In several types of coating processes a solid substrate is removed at a controlled velocity U from a liquid bath. The shape of the liquid meniscus and the thickness of the coating layer depend on U. These dependencies have to be understood in detail for non-volatile liquids to control the deposition of such a liquid and to lay the basis for the control in more complicated cases (volatile pure liquid, solution with volatile solvent). We study the case of non-volatile liquids employing a precursor film model that describes partial wettability with a Derjaguin (or disjoining) pressure. In particular, we focus on the relation of the deposition of (i) an ultrathin precursor film at small velocities and (ii) a macroscopic film of thickness h ∝ U^(2/3) (corresponding to the classical Landau Levich film). Depending on the plate inclination, four regimes are found for the change from case (i) to (ii). The different regimes and the transitions between them are analysed employing numerical continuation of steady states and saddle-node bifurcations and simulations in time. We discuss the relation of our results to results obtained with a slip model. In connection with evaporative processes, we will study the pinning of a droplet due to a sharp corner. The approach employs an evolution equation for the height profile of an evaporating thin film (small contact angle droplet) on a substrate with a rounded edge, and enables one to predict the dependence of the apparent contact angle on the position of the contact line. The calculations confirm experimental observations, namely that there exists a dynamically produced critical angle for depinning that increases with the evaporation rate. This suggests that one may introduce a simple modification of the Gibbs criterion for pinning that accounts for the non-equilibrium effect of evaporation.

Biological processes at the cellular level take place in heterogeneous environments, and usually involve only a small number of molecules. They tend to exhibit strong time dependent fluctuations, as a result, and are, therefore, intrinsically stochastic. The present thesis describes some of the efforts I have made during the course of my research work to develop simple, analytically tractable models of a selection of biologically-inspired problems in which this kind of stochasticity is a central ingredient. These problems are: (i) single molecule enzyme activity (ii) intermittency in single enzymes, (iii) liquids crystal dynamics (iv) modulation of electron transfer kinetics during photosynthesis, and (v) anomalous polymer translocation dynamics. All of these problems can be defined in terms of quantity that changes randomly in time because of environmental fluctuations with broad distributions of relaxation times. In this thesis I show that a generalization of a model that describes simple Brownian Motion can be used to understand many of the dynamical aspects of these problems.