Analysis of Smoluchowski’s Coagulation Equation with Injection

Abstract

The stationary solution of Smoluchowski’s coagulation equation with injection is found analytically with different exponentially decaying source terms. The latter involve a factor in the form of a power law function that plays a decisive role in forming the steady-state particle distribution shape. An unsteady analytical solution to the coagulation equation is obtained for the exponentially decaying initial distribution without injection. An approximate unsteady solution is constructed by stitching the initial and final (steady-state) distributions. The obtained solutions are in good agreement with experimental data for the distributions of endocytosed low-density lipoproteins.

The boundary integral equation describing a curvilinear solid/liquid interface with nonlinear phase transition temperature

“…An important research direction related to the development of the BIE theory is the simultaneous consideration of directional interface motion deep into the melt and volumetric crystal growth in front of this boundary. Such an analysis, which takes into account the partial removal of undercooling by crystal growth, should be based on a combination of this theory and the theory of volumetric crystallization [47–55]. Another important area of research is the derivation of BIEs for directional solidification with a mushy region when two moving boundaries exist: “solid phase ‐ mushy region” and “mushy region ‐ liquid phase.” Such a problem should be solved by combining the present approach with a two‐phase region theory [56–63].…”

Desupersaturation dynamics in solutions with applications to bovine and porcine insulin crystallization

“…Summing up this section, it should be noted that the analytical solution is found in a parametric form (the modified time variable y represents the parameter). Namely, the crystal-radius distribution function ϕ (ξ, y) is given by expression (22), the solution supersaturation w(y) is defined by formula (24), and the crystallization time t(y) is determined by expression (30). Note that the obtained analytical solution can be simplified by calculating integrals over the spatial variable ξ for some initial crystal-size distributions ϕ o (ξ).…”

“…When describing the final stage, a more detailed model of the dominant process (e.g. Ostwald ripening, coagulation, fragmentation, clustering of particles or aggregates) is usually used [15][16][17][18][19][20][21][22].…”

An exact analytical solution to the nonstationary coagulation equation describing the endosomal network dynamics

“…self-similar, and stationary solutions of some particular cases has been discussed by Herrmann et al, 6 Crump and Seinfeld, 7 Simons, 8 Alexandrov et al, 9 and Makoveeva et al 10 ; the global existence of solutions has been proved for a large class of data by Laurençot 11 ; self-similar solutions and the behavior of solutions over long times have been investigated by Alyab'eva et al 12 and Alexandrov. 13,14 The coagulation equation derived by Smoluchowski 1,2 accounts for the basic mechanism of the process: the merging of two closely approaching particles leads to the formation of a new greater particle.…”

“…Thus, considering proper initial and boundary conditions, we have a strongly nonlinear problem with no common methods for solving it. As a special note, the existence of exponentially damped, self‐similar, and stationary solutions of some particular cases has been discussed by Herrmann et al, 6 Crump and Seinfeld, 7 Simons, 8 Alexandrov et al, 9 and Makoveeva et al 10 ; the global existence of solutions has been proved for a large class of data by Laurençot 11 ; self‐similar solutions and the behavior of solutions over long times have been investigated by Alyab'eva et al 12 and Alexandrov 13,14 …”