Emmanuel Kwame Essel

Reiterated homogenization applied in hydrodynamic lubrication

This work is devoted to studying the combined effect that arises due to surface texture and surface roughness in hydrodynamic lubrication. An effective approach in tackling this problem is by using the theory of reiterated homogenization with three scales. In the numerical analysis of such problems, a very fine mesh is needed, suggesting some type of averaging. To this end, a general class of problems is studied that, e.g. includes the incompressible Reynolds problem in both Cartesian and cylindrical coordinate forms. To demonstrate the effectiveness of the method several numerical results are presented that clearly show the convergence of the deterministic solutions towards the homogenized solution. Moreover, the convergence of the friction force and the load-carrying capacity of the lubricant film is also addressed in this paper. In conclusion, reiterated homogenization is a feasible mathematical tool that facilitates the analysis of this type of problem.

Multiscale homogenization of a class of nonlinear equations with applications in lubrication theory and applications

We prove a homogenization result for monotone operators by using the method of multiscale convergence. More precisely, we study the asymptotic behavior as e→0 of the solutions ue of the nonlinear equation div⁡ae(x,∇ue)=div⁡be, where both ae and be oscillate rapidly on several microscopic scales and ae satisfies certain continuity, monotonicity and boundedness conditions. This kind of problem has applications in hydrodynamic thin film lubrication where the bounding surfaces have roughness on several length scales. The homogenization result is obtained by extending the multiscale convergence method to the setting of Sobolev spaces W01,p(Ω), where 1<p<∞. In particular we give new proofs of some fundamental theorems concerning this convergence that were first obtained by Allaire and Briane for the case p=2.

Stability in delay Volterra difference equations of neutral type

Sufficient conditions for the zero solution of a certain class of neutral Volterra difference equations with variable delays to be asymptotically stable are obtained. The Banach’s fixed point theorem is employed in proving our results.