Some class of iterative TVD scheme for solving variational inequality appear in Elastohydrodynamic lubrication problems
Speaker 
Dr. Peeyush Singh, IIT, Kanpur


When 
Sep 08, 2017
from 04:00 PM to 05:00 PM 
Where  LH 006 
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Abstract: This talk is motivated by presenting some class of iterative TVD scheme for solving variational inequality appear in Elastohydrodynamic lubrication problem in Tribology.
Discussion will be start by briefly examine a linear quadratic problem of the following form
\(\min J(y)\) = \(\frac{1}{2}\)\(\langle y, Ay \rangle \langle f, y \rangle\)
\(y \ge \phi.\)
Further study involves in developing a solution procedure and convergence analysis of the problem and giving an algorithm for solving our model problem of the type (2)(7)
\(\frac{\partial }{\partial x}\) \(\Big(\epsilon^{*}\) \(\frac {\partial p}{\partial x}\Big)+\) \(\frac{\partial }{\partial y} \)\(\Big(\epsilon^{*}\)\( \frac {\partial p}{\partial y}\Big)\)\(\le \frac {\partial (\rho h)}{\partial x},\)
\(p\ge 0,\)
\(p.\Big[\frac{\partial }{\partial x}\) \(\Big(\epsilon^{*}\) \(\frac {\partial p}{\partial x}\Big)+\) \(\frac{\partial }{\partial y} \)\(\Big(\epsilon^{*}\)\( \frac {\partial p}{\partial y}\Big)\)\(\) \(\frac {\partial (\rho h)}{\partial x}\) \( \Big] \) = 0,
where p,\(\rho\) are pressure and density of the lubricant and diffusive coefficient \(\epsilon^{*}=\frac{\rho h^{3}}{\eta \lambda}\)
\(\eta(p) = \eta_{0}e^{l_{2}p},\epsilon^{*} = \frac{\rho h^{3}}{\eta \lambda}, \lambda = \frac{12\mu v(2R)^{3}}{\pi E}.\)
Above nonlinear variational inequality (2)(4) is defined in a bounded, but large domain \(\Omega\) with natural boundary condition
\(p= 0 \quad \text{on} \quad \partial \Omega.\)
Dimensionless film thickness h(x,y) is written as follows
\(h(x,y) = h_{00}+\frac{x^{2}}{2}+\frac{y^{2}}{2}\)+\(\frac{2}{\pi^{2}}\) \(\int_{\Omega}\) \(\frac{p(x^{'},y^{'})dx^{'}dy^{'}}{\sqrt{(xx^{'})^2\)+(yy^{'})^2}},\)
The dimensionless force balance equation is defined as follows
\(\int_{\infty}^{\infty} \int_{\infty}^{\infty}p(x',y') dx'dy' = \frac{3\pi}{2}.\)