Large Peclet Numbers Research Articles

Fast and Slow Crystallization-driven Convection in White Dwarfs

We investigate crystallization-driven convection in carbon–oxygen white dwarfs. We present a version of the mixing length theory that self-consistently includes the effects of thermal diffusion and composition gradients, and provides solutions for the convective parameters based on the local heat and composition fluxes. Our formulation smoothly transitions between the regimes of fast adiabatic convection at large Peclet number and slow thermohaline convection at low Peclet number. It also allows for both thermally driven and compositionally driven convection, including correctly accounting for the direction of heat transport for compositionally driven convection in a thermally stable background. We use the MESA stellar evolution code to calculate the composition and heat fluxes during crystallization in different models of cooling white dwarfs, and determine the regime of convection and the convective velocity. We find that convection occurs in the regime of slow thermohaline convection during most of the cooling history of the star. However, at the onset of crystallization, the composition flux is large enough to drive fast overturning convection for a short time (∼10 Myr). We estimate the convective velocities in both of these phases and discuss the implications for explaining observed white dwarf magnetic fields with crystallization-driven dynamos.

Development and application of turbulent heat flux model for lead-bismuth eutectic based on deep learning

Existing turbulent heat flux (THF) closure models in Reynolds-averaged Navier–Stokes simulation of scalar transport are insufficient on accuracy for fluids with a low Prandtl number (Pr), such as lead–bismuth eutectic (LBE), especially under complex operating conditions like in heat exchangers of nuclear reactors. This paper aims to propose a THF model suitable for such conditions. Firstly, we established a high-fidelity database of forced convection flow past two tandem cylinders under a Reynolds number (Re) of 1 × 105 and various low Pr conditions by hybrid large eddy simulation and direct numerical simulation algorithm. Based on the database, a deep learning architecture was then built and trained to predict dimensionless THF. Finally, a posteriori test proved that the proposed THF model helps to improve the prediction performance for temperature fields and also performs well when extrapolated to cases of LBE past the tube bundles, especially under the case with large Peclet numbers.

Simulation of heat transfer in Poiseuille pipe flow via generalized finite difference method with a space stepping algorithm

The aim of this paper is to propose an efficient numerical scheme to deal with heat transfer problems in pipe flow with a large length/diameter ratio. The generalized finite difference method (GFDM) is combined with a space stepping algorithm (SSA) for the solution. The SSA divides the solution domain into a number of overlapping sections in the length direction of the pipe. Due to the large Peclet number, the heat transport process is dominated by advection, which allows an approximate boundary condition to be applied at the downstream cross section. The problem is then solved section by section. Using the uniform distributed points, the solution matrix for each section does not change, and only the right hand side needs to be refreshed for the changing boundary conditions. This leads to a highly efficient scheme for problems with a large pipe length. To show the accuracy of the numerical scheme, a problem with available analytical solution is studied. Then the method is applied to problems of single pipe with different pipe wall temperature and flow Reynolds number, and concentric annular pipe. Numerical results confirm the stability and efficiency of the GFDM-SSA. The method can be applied to many real world transient heat transfer problems in which the pipe is heated or cooled along its length with arbitrary wall temperature for heat exchange and other purposes.

Специальная разностная схема для решения жестких краевых задач конвективно-диффузионного переноса

The convective-diffusion transfer equation is often found in problems of hydromechanics and heat and mass transfer. The dominance of convection over diffusion and the change in sign of the coefficient at the first derivative lead to the formation of boundary and internal layers with high gradients of the function. This creates serious difficulties in numerical analysis of the problem using traditional difference schemes. The traditional method of approximating the first derivative using central differences at high Peclet numbers can lead to oscillations and violate the monotonicity of the numerical solution. To avoid this problem, it is necessary to significantly reduce the size of grid cells in narrow areas with large gradients of the unknown function. The use of one-sided differences significantly smears the desired solution, due to the viscosity of the scheme, and leads to loss of accuracy. The practical need to solve stiff boundary value problems requires the development and use of computational technologies that guarantee monotonicity, accuracy, and cost-effectiveness in numerical analysis. In this paper, a new special difference scheme is proposed for the numerical solution of a stiff equation of convective-diffusion transfer. The dominant convective term is eliminated from explicit consideration by transforming the equation into self-adjoined form, which permits the use of well-known numerical approximation techniques. The control volume method is used to construct a difference analogue of a differential equation on a three-point template. The resulting scheme is monotonic and conservative. The test examples show great possibilities of the proposed difference scheme for large Peclet numbers on coarse grids in solving stiff boundary value problems of convective diffusion transfer.

