Small Peclet Numbers Research Articles

Technical note: The Graetz problem at small Peclet number and its application to sizing condensation particle counters

We extend to small Peclet number Pe the familiar high-Pe solution to the Graetz problem of heat/mass transfer in a tube with parabolic flow with a wall temperature discontinuity at an axial position z = 0. Retaining diffusion in the axial direction results only in slight shifts in the eigenfunctions describing the problem. This enables the characterization of the saturation field S by separation of variables, which we implement for Pe = 0, 0.1, 0.25. Calculations based on discretization of the transport partial differential equations are also used to analyze the problem at Pe up to 5. As Pe tends to 0, the heat and mass transfer problems are both described by the same dimensionless function θ(z,r) of the axial and radial coordinates z and r. S then depends on the single quantity θ, and its maximum value Smax is exactly the same along all streamlines. This would result in sharp activation curves in a hypothetical low-Pe condensation particle counter, similarly as in the special circumstance when the ratio Le between heat and mass diffusivities is unity. Calculations are run to determine how the radial uniformity of Smax is degraded at increasing Pe. When accounting for axial diffusion, Smax at the wall is independent of Pe or Le. The commonly considered high Pe limit fails to satisfy this requirement, and therefore predicts an incorrect Smax at distances from the wall discontinuity small compared to Pe−1/2.

Use of tritium in characterizing the leachate migration from a landfill – Central de tratamento de resíduos sólidos (CTRS) BR040 (Belo Horizonte, Minas Gerais, Brazil)

Landfills often become a source of environmental impact due to the percolation of the leachate (liquid resulting from the decomposition of buried waste), which can eventually disperse and contaminate the soil and water bodies. This study aimed to evaluate quantitatively and spatially the leachate plume from the former Belo Horizonte landfill in the Coqueiros and Taiobas watersheds by using tritium (3H) concentration in nineteen (19) sampling points: 14 groundwater, 4 surface water and 1 leachate. Among the methodologies applied to determine the 3H content in the leachate, pre-filtration proved to be the most effective due to the ease treating of the sample, as well as the identical results obtained compared to the traditional method. The 3H contents resulted in concentrations up to thirty times higher in the leachate (=96,19 to 111,87 TU) and ten times in groundwater (PM23/3 = 45,06 TU), compared to the calculated threshold (TS) of rainfall tritium (TS_3Hrain = 3,41 TU). Temporal series results indicate no trend and no influence of seasonality regarding the entry of pluvial tritium, leachate liquid level and wells water level. The 3H contour maps shows a transport of the mixture (water and leachate) towards the Coqueiros stream, with a higher concentration of leachate in wells closer to the landfill slope (PM13, PM14, PM15, and PM23) and in the southwest margin of the Coqueiros stream (PM17). In the furthest portion of the landfill, 3H levels are higher than the base threshold only at the deepest level of PM20 (=3,90 TU), possibly due to factors such as: i) denser character of the leachate compared to groundwater; ii) greater influence of recharge for regions further away from the landfill slope; and iii) the presence of the rock about 50 m from the surface. Based on the hydrogeological conceptual model, the transport of the leachate water mixture in the groundwater water compartment of the Coqueiros watershed basin seems to follow a piston or dispersion flow, with a small Peclet number (Pd = 0,05, as an example), as indicated by the high correlation coefficient values (R2 ≈ 0,85 to 0,99) from graphic interpolation. The mixture of water and leachate results in a transient time (tt) ≈ 30 years and linear velocity (Vl) ≈ 3–14 m/year in the interpolated sections. These indicate a coherent correlation with the mean hydraulic conductivity (K ≈ 29,34 m/year), as they result in effective porosity values between (Ne) ≈ 0.16 to 0.36, which corresponds to Ne range for unconsolidated sediments, similar to the wells lithotype profiles. Time series evaluation combined with lumped parameter approach suggest to be an interesting way to better understanding the HCM, and, therefore, to determine hydrodynamic parameters using tritium as a tracer.

