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A novel numerical scheme for the analysis of effects of surface roughness on EHL line contact with couple stress fluid as lubricant

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Abstract

In this article, a Jacobian-free Newton Multigrid (JFNMG) method is used for obtaining the solution of isothermal, steady and compressible elastohydrodynamic lubrication (EHL) line contact problem with surface roughness. The lubricant is a couple stress fluid. A finite difference scheme is used for the solution of EHL equations. The proposed JFNMG method, for the solution of resulting nonlinear system of algebraic equations, comprises nonlinear Newton iterations on the outer loop and linear multigrid iterations on the inner loop. It overcomes the limitations of conventional schemes for the investigation of the problems covering wide range of physical parameters of interest. For increasing values of couple stress parameter, there is an increase in minimum film thickness and considerable decrease (in height as well as spread) of pressure spike. Also, the sensitivity of height and spread of pressure spike as a function of load, couple stress parameter and other parameters are investigated.

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Abbreviations

\( a \) :

Amplitude of roughness \( \left( {\upmu{\text{m}}} \right) \)

\( \overline{a} \) :

Dimensionless amplitude of roughness, \( \overline{a} = {{aR} \mathord{\left/ {\vphantom {{aR} {b^{2} }}} \right. \kern-0pt} {b^{2} }} \)

A:

Normalized surface roughness amplitude, \( A = {{\overline{a} } \mathord{\left/ {\vphantom {{\overline{a} } {H_{\hbox{min} }^{DH} }}} \right. \kern-0pt} {H_{\hbox{min} }^{DH} }} \)

\( b \) :

Half width of the Hertzian contact, \( b = 4R\sqrt {W/2\pi } \)

\( E^{^{\prime}} \) :

Effective elastic modulus of rollers 1 and 2 (Pa)

\( G \) :

Dimensionless materials parameter, \( \alpha E^{\prime} \)

\( h \) :

Film thickness (m)

\( H \) :

Dimensionless film thickness, \( H = hR/b^{2} \)

\( h_{0} \) :

Offset film thickness (m)

\( H_{0} \) :

Dimensionless constant/offset film thickness \( H_{0} = h_{0} R/b^{2} \)

\( K_{ij} \) :

Discrete approximation of \( K \) logarithmic kernel Eq. (4.3)

\( l \) :

Dimensionless wavelength of roughness, \( l = {\lambda \mathord{\left/ {\vphantom {\lambda b}} \right. \kern-0pt} b} \)

\( L_{m} \) :

Dimensionless couple stress parameter, \( L_{m} = {{\lambda_{a} } \mathord{\left/ {\vphantom {{\lambda_{a} } R}} \right. \kern-0pt} R} \)

\( N \) :

Number of nodes on grid

\( p \) :

Pressure (pa)

\( P_{h} \) :

Maximum Hertzian press \( P_{h} = (2w)/(\pi b) \)

\( P \) :

Dimensionless pressure, \( P = p/P_{h} \)

\( P_{0} \) :

Ambient pressure

\( R \) :

Equivalent radius of contact

\( u \) :

Velocity component

\( u_{1} ,u_{2} \) :

Velocities of lower and upper surfaces respectively (m/s)

\( u_{s} \) :

Sum velocity, \( u_{s} = \left( {u_{1} + u_{2} } \right)/2 \)

\( U \) :

Dimensionless speed parameter, \( U = (\eta_{0} u_{s} )/(E^{\prime}R) \)

\( \upsilon \) :

Surface displacement (m)

\( \mathop \upsilon \limits^{ - } \) :

Dimensionless displacement, \( \mathop \upsilon \limits^{ - } = {{\upsilon R} \mathord{\left/ {\vphantom {{\upsilon R} {b^{2} }}} \right. \kern-0pt} {b^{2} }} \)

\( w \) :

External load per unit width

\( W \) :

Dimensionless load parameter, \( W = w/(E^{\prime}R) \)

\( \Delta X \) :

Mesh size

\( x \) :

Abscissa coordinate

\( X \) :

Dimensionless coordinate \( x/b \)

\( \left[ {X_{1} ,\;X_{2} } \right] \) :

Domain of interest \( \left[ { - 2,\;1.5} \right] \)

\( X_{c} \) :

Dimensionless location of pressure spike

\( z \) :

Pressure viscosity parameter

\( \alpha \) :

Pressure viscosity index

\( \eta \) :

Fluid viscosity

\( \eta_{0} \) :

Viscosity at ambient pressure

\( \bar{\eta } \) :

Dimensionless viscosity, \( \eta /\eta_{0} \)

\( \lambda \) :

Surface roughness wavelength \( \left( {\upmu{\text{m}}} \right) \)

\( \lambda_{a} \) :

Molecular length of additives (m)

\( \xi \) :

Viscosity modification factor

\( \rho \) :

Lubricant density at local pressure

\( \rho_{0} \) :

Inlet density of the lubricant

\( \bar{\rho } \) :

Dimensionless fluid density, \( \rho /\rho_{0} \)

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Acknowledgements

Authors thank the Department of Science and Technology (SR/S4/MS: 771/12) and Indian National Science Academy (SP/HIS/2012/425), New Delhi, India for the financial support. Also, thank the reviewers for their useful suggestions and comments on the earlier draft of the manuscript.

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Authors and Affiliations

  1. KLE Technological University, Vidyanagar, Hubballi, 580 031, India

    Bharati M Shettar & P S Hiremath

  2. Department of Mathematics, Karnatak University, Dharwad, 580 003, India

    N M Bujurke

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  1. Bharati M Shettar
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  2. P S Hiremath
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  3. N M Bujurke
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Correspondence to N M Bujurke.

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Shettar, B.M., Hiremath, P.S. & Bujurke, N.M. A novel numerical scheme for the analysis of effects of surface roughness on EHL line contact with couple stress fluid as lubricant. Sādhanā 43, 122 (2018). https://doi.org/10.1007/s12046-018-0906-y

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