Bioconvection entropy optimized flow of Reiner-Rivlin nanoliquid with motile microorganisms

Present work is concerned with hydromagnetic bioconvective chemically reactive flow of Reiner-Rivlin nanoliquid. Analysis is constructed for entropy generation. Thermal relation consists of Joule heating, radiation and dissipation. Soret effect for chemically reactive flow is examined. Nonlinear governing systems after adequate transformations are solved by ND-solve technique. Graphical analysis for velocity, microorganism field, entropy rate, thermal field and concentration is analyzed. Graphical description for coefficient of skin friction, microorganism density number, heat transport rate and concentration gradient are studied. Entropy rate increase is detected for higher magnetic parameter while reverse trend holds for velocity. Clearly temperature augments against thermophoresis and radiation. Radiation intensifies entropy rate and heat transport rate. Similar impact of drag force and temperature for magnetic parameter is noticed. Concentration and mass transport rate rise against higher Soret number. Brownian motion variation results in concentration decays. Large Peclet number leads to decrease of microorganism field whereas it intensifies the microorganism density number.

Tracer dispersion due to non-Newtonian fluid flows in hydraulic fractures with different geometries and porous walls

Tracer transport in hydraulic fractures is substantially affected by the rock matrix surrounding them. Nevertheless, there is not yet an expression in literature to address the relationship among the transport coefficients, porous walls, hydraulic fracture geometries, and non-Newtonian fluid properties. Therefore, the tracer dispersion due to non-Newtonian fluid flows in hydraulic fractures with different geometries and porous walls is mathematically derived and studied in the current work. Rectangular, triangular, and elliptical models and power-law model are considered here to describe the hydraulic fracture geometry and the non-Newtonian fluid rheology, respectively. The results reveal that as the flow behavior index grows, the coefficient of the shear dispersion term in the case of the shear thinning and Newtonian fluids follows an order of triangular > elliptical > rectangular and in the case of the shear thickening while the same order is followed at the beginning, it turns first to triangular > rectangular > elliptical, then to rectangular > triangular > elliptical, and finally to rectangular > elliptical > triangular. The coefficient of the shear dispersion term increases sharply for each geometrical model with the flow behavior index when the fluid is shear thinning while it first increases slightly and then flattens when the fluid is shear thickening. However, the coefficient of the shear dispersion term for porous walls is lower than the one for nonporous walls. It is also found that the average tracer dispersion coefficients in rectangular, triangular, and elliptical nonporous-walled and porous-walled hydraulic fractures and their ratio dictate that the hydraulic fracture-rock matrix communication through the continuity of the tracer concentration and the mass flux needs to be considered for determination of the tracer dispersion during the transition for intermediate Peclet numbers and the advection-dominated tracer transport for large Peclet numbers. This study along with its findings can pave the way for future investigations on the tracer dispersion in a network of hydraulic fractures with rough and porous walls.

Forchheimer–Bénard Instability of the Non-Newtonian Fluid

The present paper examines the effect of vertical throughflow on the onset of convective instability in a horizontal porous layer filled with a non-Newtonian power-law fluid (PL). The permeable boundary layers are exposed to two different uniform constant temperature conditions, The Oberbeck-Boussinesq hypothesis is considered with the Darcy-Forchheimer model. A fourth-order eigenvalue problem is stemmed from the performance of the linear stability analysis, and the critical values are obtained using the shooting method combined with the Runge-Kutta method. The non-Newtonian Darcy-Rayleigh number (R), the Péclet number (Pe), the Forchheimer number (G), and the power-law index (n) are the parameters whose value play a crucial role in the onset of instability. The finding shows more stabilizing effects arise in pseudoplastic fluid than dilatant one at Peclet number Pe << 1 where the inverse behaviour takes place at large Peclet number even with the existence of the drag number or without it.

Mathematical model of process of sedimentation of multicomponent suspension on the bottom and changes in the composition of bottom materials

The paper considers 2D and 3D models of transport of suspended particles, taking into account the following factors: movement of aqueous medium; variable density depending on the suspension concentration; multicomponent character of suspension; changes in bottom geometry as a result of suspension sedimentation. The approximation of the three-dimensional diffusion-convection equation is based on splitting schemes into two-dimensional and one-dimensional problems. In this work, we use discrete analogues of convective and diffusion transfer operators in the case of partial cell occupancy. The geometry of the computational domain is described based on the occupancy function. The difference scheme used is a linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients obtained by minimizing the approximation error. This scheme is designed to solve the problem of impurity transfer at large grid Peclet numbers. Based on the results of numerical experiments, conclusions are drawn about the advantage of the 3D model of multicomponent suspension transport in comparison with the 2D model. Computational experiments have been performed to simulate the process of sedimentation of a multicomponent suspension, as well as its effect on the bottom topography and changes in its composition.

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.

