We present an asymptotic theory that includes in a perturbative expansion the coupling effects between the director dynamics and the velocity field in a nematic liquid crystal. Backflow effects are most significant in the presence of defect motion, since in this case the presence of a velocity field may strongly reduce the total energy dissipation and thus increase the defect velocity. As an example, we illustrate how backflow influences the speeds of opposite-charged defects.

In this work we derive a formula for the maximum value of the voltage drop that takes place in power grids of integrated circuits with an array bonding package. We consider a simplified model for the power grid where the voltage is represented as the solution of the Poisson's equation in an infinite planar domain with a regularly aligned array of small square pads where the voltage is set to be zero. We study the singular limit where these pads' size tends to be zero and we derive an asymptotic formula in terms of a power series in ε, being 2ε the side of the square. We also discuss pads of more general shapes, for which we provide an expression for the leading order term. To deduce all these formulae we use the method of matched asymptotic expansions using an iterative scheme, along with conformal maps to compare this problem with the corresponding one when the pads are circular.