Contribution of electromagnetism in lowering entropy in microchannel

The study examined how entropy can be reduced in an asymmetric microchannel with sinusoidal walls by analyzing the flow of an incompressible micropolar fluid that is electrokinetically driven. The study takes into account the effects of Ohmic and viscous dissipation of energy in a magnetohydrodynamic environment. To model electroosmotic phenomena, the Poisson equation and Nernst–Planck equation were used, which were simplified using Debye–Huckel linearization and a small ionic Peclet number. The energy, rotational momentum, mass, and momentum equations for the micropolar fluid were constructed and linearized using a lubrication approach before being solved analytically. The resulting expressions for velocity microrotation and temperature were then used to compute the Bejan number and the formation of entropy number The study also examined the effects of various thermophysical parameters and discussed their potential applications in fields such as biomedical engineering, pharmaceuticals, energy conservation, cancer treatment, and manufacturing. The findings suggest that entropy generation can be reduced by increasing the micropolar coupling number and temperature difference parameter This study provides a theoretical guide for the operation and thermal control of flow actuation systems with asymmetric geometry.

Electrokinetic peristaltic transport of Bingham‐Papanastasiou fluid via porous media

AbstractAn important rheological mathematical model is created to investigate the rheological impacts of slip velocity and varied zeta potentials in an inclined asymmetric channel. The flow is taken in an isotropic porous medium and is governed by Bingham‐Papanastasiou model. The membrane based pumping analysis is done in a wave frame of reference moving with the speed of the wave. Flow model is simplified by considering small wave number δ, small Reynolds number and small Peclet number . The emerging linearized non‐dimensional system of equations is evaluated for analytical and numerical methods. The effects of sundry parameters on pumping, temperature θ, axial velocity u and trapping have been studied graphically. The viscous model is retrieved for Bingham number , or stress growth parameter . Finally, the effects of relevant parameters on heat transfer rate and shear stress at walls are discussed numerically. The results show that more pressure is required to flow same amount of fluid in an inclined channel. The temperature field θ is boosted by both the Bingham number and the continuation parameter M. It is also observed that different zeta potentials and velocity slip conditions are significant phenomena to influence channel flow. A pumping‐based device can be built using the existing model to combine and filter physiological samples and chemicals as well as to visualize the transit of physiological fluids.

Anisotropic Transport of Dielectric Particles by a Uniform Electric Field in an Inhomogeneously Heated Viscous Fluid

The anisotropic transfer of dielectric particles by a uniform electric field in a nonuniformly heated fluid is modeled. The transport anisotropy is determined by the mechanism of interaction between particles whose permittivity depends on temperature. The temperature distribution in the particles and in the fluid is determined by their thermal diffusivity and does not depend on the motion of the fluid, thus corresponding to small Peclet numbers. The fluid flow is considered in the approximation of small Reynolds numbers. The transfer of particles is due to the action of an anisotropic force exerted by applied uniform electric field and friction forces exerted by the fluid. The interaction of particles is taken into account. Numerical modeling of anisotropic transport dynamics of two dielectric particles is carried out. The process mentioned depends on the mutual orientation of electric field vector, temperature gradient, and initial orientation of the vector connecting the particle centers. For the case of a large number of particles, an anisotropic equilibrium distribution of the particle concentration in an external electric field is found taking into account the mechanisms of their diffusion and interaction.

A 3D Monte Carlo Simulation of Convective Diffusional Deposition of Ultrafine Particles on Fiber Surfaces

The microscale simulation of ultrafine particle transport and deposition in fibrous filtration media was achieved with a novel particle tracking model using a 3D Monte Carlo model. The particle deposition process is governed by the convection–diffusion field. Simulations were performed by considering the fibrous filtration media as an array of identical parallel fibers, in which the flow field was accurately described by an analytical solution. The model of particle movement was described by the random probability distribution characterized by a dimensionless factor, the Peclet number (Pe), based on a convection–diffusive equation of particle transport in fluid. The influence of the particle Peclet number (Pe) on the particle deposition process and the resulting deposition morphology was investigated. The results were analyzed in terms of dust layer growth, particles’ trajectories and dust layer porosity for a vast range of Peclet numbers. The development of distinct deposition morphologies was found by varying the Peclet number (Pe). With a small Peclet number, diffusion dominated deposition and led to the formation of a more open and looser dust layer structure, while with larger Peclet numbers, convection dominated deposition and was found to form compact deposits. According to the change in the location of the packing densities along the dust layer height direction, the dust layer structure could be divided into three typical parts: the substructure, main profile and surface layer. In addition, the deposit morphologies observed for a high Pe were in good agreement with the experimental results found in the literature.

Taylor dispersion of elongated rods at small and large rotational Péclet numbers

This work analyzes the Taylor dispersion of elongated rods in a two-dimensional Poiseuille flow between parallel, rigid walls at small and large rotational Peclet numbers. Asymptotic expressions for the dispersivity and mean advection speed in these limits are derived. The approach could be used to analyze dispersion of rods in other types of unidirectional flows.