Applicability of the Time Fractional Derivative Model on Fickian and Non-Fickian Transport in the Single Fracture: An Experimental Investigation

The understanding of solute transport in rock fractures is of great importance in many engineering fields. In this study, two groups of experiments on artificial and natural single fractures with different fracture apertures and roughness were conducted to investigate the changes of solute transport regimes. The time fractional advection–dispersion equation (tFADE) as a promising model was applied to describe the anomalous transport. The performance of the classical advection-dispersion equation (ADE) and tFADE was compared according to the fitting precision of breakthrough curves (BTCs). The responses of the fitting parameters in the tFADE to the experimental conditions were also discussed. The results indicated that the non-Fickian transport more likely occurred in the short distance transport, and the larger Peclet number ( Pe ) led to the increase of the exponent of the power-law function in the phase of concentration decline. The tFADE was superior to ADE in capturing the non-Fickian transport especially the tailing behavior. The fractional order of time α in the tFADE was the key parameter to describe the anomalous transport process, and its responding mechanisms of were revealed: the best-fit α decreased with the increase of flow velocity and the decrease of the fracture aperture. The roughness of the single fracture which leads to a complex flow field had a significant effect on the best-fit α . The findings of this study can help for better understanding the effectiveness and physical significance of the tFADE.

An Analytical Heat Transfer Model in Oil Reservoir during Long-Term Production

Contrary to the assumption of previous researchers, the radial temperature in the petroleum reservoir during production is non-isothermal because several heat transfer mechanisms change the radial temperature in reservoirs. As there have been few studies, especially after long-term production, this work derives steady-state analytic solutions considering the long-term production. It also presents sensitivity analysis with the various production conditions to investigate heat transfer between the producing fluids and surrounding formations during fluids flowing (hereafter, system heat transfer) in a steady-state. For oil production, the system heat transfer induces a cooling effect on the radial temperature in the reservoir, reducing the temperature rise due to the Joule–Thomson (J–T) heating. This cooling effect increases with the larger Peclet number, however, the relative contribution of the cooling effect to the radial temperature change diminishes. The equations explain that the cooling effect is proportional to the temperature increase due to J–T heating. With a larger permeability, a more convective-dominant phase causes more heat transfer actively. Although the cooling effect itself is amplified with the larger permeability, its relative contribution to the temperature change decreases. From the analysis, the cooling effect of system heat transfer is significant in the low-permeability reservoirs with large drawdown. The system heat transfer is confirmed to be an essential factor in measuring the accurate productivity index of unconventional reservoirs.

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.

External mass transfer from/to a single sphere in a nonlinear uniaxial extensional creeping flow

This work systematically simulates the external mass transfer from/to a spherical drop and solid particle suspended in a nonlinear uniaxial extensional creeping flow. The mass transfer problem is governed by three dimensionless parameters: the viscosity ratio ( λ ), the Peclet number ( Pe ), and the nonlinear intensity of the flow ( E ). The existing mass transfer theory, valid for very large Peclet numbers only, is expanded, by numerical simulations, to include a much larger range of Peclet numbers (1 ≤ Pe ≤ 10 5 ). The simulation results show that the dimensionless mass transfer rate, expressed as the Sherwood number ( Sh ), agrees well with the theoretical results at the convection-dominated regime ( P e > 10 3 ). Only when E > 5/4, the simulated Sh for a solid sphere in the nonlinear uniaxial extensional flow is larger than theoretical results because the theory neglects the effect of the vortex formed outside the particle on the rate of mass transfer. Empirical correlations are proposed to predict the influence of the dimensionless governing parameters ( λ , Pe , E ) on the Sherwood number ( Sh ). The maximum deviations of all empirical correlations are less than 15% when compared to the numerical simulated results.

Taylor dispersion of elongated rods

Particles transported in fluid flows, such as cells, polymers, or nanorods, are rarely spherical. In this study, we numerically and theoretically investigate the dispersion of an initially localized patch of passive elongated Brownian particles constrained to one degree of rotational freedom in a two-dimensional Poiseuille flow, demonstrating that elongated particles exhibit an enhanced longitudinal dispersion. In a shear flow, the rods translate due to advection and diffusion and rotate due to rotational diffusion and their classical Jeffery's orbit. The magnitude of the enhanced dispersion depends on the particle's aspect ratio and the relative importance of its shear-induced rotational advection and rotational diffusivity. When rotational diffusion dominates, we recover the classical Taylor dispersion result for the longitudinal spreading rate using an orientationally averaged translational diffusivity for the rods. However, in the high-shear limit, the rods tend to align with the flow and ultimately disperse more due to their anisotropic diffusivities. Results from our Monte Carlo simulations of the particle dispersion are captured remarkably well by a simple theory inspired by Taylor's original work. For long times and large Peclet numbers, an effective one-dimensional transport equation is derived with integral expressions for the particles' longitudinal transport speed and dispersion coefficient. The enhanced dispersion coefficient can be collapsed along a single curve for particles of high aspect ratio, representing a simple correction factor that extends Taylor's original prediction to elongated particles.

