The deposition of solids from mixtures of a paraffinic wax (C20−C40) dissolved in a multicomponent solvent (C9−C16) was studied under laminar flow conditions. A novel benchscale flow loop was developed, which consisted of a jacketed heat-exchange section for solids deposition on the inner surface of an aluminum tube. Experiments were performed to investigate the effects of the wax−solvent mixture composition, hot and cold stream temperatures, flow or shear rate, deposition residence time, and hydrodynamic entry length on the deposition process. The data were analyzed with a pseudo-steady-state heat-transfer model, which validated the solids deposition process to be controlled primarily by heat transfer. The mass of deposited solids was related to the ratio of temperature difference across the deposit layer and the overall temperature difference. Gas chromatography (GC) analyses of the deposited layer showed significant shifts in the carbon number distribution. The C20+ content of the deposit layer was observed to be higher, by ∼70%−200%, than that of the corresponding wax−solvent mixture.
For two-dimensional, creeping flow in a half-plane, we consider the singularity that arises at an abrupt transition in permeability from zero to a finite value along the wall, where the pressure is coupled to the seepage flux by Darcy’s law. This problem represents the junction between the impermeable wall of the inflow section and the porous membrane further downstream in a spiral-wound desalination module. On a macroscopic, outer length scale the singularity appears like a jump discontinuity in normal velocity, characterized by a non-integrable $1/ r$ divergence of the pressure. This far-field solution is imposed as the boundary condition along a semicircular arc of dimensionless radius 30 (referred to the microscopic, inner length scale). A preliminary numerical solution (using a least-squares variant of the method of fundamental solutions) indicates a continuous normal velocity along the wall coupled with a weaker $1/ \sqrt{r} $ singularity in the pressure. However, inconsistencies in the numerically imposed outer boundary condition indicate a very slow radial decay. We undertake asymptotic analysis to: (i) understand the radial decay behaviour; and (ii) find a more accurate far-field solution to impose as the outer boundary condition. Similarity solutions (involving a stream function that varies like some power of $r$) are insufficient to satisfy all boundary conditions along the wall, so we generalize these by introducing linear and quadratic terms in $\log r$. By iterating on the wall boundary conditions (analogous to the method of reflections), the outer asymptotic series is developed through second order. We then use a hybrid computational scheme in which the numerics are iteratively patched to the outer asymptotics, thereby determining two free coefficients in the latter. We also derive an inner asymptotic series and fit its free coefficient to the numerics at $r= 0. 01$. This enables evaluation of the singular flow field in the limit as $r\ensuremath{\rightarrow} 0$. Finally, a uniformly valid fit is obtained with analytical formulas. The singular flow field for a solid–porous abutment and the general Stokes flow solutions obtained in the asymptotic analysis are programmed in Fortran for future use as local basis functions in computational schemes. Numerics are required for the intermediate-$r$ regime because the inner and outer asymptotic expansions do not extend far enough toward each other to enable rigorous asymptotic matching. The logarithmic correction terms explain why the leading far-field solution (used in the preliminary numerics) was insufficient even at very large distances.
Swarms of Stokeslets have previously been shown to be effective for simulating three-dimensional, free-surface, buoyancy-driven drop flows at unit viscosity ratio, including interfacial rupture and pass-through phenomena. This article presents an efficient marker-and-cell=fast Fourier transform (MAC-FFT) algorithm that yields N log N scaling of the operations, thereby enabling accurate simulations involving millions of particles. The formerly separate steps of regularization of the Green's function (commonly used in vortex blob methods for inviscid flow) and discretization are unified by regularizing the Stokeslet over the cubical cells that form the underlying grid-as opposed to the spherical blobs used in the past. A piecewise-constant drop phase function, obtained by simple binning of the particles, thereby yields second-order quadrature of the Green's function to obtain the velocity field. An iterative cascade algorithm (adapted from Daubechies wavelets) allows the Stokes ''cubelet'' field to be calculated very efficiently. A similar cubelet approach is used for the cohesive forces that mimic interfacial tension. A user-friendly library of Fortran 95 subroutines (DropLib, www.droplib.org) has been developed to carry out the simulations and visualize drop shape evolutions in three dimensions.
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