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# Some class of iterative TVD scheme for solving variational inequality appear in Elasto-hydrodynamic lubrication problems

Dr. Peeyush Singh, IIT, Kanpur
 Speaker Dr. Peeyush Singh, IIT, Kanpur Sep 08, 2017 from 04:00 PM to 05:00 PM LH 006 vCal iCal

Abstract: This talk is motivated by presenting some class of iterative TVD scheme for solving variational inequality appear in Elasto-hydrodynamic 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

$$h(x,y) = h_{00}+\frac{x^{2}}{2}+\frac{y^{2}}{2}$$+$$\frac{2}{\pi^{2}}$$ $$\int_{\Omega}$$ $$\frac{p(x^{'},y^{'})dx^{'}dy^{'}}{\sqrt{(x-x^{'})^2$$+(y-y^{'})^2}},\)

The dimensionless force balance equation is defined as follows

$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}p(x',y') dx'dy' = \frac{3\pi}{2}.$$

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