Unidirectional consolidation of clay soils and flocculated suspensions

A model of the unidirectional consolidation of a clay soil or flocculated suspension between a series of parallel drains is developed. A convective-diffusion equation for the pore pressure is derived, and an equal-strain approximation leads to an expression for the average effective stress profile between the drains. The solution depends on a Peclet number quantifying the ratio of the bulk soil flow rate to the rate of consolidation. By adjusting the number, height and spacing of the drains, very high rates of dewatering can be achieved. A potential application of the method to the rapid dewatering of mine tailings is described.

Boussinesq system with measure forcing

The paper analyzes the Navier–Stokes system coupled with the convective-diffusion equation responsible for thermal effects. It is a version of the Boussinesq approximation with a heat source. The problem is studied in the two dimensional plane and the heat source is a measure transported by the flow. For arbitrarily large initial data, we prove global in time existence of unique regular solutions. Measure being a heat source limits regularity of constructed solutions and it requires a non-standard framework of inhomogeneous Besov spaces of the $$L^\infty (0,T;B^s_{p,\infty })$$ L ∞ ( 0 , T ; B p , ∞ s ) -type. The uniqueness of solutions is proved by using the Lagrangian coordinates.

Transport and Deposition of Large Aspect Ratio Prolate and Oblate Spheroidal Nanoparticles in Cross Flow

The objective of this paper was to study the transport and deposition of non-spherical oblate and prolate shaped particles for the flow in a tube with a radial suction velocity field, with an application to experiments related to composite manufacturing. The transport of the non- spherical particles is governed by a convective diffusion equation for the probability density function, also called the Fokker–Planck equation, which is a function of the position and orientation angles. The flow is governed by the Stokes equation with an additional radial flow field. The concentration of particles is assumed to be dilute. In the solution of the Fokker–Planck equation, an expansion for small rotational Peclet numbers and large translational Peclet numbers is considered. The solution can be divided into an outer region and two boundary layer regions. The outer boundary layer region is governed by an angle-averaged convective-diffusion equation. The solution in the innermost boundary layer region is a diffusion equation including the radial variation and the orientation angles. Analytical deposition rates are calculated as a function of position along the tube axis. The contribution from the innermost boundary layer called steric- interception deposition is found to be very small. Higher order curvature and suction effects are found to increase deposition. The results are compared with results using a Lagrangian tracking method of the same flow configuration. When compared, the deposition rates are of the same order of magnitude, but the analytical results show a larger variation for different particle sizes. The results are also compared with numerical results, using the angle averaged convective-diffusion equation. The agreement between numerical and analytical results is good.

An Analytical Solution Of 1-D Pseudo Homoneneous Model For Oxidation Reaction Using Homotopy Perturbation Method

In this paper, we propose an analytical solution of convective-diffusion equation that derived from an oxidation reaction in a chemical reactor. Here, concentration of feed gas as dependent variable. In this study, the reaction are assumed to be a one-dimensional pseudo homogeneous model and it is evaluated at a certain reaction rate. By rescaling process, the nonlinear term of the reaction rate can be approximated by a linear term, resulting a linear convective-diffusion equation with an initial condition and a set of boundary conditions. Here, we present an analytic solution of the initial condition and the boundary conditions using the homotopy perturbation method. The results show that at the end of the reactor, the solution is in agreement with numerical solution of the initial and boundary conditions.

Solutions of Navier-Stokes Equation with Coriolis Force

We investigate the Navier-Stokes equation in the presence of Coriolis force in this article. First, the vortex equation with the Coriolis effect is discussed. It turns out that the vorticity can be generated due to a rotation coming from the Coriolis effect, Ω. In both steady state and two-dimensional flow, the vorticity vector ω gets shifted by the amount of -2Ω. Second, we consider the specific expression of the velocity vector of the Navier-Stokes equation in two dimensions. For the two-dimensional potential flow v→=∇→ϕ, the equation satisfied by ϕ is independent of Ω. The remaining Navier-Stokes equation reduces to the nonlinear partial differential equations with respect to the velocity and the corresponding exact solution is obtained. Finally, the steady convective diffusion equation is considered for the concentration c and can be solved with the help of Navier-Stokes equation for two-dimensional potential flow. The convective diffusion equation can be solved in three dimensions with a simple choice of c.

A Numerical Study on the Ventilation Coefficients of Falling Hailstones

The ventilation coefficients that represent the enhancement of mass transfer rate due to the falling motion of spherical hailstones in an atmosphere of 460 hPa and 248 K are computed by numerically solving the unsteady Navier–Stokes equation for airflow past hailstones and the convective diffusion equation for water vapor diffusion around the falling hailstones. The diameters of the hailstones investigated are from 1 to 10 cm, corresponding to Reynolds number from 5935 to 177 148. The calculated ventilation coefficients vary approximately linearly with the hailstone diameter, from about 19 for a 1-cm hailstone to about 208 for a 10-cm hailstone. Empirical formulas for ventilation coefficient variation with hailstone diameter as well as Reynolds and Schmidt numbers are given. Implications of these ventilation coefficients are discussed.

Diffusion in Replica Healthy and Emphysematous Alveolar Models Using Computational Fluid Dynamics

Deposition of nanosized particles in the pulmonary region has the potential of crossing the blood-gas barrier. Experimental in vivo studies have used micron-sized particles, and therefore nanoparticle deposition in the pulmonary region is not well understood. Furthermore, little attention has been paid to the emphysematous lungs, which have characteristics quite different from the healthy lung. Healthy and emphysematous replica acinus models were created from healthy and diseased human lung casts using three-dimensional reconstruction. Particle concentration and deposition were determined by solving the convective-diffusion equation numerically for steady and unsteady cases. Results showed decreased deposition efficiencies for emphysema compared to healthy lungs, consistent with the literature and attributed to significant airway remodeling in the diseased lung. Particle diffusion was found to be six times slower in emphysema compared to healthy model. The unsteady state simulation predicted deposition efficiencies of 96% in the healthy model for the 1 nm and 3 nm particles and 94% and 93% in the emphysema model for the 1 nm and 3 nm particles, respectively. Steady state was achieved in less than one second for both models. Comparisons between steady and unsteady predictions indicate that a steady-state simulation is reasonable for predicting particle transport under similar conditions.