The slow flow of thin liquid films on solid surfaces is an important phenomenon in nature and in industrial processes, and an intensive effort has been made to investigate it. It is well known that the contact line of currents on an inclined surface may become unstable and then a pattern of "fingers" develops that affects the quality of the coatings. This instability has been intensively studied due to its relevance for the technology of various industrial processes. So far the theoretical and numerical research has been focused on Newtonian fluids, notwithstanding that often in the real situations as well as in the experiments, the rheology of the involved liquid is non-Newtonian. Using the lubrication approximation, we derive the governing equations for a current of a power law non-Newtonian fluid on an inclined plane under the action of gravity and the viscous stresses. We show that surface tension effects can be included in the theory by a slight modification of the governing equations, that can then be used as a starting point to investigate the influence of rheology on the fingering instability and other phenomena of interest. We consider the one-dimensional case and we present three families of traveling wave solutions: two running downwards and the other upwards.

We investigate the solutions of the Navier–Stokes equations that describe the steady flow of rivulets down an inclined surface. We find that the shape of the free surface is given by an analytic formula obtained by solving the equation that expresses the condition of static equilibrium under the action of gravity and surface tension, independently of the velocity field and of any assumption concerning the rheology of the liquid. The velocity field is then obtained by solving (in general numerically) a Poisson equation in the domain defined by the cross-section of the rivulet. The isovelocity contours are perpendicular to the free surface. Various properties of the solutions are given as functions of the parameters of the problem. Two special analytic solutions are presented. The exact solutions suggest that the lubrication approximation, frequently employed to investigate problems similar to the present one, predicts reasonably well the global properties of the rivulet provided the static contact angle is not too large.

We study the thermocapillary migration of two-dimensional droplets of partially wetting liquids on a non-uniformly heated surface. The effect of a non-zero contact angle is imposed through a disjoining–conjoining pressure term. The numerical results for two different molecular interactions are compared: on the one hand, London–van der Waals and ionic–electrostatics molecular interactions that account for polar liquids; on the other hand, long- and short-range molecular forces that model molecular interactions of non-polar fluids. In addition, the effect of gravity on the velocity of the drop is analysed. We find that for small contact angles, the long-time dynamics is independent of the molecular potential, and the footprint of the droplet increases with the square root of time. For intermediate contact angles we observe that polar droplets are more likely to break up into smaller volumes than non-polar ones. A linear stability analysis allows us to predict the number of droplets after breakup occurs. In this regime, the effect of gravity is stabilizing: it reduces the growth rates of the unstable modes and increases the shortest unstable wavelength. When breakup is not observed, the droplet moves steadily with a profile that consists in a capillary ridge followed by a film of constant thickness, for which we find power law dependencies with the cross-sectional area of the droplet, the contact angle and the temperature gradients. For large contact angles, non-polar liquids move faster than polar ones, and the velocity is proportional to the Marangoni stress. We find power law dependencies for the velocity for the different regimes of flow. The numerical results allow us to shed light on experimental facts such as the origin of the elongation of droplets and the existence of saturation velocity.

Total dialysis dose (Kt/V) is considered to be a major determinant of morbidity and mortality in hemodialyzed patients. The continuous growth of the blood urea concentration over the 30- to 60-min period following dialysis, a phenomenon known as urea rebound, is a critical factor in determining the true dose of hemodialysis. The misestimation of the equilibrated (true) postdialysis blood urea or equilibrated Kt/V results in an inadequate hemodialysis prescription, with predictably poor clinical outcomes for the patients. The estimation of the equilibrated postdialysis blood urea (eqU) is therefore crucial in order to estimate the equilibrated (true) Kt/V. In this work we propose a supervised neural network to predict the eqU at 60 min after the end of hemodialysis. The use of this model is new in this field and is shown to be better than the currently accepted methods (Smye for eqU and Daugirdas for eqKt/V). With this approach we achieve a mean difference error of 0.22 ± 7.71 mg/ml (mean % error: 1.88 ± 13.46) on the eqU prediction and a mean difference error for eqKt/V of –0.01 ± 0.15 (mean % error: –0.95 ± 14.73). The equilibrated Kt/V estimated with the eqU calculated using the Smye formula is not appropriate because it showed a great dispersion. The Daugirdas double-pool Kt/V estimation formula appeared to be accurate and in agreement with the results of the HEMO study.