Modeling and Numerical Solution of the Fate of Multicomponent Substrate in Porous Media

A mathematical model was proposed to appropriately describe the fate of multicomponent substrates in porous media especially soil. The model utilised appropriate biodegradation kinetic expressions that better describe the consumption or degradation rate of the substrate. The Equation, with the second and third type boundary conditions in non-dimensionalised form was solved using the Finite Volume method and simulated in the Matlab environment. An experiment, using a 5 cm (inside diameter) x 60 cm (height) glass column packed with severally autoclaved soil spiked with 2 % substrate (a mixture of hexadecane, heneicosane, 1-methylnaphthalene, 2- methylnaphthalene and 1, 3-dimethylnapthalene) and a consortium of organisms (Providential rettgeri, Streptococcus salivarius, Trichoderma harzianum, Aspergillus flavipes, and Candida famata) was set up to validate the model. The result showed that the model describes the fate of each component within the multicomponent substrate. It also indicates that both Peclet and Thiele numbers affect the biodegradation of the substrate. It was observed that small Peclet number should be allowed for effective biodegradation of the substrate. The model was validated with data obtained for an experiment where a mixture of hydrocarbons was degraded with a mixed culture of microorganisms. The results of the experiment were well described by the model indicating that the model can be used to predict compositions of components of a mixture during biodegradation.

Analytical and Computational Modeling of Sustained-Release Drug Implants in the Vitreous Humor.

Sustained ocular drug delivery systems are necessary for patients needing regular drug therapy since frequent injection is painful, undesirable, and risky. One type of sustained-release systems includes pellets loaded with the drug, encapsulated in a porous shell that can be injected into the vitreous humor. There the released drug diffuses while the physiological flow of water provides the convective transport. The fluid flow within the vitreous is described by Darcy's equations for the analytical model and Brinkman flow for the computational analysis while the drug transport is given by the classical convection-diffusion equation. Since the timescale for the drug depletion is quite large, for the analytical model, we consider the exterior surrounding the capsule to be quasi-steady and the interior is time dependent. In the vitreous, the fluid-flow process is relatively slow, and meaningful results can be obtained for small Peclet number whereby a perturbation analysis is possible. For an isolated capsule, with approximately uniform flow in the far field around it, the mass-transfer problem requires singular perturbation with inner and outer matching. The computational model, besides accommodating the ocular geometry, allows for a fully time-dependent mass-concentration solution and also admits moderate Peclet numbers. As expected, the release rate diminishes with time as the drug depletion lowers the driving potential. The predictive results are sufficient general for a range of capsule permeability values and are useful for the design of the sustained-release microspheres as to the requisite permeability for specific drugs.

An efficient high-order meshless method for advection-diffusion equations on time-varying irregular domains

We present a high-order radial basis function finite difference (RBF-FD) framework for the solution of advection-diffusion equations on time-varying domains. Our framework is based on a generalization of the recently developed Overlapped RBF-FD method that utilizes a novel automatic procedure for computing RBF-FD weights on stencils in variable-sized regions around stencil centers. This procedure eliminates the overlap parameter δ, thereby enabling tuning-free assembly of RBF-FD differentiation matrices on moving domains. In addition, our framework utilizes a simple and efficient procedure for updating differentiation matrices on moving domains tiled by node sets of time-varying cardinality. Finally, advection-diffusion in time-varying domains is handled through a combination of rapid node set modification, a new high-order semi-Lagrangian method that utilizes the new tuning-free overlapped RBF-FD method, and a high-order time-integration method. The resulting framework has no tuning parameters and has O(Nlog⁡N) time complexity. We demonstrate high-orders of convergence for advection-diffusion equations on time-varying 2D and 3D domains for both small and large Peclet numbers. We also present timings that verify our complexity estimates. Finally, we utilize our method to solve a coupled 3D problem motivated by models of platelet aggregation and coagulation, once again demonstrating high-order convergence rates on a moving domain.

Entropy generation and mechanical efficiency in laminar peristaltic flow through an elliptical duct

AbstractHeat transfer analysis coupled with peristaltic transport is important in many real‐world application areas varying from microchannels to spacecrafts. Power production, chemical, and food industries, electronics, and environmental engineering are some examples of applications. In thermal devices, the overall performance of a heat exchanger depends on heat exchanger efficiency and entropy generation. The main purpose of this paper is to study a mathematical model coupling the peristaltic pumping with the heat transfer phenomenon for an incompressible Newtonian fluid in an elliptical tube. The Navier–Stokes and energy equations have been analytically solved for long wavelength, small Reynolds, and small Peclet numbers approximations. Exact expressions of velocity profile and temperature distribution have been found in the wave frame analysis. The impacts of pertinent parameters on the physical quantities of the problem have been analyzed with the help of graphs. We concluded that the geometrical parameters (occlusion, aspect ratio) enhance the pressure rise and the mechanical efficiency. It should be noted that the best way for minimizing entropy generation is decreasing occlusion, aspect ratio, flow rate, or Brinkman number.

