Viscous fluid flow past an infinite periodic array of rigid spheres of the same radius is considered. A solution of the Stokes equations periodic in three variables is obtained for viscous incompressible flow with a linear velocity profile. The solution takes into account the hydrodynamic interaction of an infinite number of particles in the array. An expression for the effective viscosity of a suspension with a cubic array of particles is obtained.
One of modern dosage forms is a medicine-saturated organic film: after putting this film onto a skin the medicine releases thus providing healing effect. Present article concerns films based on chitosan and containing amikacinum or cefazolinum. The most important characteristic of such film is rate of medicine release described by diffusion coefficient. To find it the film is placed in water and the average medicine concentration in the film is measured at different time moments. Two problems arise here. First, the film properties change because of its swelling. Second, diffusion is not the only process that takes place inside the film. To deal with these effects, authors suppose diffusion coefficient to be time-variable and complete the mathematical model with ODE describing detachment of medicine molecules from high-molecular matrix. All the equations in the model are solved analytically, so average medicine concentration in the film is known function of time. Thus, to solve stated inverse problem it is sufficient to find unknown scalar parameters of known functions using least-squares framework. Expressions arising in the solution are complicated so non-gradient methods are preferrable for optimization. Applying described procedure to experimental data leads to a good accuracy and the results may be explained from physicochemical point of view. In particular, the film swelling doesn’t influence release rate. In fact, the diffusion rate during first hours of experiment is large, and the main part of the medicine is released before swelling starts to play important role.
Author solves problems about interaction of two spherical particles with different radii and also about interaction of a sphere and a plane that are immersed in electrolyte. Double electric layer near the objects' surfaces is supposed to be wide, so Poisson -- Boltzmann equation describing the distribution of electric potential in the medium may be linearized. The problems stated are solved by multipole expansion method; the plane is modelled by a dummy particle. Asymptotic expressions are obtained for the coefficients of the expansion. Basing on this solution, forces acting between bodies in electrolyte are found. The particular case when the size of one sphere is much larger than the size of another particle is examined. Author shows that this case can't transform to interaction of a sphere and a plane. The unexpected result of calculation is that under certain conditions the plane may attract spherical particle which has potential of the same sign on its surface, while the interaction between two spheres having potentials of the same sign is always repulsion.
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