (t) = W (t) - F_{0},$ where W(t) is the load on the disc due to the fluid film forces. This force on the upper disc is a function of time, and is given by integrating the pressure field of fluid film on the contact surface: $W (t) = 2 π \int_{0}^{R} p r d r,$

Equation of fluid film thickness

The film thickness, taking into account the elastic deformation of the porous interface under the effect of hydrodynamic pressure, is written as follows (see Fig. 1): $h (r, t) = g (r) - H + δ (r, t)$

Equation of the porous interface deformation: Elastic thin layer model

Elastic deformations are often calculated throughout the porous matrix. This requires a structural analysis by the finite element method, generally expensive in computation time. A possible alternative is the use of elastic thin layer model based on the assumption of a thin porous layer of thickness H compared to the contact radius R. This simplification, introduced by Winkler [21], was justified by Higginson [22], Medley [23], and by Dowson and Jin [24]. The deformation of the porous interface

Initial conditions

In the fluid film, we choose the solution to the classical Reynolds and continuity equations as the initial conditions for velocity and pressure. For a Newtonian fluid film located between two rigid discs with the same initial thickness h₀, we have: $u (r, z, t = 0) = \frac{3}{{h_{0}}^{3}} r (z - H) (z - g) \frac{d g}{d t} (0),$ $w (r, z, t = 0) = - \frac{1}{{h_{0}}^{3}} {(z - H)}^{2} (2 (z - H) - 3 h_{0}) \frac{d g}{d t} (0)$ $p (r, z, t = 0) = \frac{3 μ}{{h_{0}}^{3}} (r^{2} - R^{2}) \frac{d g}{d t} (0)$

In the porous medium, we assume that at the initial time there is no flow: $u^{(⁎)} (r, z, t = 0) = w^{(⁎)} (r, z, t = 0) = 0$

In addition, we assume that the pressure

Pressure boundary conditions

On the axis of symmetry, r=0, we have: $\frac{d p}{d r} = \frac{\partial p^{(⁎)}}{\partial r} = 0$

On the lubricated contact and porous disc boundaries, r=R, the pressure is taken equal to the ambient pressure: $p = p^{(⁎)} = 0$

On the porous interface, z=g−h, the pressure is assumed to be continuous: $p = p^{(⁎)}$

The condition of impermeability on the lower wall of the porous disc, z=0, is written: $\frac{\partial p^{(⁎)}}{\partial z} = 0$

Velocity boundary conditions

The fluid particles satisfy the following conditions at the surfaces of solid impermeable walls:

−
On the upper disc, z=g(t): $u = 0, w = \frac{d g}{d t}$
−
On the lower wall, z=0:

Dimensionless equations, initial and boundary conditions

We consider R a characteristic length along (O, r), h₀ along (O, z), P₀ a reference pressure chosen equal to F₀/π and R², a reference velocity V₀ according to (O, z) for which the expression must be determined, and h₀/V₀ a reference time. Dimensionless variables are: $\bar{r} = \frac{r}{R}, \bar{u} = \frac{h_{0}}{R V_{0}} u, \bar{w} = \frac{w}{V_{0}}, \bar{p} = \frac{p}{P_{0}}, \bar{u^{(⁎)}} = \frac{h_{0}}{R V_{0}} u^{(⁎)}, \bar{w^{(⁎)}} = \frac{w^{(⁎)}}{V_{0}}, \bar{p^{(⁎)}} = \frac{p^{(⁎)}}{P_{0}}, \bar{t} = \frac{V_{0}}{h_{0}} t, \bar{h} = \frac{h}{h_{0}}$

The deformable porous interface causes a complication in the boundary conditions when the finite difference method is used. A change of variables

Finite difference discretization and the solution technique

The RNSP equations in the film (9.3), (9.4), those of generalized Darcy–Brinkman–Forchheimer (9.6), (9.7) and Poisson (9.8) in the porous disc are coupled at the porous interface by the continuity of pressure, velocity, and of normal and tangential stresses. The solution of this problem of fluid—porous medium interaction requires numerical solution of these equations accounting for initial (9.14), (9.15), (9.16), (9.17), (9.18), (9.19), (9.20) and boundary conditions (9.21), (9.22), (9.23),

Solution algorithm

Initially, velocity initial conditions (9.14), (9.15), (9.17) and pressure (9.16), (9.19) are specified. At each time step, the squeezing velocity and position of the upper disc are calculated from its equation of motion written in the previous time step by an explicit time scheme (9.13). The global iterative process begins by fixing a pressure field within the fluid film. A good estimate is to consider that of the previous time step. This pressure field permits the calculation of the thickness

Results and discussion

The numerical model developed in this work is used to study the behavior of a non-Newtonian lubricant, the effects of elastic deformation of the porous layer, and the permeability of the porous medium—all on the performance of the squeezing fluid film. Thus, different values of the couple stress parameter, which is a characteristic property of the non-Newtonian behavior of the lubricant; and the compliance coefficient, which controls the deformation of the elastic porous layer, are considered.

Conclusion

In this paper, we have investigated the influence of non-Newtonian effects and elastic deformation of the porous interface on the hydrodynamic performances of a poroelastic contact lubricated by a couple stress fluid. The Reduced Navier–Stokes equations were written in the fluid film with an additional term that accounts for the couple stresses. The flow in the porous disc is described by the generalized Darcy–Brinkman–Forchheimer and Poisson equations for a Newtonian fluid of dynamic viscosity

Acknowledgements

The authors would like to thank the French-Moroccan Mixed Inter-University Committee for supporting this work under Grant number SPI 06/12 and Professor John Tichy (RPI Troy USA) for the improvement of the English writing

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¹: Present address: Direction de la Météorologie Nationale, BP 8106, Casablanca, Morocco.

View full text

Non-Newtonian couple stress poroelastic squeeze film

Highlights

Abstract

Introduction

Section snippets

Governing equations in the fluid film for couple stress fluids

Governing equations in the porous medium

Equation of motion of the upper disc

Equation of fluid film thickness

Equation of the porous interface deformation: Elastic thin layer model

Initial conditions

Pressure boundary conditions

Velocity boundary conditions

Dimensionless equations, initial and boundary conditions

Finite difference discretization and the solution technique

Solution algorithm

Results and discussion

Conclusion

Acknowledgements

Tribology International

Journal of Biomechanics

Tribology International

Tribology International

Tribology International

Matrix Biology

Journal of Biomechanics

Journal of Computational and Applied Mathematics

Journal of Biomechanics

Journal of Biomechanics

Computers and Structures

Electrophoretic separation of the synovial fluid proteins in patients with temporomandibular joint disorders

Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontics

Measurement of total and differential white blood cell counts in synovial fluid by means of an automated hematology analyzer

Journal of Laboratory and Clinical Medicine

Lubrication theory in highly compressible porous media: the mechanics of skiing, from red cells to humans

Journal of Fluid Mechanics

In vitro measurement of articular cartilage deformations in the intact human hip joint under load

The Journal of Bone & Joint Surgery

Articular cartilage deformation determined in an intact tibiofemoral joint by displacement-encoded imaging

Magnetic Resonance in Medicine