Peristaltic activity for electro-kinetic complex driven cilia transportation through a non-uniform channel

Motivations: Now-a-days in medical science, the transport study of biological fluids through non-uniform vessels are going to increase due to their close relation to the reality. Motivated through such type of complex transportation, the current study is presented of cilia hydro-dynamics of an aqueous electrolytic viscous fluid through a non-uniform channel under an applied axial electric field.Mathematical Formulations: Because of the complexity shape and nature of flow channel, we have used curvilinear coordinates in the derivation of continuity and momentum equationsin a fixed frame of reference. A linear transformation is used to renovate the flow system of equations from fixed (laboratory) to moving (wave) frame. For further simplification, the dimensionless variables are introduced to make the flow system of equations into the dimensionless form and at last convert these equations in term of stream function by using the mathematical terminologies of streamlines. The whole analysis is performed under (low Reynolds number) creeping phenomena and long wavelength approximation, respectively. Additionally, small ionic Peclet number and Debye–Huckel linearization are used to simplify the Nernst–Planck and Poisson–Boltzmann equations. The BVP4C technique is used to obtain the numerical solution for velocity distribution, pressure gradient, pressure rise and stream function through MATLAB.Main Outcomes: The amplitude of velocity distribution is increased (decreased) at larger values of non-uniform parameter (cilia length). The non-uniform parameter played a vital role not only in the enhancement of circulation at the upper half of the channel but also the length of bolus increased. Results of straight channel are gained for larger value of the dimensionless radius of curvature parameter as well as cilia length.

Thermophoresis of a cylindrical particle at small finite Péclet numbers

The thermophoretic motion of a circular cylindrical particle in gaseous media with a temperature gradient perpendicular to its axis is analyzed for a small but nonzero Peclet number P e . The Knuds...

Swimming sheet in a viscosity-stratified fluid

In this work, we theoretically investigate the motion of a Taylor swimming sheet immersed in a viscosity-stratified fluid. The propulsion of the swimmer disturbs the surrounding fluid, which influences the transport of the stratifying agent described by the advection–diffusion equation. We employ a regular perturbation scheme to solve the coupled differential equations of motion up to the second order with the small parameter given by the ratio of the wave amplitude and the wavelength. The expression for the swimming velocity is linear in the magnitude of the viscosity gradient, while depending on the Peclet number in a non-monotonic way. Interestingly, we find that the Peclet number governs the propensity of the sheet to propel towards regions of favourable viscosities. In particular, for small Peclet numbers ( $0<Pe<3$ ), the swimmer prefers regions of low viscosity, while for high Peclet numbers ( $Pe>3$ ), the swimmer prefers regions of high viscosity. Our analysis shows that purely hydrodynamic effects might be responsible for the experimentally observed accumulation of swimmers near favourable viscosity regions. We find that viscosity gradients influence other motility characteristics of the swimmer, such as power expenditure and hydrodynamic efficiency, and provide analytical expressions for both.

Forced Convection Heat Transfer From a Particle at Small and Large Peclet Numbers

Abstract We theoretically study forced convection heat transfer from a single particle in uniform laminar flows. Asymptotic limits of small and large Peclet numbers Pe are considered. For Pe≪1 (diffusion-dominated regime) and a constant heat flux boundary condition on the surface of the particle, we derive a closed-form expression for the heat transfer coefficient that is valid for arbitrary particle shapes and Reynolds numbers, as long as the flow is incompressible. Remarkably, our formula for the average Nusselt number Nu has an identical form to the one obtained by Brenner for a uniform temperature boundary condition (Chem. Eng. Sci., vol. 18, 1963, pp. 109–122). We also present a framework for calculating the average Nu of axisymmetric and two-dimensional (2D) objects with a constant heat flux surface condition in the limits of Pe≫1 and small or moderate Reynolds numbers. Specific results are presented for the heat transfer from spheroidal particles in Stokes flow.

