Exact analytic solutions of the lubrication equations for squeeze-flow of a biviscous fluid between two parallel disks

doi:10.1016/j.jnnfm.2011.08.003

Journal of Non-Newtonian Fluid Mechanics

Volume 166, Issues 21–22, November 2011, Pages 1253-1261

https://doi.org/10.1016/j.jnnfm.2011.08.003 Get rights and content

Abstract

Based on lubrication approximation, the squeezing flow of biviscous fluids between two parallel disks with partial slip boundary condition was investigated. In addition to the solution of the kinematics in the bi-viscosity region leading to a cubic equation of the yield surface, the full explicit expressions of radial pressure gradient, pressure and squeeze force are given in the exact relationships. According to three dimensionless numbers, different behaviors are covered including Bingham fluid as a limiting case of bi-viscosity model. Besides, a critical force separating the Newtonian and biviscous regions of the flow is provided. However, for a flow of a Bingham fluid without wall slip, the expression of the applied force may be expressed or not according to the yield stress. This depends on the ratio value of the characteristic time of the fluid to the time scale of observation if it is very lower or higher than unity.

Highlights

► Exact solution of the squeeze force for a biviscous fluid is obtained. ► Bingham fluid is presented as a limiting case of bi-viscosity model. ► Relationships of the force are given according to slip coefficient and yield number. ► The squeeze force for a Bingham fluid should be larger than a threshold force. ► Below this force the fluid is Newtonian and this is linked to a dimensionless number.

Introduction

Since the study of the squeezing flow of fluid between parallel disks has been of great interest due to its wide application in engineering practice, several authors are concerned in its theoretical development. For example, Stefan [1] (quoted in Ref. [2]) can be considered as the first to establish an exact solution for the squeezing flow of viscous Newtonian fluid using the “lubrication approximation”. Scott [3] (quoted in Ref. [2]) extended such development to power-law fluids using a no-slip boundary condition between the wall and the fluid. As for Laun et al. [4], they studied the partial slip squeeze flow of Newtonian and power-law fluids using the lubrication approach. It is worth mentioning that the slip condition had already been considered in the classical solution of Stefan. The no-slip squeeze flow of yield stress fluids was originally examined by Scott [3] and Peek [5] (quoted in Ref. [6]) with different rigid cores in the flow region. Later, Covey and Stanmore [7] gave simplified solutions for Bingham and Herschel–Bulkley fluids using lubrication theory. Besides, Meeten [8] carried out both theoretical and experimental investigations of the squeeze flow of Herschel–Bulkley fluid, but he did not account for the slip at the walls. Taking into consideration the partial slip at the plate-sample interface, Sherwood and Durban [9], [10] studied the squeeze flow of Bingham and Herschel–Bulkley fluids using asymptotic expansions. With regard to Lawrence and Corfield [11] and Adams et al. [12], they proposed analytical expressions using a slip law which in turn had an “interfacial” yield stress, whose fluids are admitted to behave like a solid in a unidirectional flow. Lipscomb and Denn [13] pointed out that, generally, complex flows cannot admit unyielded zones. According to them, there is a difficulty pertaining to the existence of yield surfaces in the interior of the fluid which must be solid and cannot move along the radial direction, simply because this is kinematically impossible. Other researchers [13], [14], [15] advocated that the Bingham model can be viewed as the limiting case of a biviscous fluid and therefore the squeeze flow paradox counteracts. They analyzed the problem using lubrication approximation with no-slip boundary conditions at the walls. Recently, Yang and Zhu [16] have theoretically analyzed the squeeze flow of the Bingham material described by the bi-viscosity model in the small gap between parallel disks with the Navier slip condition. They approximated the pressure gradient by a linear function and obtained an approximate expression for the squeeze force.

The present paper undertakes the theoretical study of squeeze flow between two parallel disks of a Bingham fluid, described by the bi-viscosity model with a partial slip at the wall. The exact solution of squeeze force is obtained by means of lubrication theory. Actually, to the best of author’s knowledge, there is no exact solution for this problem available in the open literature. Then, the effects of fluid properties, disks velocity, gap width and slip coefficient upon pressure, pressure gradient and squeeze force are presented according to the three dimensionless variables (viscosity ratio, slip coefficient and yield number).

The paper is organized as follows. First, the mathematical models are presented in Section 2. Afterwards, the analytical solutions are detailed in Section 3. Next, the results are discussed in Section 4. Finally, concluding remarks are given in Section 5.

Section snippets

Mathematic models

The flow system shown in Fig. 1 consists of the squeeze flow of a fluid within a narrow gap between two parallel disks. The plates of radius R_a are separated by 2h and translated towards each other with a relative velocity $2 \dot{h}$ . This squeezing speed is obtained by the squeeze force F applied to the disks. The governing differential equations are written in cylindrical polar coordinate system(r, θ, z). For symmetry reasons, the dependence on coordinate θ terms can be omitted. Besides, for a small

Analytical solutions

A Newtonian region of r ⩽ R₀ and a bi-viscosity region of r > R₀ are separately considered with different solutions.

Discussion

The effects of the three dimensionless parameters (viscosity ratio γ, slip coefficient σ and yield number ψ_n) on dimensionless functions (dimensionless pressure gradient Θ, pressure distribution Π and squeeze force $F$ ) will be examined later. However, radial velocity, shear stress distributions and yield surface in the flow field are not discussed in this study because they have been presented in the literature [16] and there are no significant changes.

Conclusions

The squeeze flow of biviscous and Bingham fluids was analyzed under the partial slip condition between the wall and the fluid. The solving of the cubic equation of the yield surface of the biviscous region has led to continuous and smooth curves of the kinematics equations with the Newtonian region. After switching to dimensionless variables, the analytical equations of pressure gradient, pressure and squeeze force are given according to viscosity ratio, slip coefficient and yield number.

Acknowledgements

The author would like to thank Professor Mounir Baccar (CFDTP University of Sfax, Tunisia) and Professor Sami Elborgi (EPT University of Carthage Tunis, Tunisia) for their insightful comments during the drafting of this paper.

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Exact analytic solutions of the lubrication equations for squeeze-flow of a biviscous fluid between two parallel disks

Abstract

Highlights

Introduction

Section snippets

Mathematic models

Analytical solutions

Discussion

Conclusions

Acknowledgements

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newton Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.