Numerical solution of thermal EHL line contact with bio-based oil as lubricant

ABSTRACT

The paper presents the numerical solution of line contact thermal elastohydrodynamic lubrication (EHL) with bio-based lubricant. The model comprises Reynolds equation, film thickness, load balance and energy equations with appropriate boundary conditions by incorporating viscosity–pressure–temperature and density–pressure–temperature relations. Second-order finite difference scheme is used for the discretised form and their equation. The multigrid method with full approximation scheme is used to solve the Reynolds equation along with multilevel multi-integration method for film thickness equation. The pressure, film thickness and temperature distributions for two rolling velocities and various loads with a bio-based lubricant are presented in detail. The present findings yield a reduction in the minimum film thickness for high speed. Details of pressure spike as a function of relevant parameters are given. The results are compared with earlier findings based on different methods.

KEYWORDS:

• Thermal
• elastohydrodynamic lubrication
• line contact
• multigrid method
• bio-based oil

Nomenclature

$A$	=	amplitude of the wave
$b$	=	half width of Hertzian contact zone, $b={\left[8w{R}_x/(\pi E^{\prime })\right]}^{1/2}$
$E^{\prime }$	=	effective elastic modulus of roller and disc, $\frac{1}{E^{\prime }}=\frac{1}{2}\left(\frac{1-v_1^2}{E_1}+\frac{1-v_2^2}{E_2}\right)$
$E_{1,2}$	=	elastic modulus of solids
$h$	=	film thickness
$H$	=	dimensionless film thickness
$P$	=	dimensionless pressure
$p$	=	pressure
$p_H$	=	maximum Hertzian pressure, $p_H=2w/\pi b$
$R_x$	=	reduced radius of curvature in the $x-$direction
$T_0$	=	inlet temperature of lubricant
$T$	=	dimensionless temperature
$U$	=	dimensionless speed parameter $U={\eta }_0{u}_m/(E^{\prime }R)$
$u_m$	=	average entrainment speed $u_m=(u_a+u_b)/2$
$u_{a,b}$	=	lower and upper surface speed.
$s$	=	sliding ratio
$w$	=	applied load per unit length
$W$	=	dimensionless load parameter $W=w/(E^{\prime }{R}^2)$
$-z$	=	pressure viscosity parameter
$X$	=	dimensionless coordinate
$\mathrm{\Delta }X$	=	space increment
$\mathrm{\Delta }\bar{Z}$	=	increment along z-axis
$\alpha$	=	pressure viscosity index
$\beta$	=	thermal-density coefficient
$\gamma$	=	thermal-viscosity coefficient of lubricant
$\xi$	=	wavelength
$\bar{\xi }$	=	dimensionless wavelength
$\lambda$	=	dimensionless parameter
$\eta$	=	viscosity of lubricant
${\eta }_0$	=	viscosity at ambient pressure
$\bar{\eta }$	=	dimensionless viscosity
$\rho$	=	density of lubricant
${\rho }_0$	=	inlet density of pressure
$\bar{\rho }$	=	dimensionless density
${{\upsilon }_{1,}}_2$	=	Poisson’s ratio of solids
$c_p$	=	specific heat of lubricant
$k_{1,2}$	=	thermal conductivity of rollers
$k_f$	=	thermal conductivity of lubricant
${\rho }_{a,b}$	=	density of solids

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Vishwanath B. Awati

Vishwanath B. Awati is Professor of department of Mathematics at Rani Channamma University, Belagavi, India. He has obtained his Ph.D degree in Mathematics from Karnatak University, Dharwad. His research areas includes the tribology and contact mechanics, stretching/ shrinking sheet, heat and mass transfer, Dirichlet series, multigrid and wavelet method.

Mahesh Kumar N

Mahesh Kumar N is Assistant Professor of department of Mathematics at Rani Channamma University, Belagavi, India. His research areas includes Elasto-hydrodynamic lubrication, stretching/ shrinking sheet,  multigrid and wavelet method.

N.M. Bujurke

N.M. Bujurke is INSA Senior scientist and Professor of department of Mathematics at Karnatak University, Dharwad. His research areas includes the tribology and contact mechanics, EHL,  MHD, stretching/ shrinking sheet, heat and mass transfer, Dirichlet series, multigrid, wavelet method.