Heat transfer in laminar viscous flow in a channel with one porous wall

Similarity solutions to the two-dimensional steady Navier–Stokes and energy equations for viscous flow in a semi-infinite horizontal channel with upper porous and lower impermeable walls are presented. The finite end of the channel is closed. Both fluid suction and injection through the porous boundary are considered. Two cases of thermal conditions at the walls are studied. In one case the temperature of both channel boundaries decreases linearly along the channel. In the second one, the upper wall has the same decreasing temperature, while the lower boundary is thermally insulated. By means of similarity transformations the dynamic and heat transfer equations are reduced to three ordinary differential equations for the velocity, pressure and temperature functions. The velocity equation is solved numerically for different values of the suction/injection Reynolds number. Analytical solution of the dynamic problem in a series of small Reynolds number is also found. The thermal problem is solved numerically for arbitrary Peclet number and two values of the Prandtl number. Analytical solution for small Peclet and Reynolds numbers is also obtained. The dependence of the Nusselt number on the Peclet number at both walls is studied. The stream lines and isotherms are presented graphically for some values of the Reynolds number at the Prandtl number equal one. It is shown the strong influence of the flow behavior on the heat transfer inside the channel.

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.

Active control of dispersion within a channel with flow and pulsating walls

Channels are fundamental building blocks from biophysics to soft robotics, often used to transport or separate solutes. As solute particles inevitably transverse between streamlines along the channel by molecular diffusion, the effective diffusion of the solute along the channel is enhanced - an effect known as Taylor dispersion. Here, we investigate how the Taylor dispersion effect can be suppressed or enhanced in different settings. Specifically, we study the impact of flow profile and active or pulsating channel walls on Taylor dispersion. We derive closed analytic expressions for the effective dispersion equation in all considered scenarios providing hands-on effective dispersion parameters for a multitude of applications. In particular, we find that active channel walls may lead to three regimes of dispersion: either dispersion decrease by entropic slow down at small Peclet number, or dispersion increase at large Peclet number dominated either by shuttle dispersion or by Taylor dispersion. This improves our understanding of solute transport e.g. in biological active systems such as blood flow and opens a number of possibilities to control solute transport in artificial systems such as soft robotics.

Stability of the Horizontal Throughflow in a Power-Law Fluid Saturated Porous Layer

Double-diffusive convective instability of horizontal throughflow in a power-law fluid saturated porous layer is investigated. The boundaries of this horizontal porous layer are impermeable, isothermal and isosolutal. The generalized Darcy’s equation for power-law fluid is employed in the momentum balance, along with Oberbeck–Boussinesq approximation for buoyancy effect in the system. It is considered that the viscous dissipation is non-negligible. The basic temperature and concentration profiles for the considered horizontal throughflow are determined analytically, and the linear stability analysis is performed on this basic solution. The eigenvalue problem is solved numerically using bvp4c in MATLAB. The conditions for the onset of instability of base flow in terms of critical Rayleigh number and the corresponding wave number are obtained. It is found that critical values are affected by the flow governing parameters such as the Peclet number (Pe), power-law index (n), viscous dissipation (Ge), angle of inclination $$(\alpha )$$ , buoyancy ratio (N) and Lewis number (Le). For the considered range of these parameter values, the instability of the basic flow is discussed for the convective rolls. It is observed that instability of any general convective roll lies between that of the longitudinal and transverse rolls. Also, the effect of viscous dissipation on the onset of instability for both the longitudinal and transverse rolls is discussed in detail. This source of heating inside the porous layer increases the thermal instability of base flow. Further, the asymptotic analysis for vanishingly small Peclet number is presented as $$Pe = 0$$ is a singular point. The boundary between the stationary and the oscillatory instabilities is given using the buoyancy ratio parameter and the Lewis number in this limiting case of $$Pe\rightarrow 0$$ .

Study of Heat and Mass Transfer in Electroosmotic Flow of Third Order Fluid through Peristaltic Microchannels

An analysis is carried out to evaluate the effects of heat and mass transfer in an electro-osmotic flow of third order fluid via peristaltic pumping. Solutions are derived for small wave number and Peclet number. The emerging non-linear mathematical model is solved analytically and compared numerically by the built-in scheme of working software. The table is inserted for shear stress distribution and a graph for comparison of solution techniques and accuracy of obtained results. The effects of various parameters of interest on pumping, trapping, temperature, heat transfer coefficient, and concentration distribution have been studied graphically. Electro-osmotic exchange of energy and mass has a role in reservoir engineering, chemical industry, and in micro-fabrication technologies.