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.

Mass transfer around a double emulsion droplet in a uniform flow

The transfer of mass around a spherical double emulsion droplet in a uniform creeping flow and at large Peclet numbers is the subject of this theoretical communication. The double emulsion droplet, submerged in an external fluid (fluid 1), is made from an internal spherical drop (fluid 3) covered by a spherical fluid shell (fluid 2). Four dimensionless numbers describe this situation: two viscosity ratios, shell over external (λ21) and internal over shell (λ32); the radii ratio, inner over outer (κ); and the Peclet number (Pe). The external mass transfer rate of a double emulsion having a fast external mobile surface is proportional to Pe1∕2, it increases as λ21 or κ decrease, and for a very small internal drop or a very thick shell (κ→ 0), it behaves like a single drop. On the other hand, the external rate of mass transfer of a double emulsion with a very slow external mobile surface is proportional to Pe1∕3, it increases as λ21 or κ increase, and for a very large internal drop or a very thin shell (κ→ 1), it mimics a single solid particle. The influence of the other viscosity ratio (λ32) on the external mass transfer can be ignored.

Bioconvection mechanism using third-grade nanofluid flow with Cattaneo–Christov heat flux model and Arrhenius kinetics

This paper aims to study the effects of activation energy and thermal radiation in the bioconvection flow of nanofluid (third-grade nanofluid) containing swimming microorganisms in the presence of a heat source-sink past a stretching sheet. Brownian movement and thermophoresis diffusion are used in mathematical modeling. The given flow phenomenon is modeled in the form of governing partial differential equations. Furthermore, appropriate dimensionless transformation is used to transfer the governing system of PDEs into an ordinary one. The remodeled systems of ODEs are tackled numerically by bvp4c on Matlab with a shooting scheme in computational tool MATLAB. The bearing of prominently involved parameters on the numerical solution of velocity, temperature distribution, nanoparticles concentration and concentration of microorganisms is comprehensively discussed and elaborated through figures. It is established that velocity can be improved with a mixed convection aspect. Furthermore, the temperature and concentration of nanoparticles reduce against Prandtl number, also, large Peclet number declines the microorganisms field. The work contained in this paper has applications in nanotechnology, electrical and mechanical engineering, biomedicine, biotechnology, drug delivery, cancer treatment, food processing and various industries. No such work is yet reported, and it is good for the research in applied sciences.

Heat transport in shear-driven flow with axial conduction

The classical Graetz problem of fluid entering a long channel possessing different temperature from that of fluid is extended here to the case where the upper wall of the channel is not stationary. Formulation of the current problem incorporates the simple Couette flow driven by the channel movement and taking into account the axial heat conduction, besides the usual convection. Analytical solutions in the thermally developing regime regarding the temperature distribution as well as the heat transfer rates from both the upper moving and the lower fixed channel walls are then obtained by means of the series expansion method making use of the proper orthogonal family of eigenfunctions. Simple approximate formulae for the rates of heat transfer at the channel entrance valid for large Peclet numbers and at the far downstream locations valid for all Peclet numbers are further derived. More simplistic outcomes are also provided when the axial pressure gradient is assumed to be constant. Results are then discussed from low to high Peclet numbers, which are in compliant with the well-known results available in the literature for the channel flow within stationary walls. On the other hand, the rate of heat transfer from the moving wall is shown to be of high importance when Peclet number increases, which requires special attention in technological applications.

Suspended particle transport process modeling based on 2D and 3D models

In this paper, suspended particles transport mathematical model is proposed and investigated. The considered model includes the following factors: the aquatic environment movement; variable density depending on the suspension concentration; multicomponent suspension; change in the bottom geometry as suspension sedimentation result. The 3D diffusion-convection equation approximation based on splitting schemes into 2D and one-dimensional problems. In the paper, convective and diffusive transfer operators’ discrete analogs in the case of partial fullness of the cells are used. Based on the fullness function, the calculated area geometry is described. The difference scheme, which is linear combination of the Upwind and Standard Leapfrog difference schemes with weight coefficients obtained from the condition of minimizing the approximation error, is used. This scheme is designed to solve the impurity transfer problem at large grid Peclet numbers. Based on the results of numerical experiments, conclusions are drawn about the advantages of the 3D model of suspended particle transport over the 2D model. The results of numerical experiments on modeling the multicomponent suspension deposition are presented.

Mixing in a sheared particulate suspension

Abstract