Abstract
The solution of elastohydrodynamically lubricated contacts at high loads and/or low speeds can be described as a Hertzian pressure with inlet and outlet boundary layers: zones where significant pressure flow occurs. For the soft lubrication regime (elastic-isoviscous), a self-similar solution exists in the boundary layers satisfying localized equations. In this paper, the boundary layer behaviour in the elastic-piezoviscous regime is investigated. The lengthscale of the boundary layers and the scaling of pressure and film thickness are expressed in non-dimensional parameters. The boundary layer width scales as \(1/\sqrt{M}\) (equivalent to \({\bar{\lambda }}^{3/8}\)), the maximum pressure difference relative to the Hertzian solution as \(1 / \root 3 \of {M}\) (equivalent to \({\bar{\lambda }}^{1/4}\)) and the film thickness as \(1/\root 16 \of {M}\) (equivalent to \({\bar{\lambda }}^{3/64}\)) with \(M\) the Moes non-dimensional load and \({\bar{\lambda }}\) a dimensionless speed parameter. The Moes dimensionless lubricant parameter \(L\) was fixed. These scalings differ from the isoviscous-elastic (soft lubrication) regime. With increasing load (decreasing speed), the solution exhibits an increasing degree of rotational symmetry. The pressure varies less than 10 % over an angle less than 45 degrees from the lubricant entrainment direction. The results provide additional fundamental understanding of the nature of elastohydrodynamic lubrication and give physical rationale to the finding of roughness deformation depending on the “inlet length”. The findings may contribute to more efficient numerical solutions and to improved semi-analytical prediction methods for engineering based on physically correct asymptotic behaviour.
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Abbreviations
- \(a\) :
-
Contact radius \(a=((3FR_x)/(2E'))^{1/3}\) (m)
- d :
-
Deformation (m)
- \(E'\) :
-
Equivalent Young’s modulus \(2/E'=(1-\nu _1^2)/E_1+(1-\nu _2^2)/E_2\) (Nm−2)
- F :
-
External load (N)
- h :
-
Film thickness (m)
- H :
-
Dimensionless film thickness \(H = hR_x/a^2\)
- H*:
-
Dimensionless film thickness \(H^{*}=H {\bar{\lambda }}^{-3/5}\)
- H M :
-
Moes dimensionless film thickness \(H_{\rm M}=h/R_x \cdot ((\eta _0 u_{\rm s})/( E' R_x))^{-1/2}\)
- \(\overline{H}\) :
-
Dimensionless film thickness \(\overline{H} = H\root 16 \of {M}\)
- \(\Delta H\) :
-
Dimensionless film thickness difference
- \(L\) :
-
Moes lubricant parameter \(L= \alpha \, E' \cdot (( \eta _0 u_{\rm s})/( E' R_x))^{1/4}\)
- \(M\) :
-
Moes load parameter point contact \(M = F/(E'\,R_x^2) \cdot (( \eta _0 u_{\rm s})/( E' R_x))^{-3/4}\)
- \(M_l\) :
-
Moes load parameter line contact \(M_1=w/(E' R_x) \cdot (( \eta _0 u_{\rm s})/( E' R_x))^{-1/2}\)
- \(p\) :
-
Pressure (Nm−2)
- \(P\) :
-
Dimensionless pressure \(P = p/p_h\)
- \(P^{*}\) :
-
Dimensionless film thickness \(P^{*}=P {\bar{\lambda }}^{-1/5}\)
- \(p_h\) :
-
Maximum Hertzian pressure \(p_h=(3F)/(2 \pi a^2)\)(Nm−2)
- \(P_h\) :
-
Dimensionless Hertzian pressure distribution \(P_h=\sqrt{1-X^2-Y^2}\)
- \(\Delta P\) :
-
Dimensionless pressure difference \(\Delta P\,=\,P(X,Y)-P_h\)
- \(\overline{\Delta P}\) :
-
Dimensionless pressure difference \(\overline{\Delta P}\, = \Delta P \root 3 \of {M}\)
- \(r\) :
-
Dimensionless radius \(r=sign(X) \sqrt{X^2+Y^2}\)
- \(R_x\) :
-
Reduced radius of curvature in \(x\) \(1/R_x=1/R_{1x}+1/R_{2x}\) (m)
- \(Re_x\) :
-
Local Reynolds number (introduction) \(Re_x=u_\infty x/\nu\)
- \(R_y\) :
-
Reduced radius of curvature in \(y\) \(1/R_y=1/R_{1y}+1/R_{2y}\) (m)
- \(u\) :
-
Surface velocity (ms−1)
- \(u_{rm s}\) :
-
Sum velocity \(u_{\rm s}=(u_1+u_2)\) (ms−1)
- \(u_\infty\) :
-
Freestream velocity (introduction) (ms−1)
- \(w\) :
-
External load per unit width (line contact) (Nm−1)
- \(x\) :
-
Coordinate in the direction of rolling (freestream flow) (m)
- \(y\) :
-
Coordinate perpendicular to the direction of rolling (freestream flow) (m)
- \(x'\) :
-
Coordinate in the direction of rolling (m)
- \(y'\) :
-
Coordinate perpendicular to the direction of rolling (m)
- \(X,Y\) :
-
Dimensionless coordinates \(X = x/a\), \(Y = y/a\)
- \(X',Y'\) :
-
Dimensionless coordinates \(X' = x'/a\), \(Y' = y'/a\)
- \(\overline{X}\) :
-
Dimensionless scaled coordinate \({\overline{X}} =\mp 1 + (X \pm 1) \sqrt{ M}\)
- \(X^{*}\) :
-
Dimensionless coordinate \(X^{*}=(X \pm 1) {\bar{\lambda }}^{-2/5}\)
- \(z\) :
-
Viscosity pressure index (Roelands)
- \(\alpha\) :
-
Viscosity-pressure coefficient (N−1m2)
- \({\bar{\alpha }}\) :
-
Dimensionless viscosity index \({\bar{\alpha }}=\alpha p_h\)
- \(\eta\) :
-
Dynamic viscosity (Nm−2s)
- \(\bar{\eta }\) :
-
Dimensionless viscosity \(\bar{\eta }= \eta /\eta _0\) (Nm−2s)
- \(\eta\) :
-
Boundary layer dimensionless coordinate (introduction) \(\eta =y \sqrt{u_\infty /(\nu x)}\)
- \(\nu\) :
-
Kinematic viscosity (introduction) \(\nu =\eta /\rho\) (m2s)
- \(\phi\) :
-
Angle with \(X\) axis \(\phi = \arctan (Y/X)\)
- \({\bar{\lambda }}\) :
-
Dimensionless speed parameter \({\bar{\lambda }}=(6\eta _0 u_{\rm s} R^2_x)/(p_h a^3)\)
- \(\rho\) :
-
Density (kg m−3)
- \(\bar{\rho }\) :
-
Dimensionless density \(\bar{\rho }= \rho /\rho _0\)
- \(\nu\) :
-
Poisson ratio
- \(\Delta\) :
-
Dimensionless mutual approach
- \(0\) :
-
At ambient pressure
- \(1,2\) :
-
Surface 1, surface 2
- s:
-
Sum
- \(x\) :
-
In \(x\) direction
- \(y\) :
-
In \(y\) direction
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Venner, C.H., Biboulet, N. & Lubrecht, A.A. Boundary Layer Behaviour in Circular EHL Contacts in the Elastic-Piezoviscous Regime. Tribol Lett 56, 375–386 (2014). https://doi.org/10.1007/s11249-014-0415-z
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DOI: https://doi.org/10.1007/s11249-014-0415-z