A numerical method for steady and nonisothermal viscoelastic fluid flow for high Deborah and Péclet numbers

A combined finite element/streamline integration method is presented for nonisothermal flows of viscoelastic fluids. The attention is focused on some characteristic problems that arise for numerical simulation of flows with high Deborah and Péclet numbers. The two most important problems to handle are the choice of an outflow boundary condition for not completely developed flow and the treatment of the dissipative term in the temperature equation. The ability of the numerical method to handle high Deborah and Péclet numbers will be demonstrated on a contraction flow of an LDPE melt with isotropic and anisotropic heat conductivity. The influence of anisotropic heat conduction and the difference between the stress work and mechanical dissipation will be discussed for contraction flows. Received:  4 February 1997 Accepted: 23 October 1997     

Closed streamline flows past small rotating particles: Heat transfer at high péclet numbers

A heat transfer problem is solved, first for an infinitely long heated cylinder and then for a small heated sphere, each freely suspended in a general linear flow at Reynolds numbers $\text{Re ? 1}$. Asymptotic solutions to the convection problem are developed for very large values of the Péclet number $\text{Pe}$, and expressions are obtained for the asymptotic Nusselt number for two-dimensional flows ranging from solid body rotation to hyperbolic flow. Since the objects in these cases are surrounded by a region of effectively isothermal closed streamlines, the asymptotic Nusselt number becomes independent of the Péclet number in the limit $\text{Pe → ∞}$.     

Buoyancy-Opposed Darcy’s Flow in a Vertical Circular Duct with Uniform Wall Heat Flux: A Stability Analysis

An analytical and numerical study is presented to show that buoyancy-opposed mixed convection in a vertical porous duct with circular cross-section is unstable. The duct wall is assumed to be impermeable and subject to a uniform heat flux. A stationary and parallel Darcy’s flow with a non-uniform radial velocity profile is taken as a basic state. Stability to small-amplitude perturbations is investigated by adopting the method of normal modes. It is proved that buoyancy-opposed mixed convection is linearly unstable, for every value of the Darcy–Rayleigh number, associated with the wall heat flux, and for every mass flow rate parametrised by the Péclet number. Axially invariant perturbation modes and general three-dimensional modes are investigated. The stability analysis of the former modes is carried out analytically, while general three-dimensional modes are studied numerically. An asymptotic analytical solution is found, suitable for three-dimensional modes with sufficiently small wave number and/or Péclet number. The general conclusion is that the onset of instability selects the axially invariant modes. Among them, the radially invariant and azimuthally invariant mode turns out to be the most unstable for all possible buoyancy-opposed flows.     

A 1D-Averaged Model for Stable and Unstable Miscible Flows in Porous Media with Varying Péclet Numbers and Aspect Ratios

This paper deals with reducing the number of spatial dimensions of the models used to solve stable and unstable miscible flows in saturated and homogeneous porous media. Unstable miscible displacements occur when a fluid displaces another fluid of higher viscosity, with which it can fully mix. Stable flows occur if the displaced fluid is less viscous than the displacing one. First, a 1D-averaged model is identified, capable of accurately describing unstable flows at high Péclet numbers. Second, another 1D-averaged model is determined, capable of accurately predicting miscible displacements at low Péclet numbers. Third, a new model is proposed, for any Péclet number and for both stable and unstable flows, as a combination of the previous two models. This combination involves three parameters whose values depend on the dimensionless numbers of the problem, namely, the viscosity ratio M, the Péclet number P_e, the aspect ratio A, and the dispersion length ratio ε. These parameters are computed for several values of M, P_e, A with ε=1 by matching results from direct 2D simulations, obtained from a numerical model previously validated against experimental data. It is found that a 1D-averaged model combining an extended version of the Todd–Longstaff model and the diffusive term of the 1D-miscible model forms an accurate general model for miscible displacements in homogeneous porous media. This paper also provides a large set of data computed from high-resolution 2D simulations of unstable miscible displacements.     

Effect of motion of the medium on the photophoresis of hot hydrosol particles

The photophoretic motion of a solid spherical particle in a viscous fluid is described theoretically in the Stokes approximation for small Péclet and Reynolds numbers and large temperature differences near the particle. In solving the hydrodynamic equations, an exponential-power law is used for the temperature dependence of the viscosity. The heat transfer equations are solved using the method of matched asymptotic expansions. The possibility of the experimental observation of photophoresis in liquids is discussed.     

Instability Analysis of Gyrotactic Microorganisms: A Combined Effect of High-Frequency Vertical Vibration and Porous Media

A linear stability analysis is carried out to predict the instability analysis in a dilute suspension of gyrotactic microorganisms in horizontal fluid-saturated porous layer influenced by high-frequency vertical vibration. The governing equations, describing the mean flow, are the time-averaged Boussinesq equations and the analytical solution of the problem has been obtained using Galerkin method. A secular relation involving bioconvection Rayleigh number and its vibrational analogs and other parameters have been established. The graphical interpretations for dependence of bioconvection Rayleigh–Darcy number and corresponding wave number, on gyrotactic number and bioconvection Péclet number in the presence of vibration are utilized to understand the problem.     

Mass transfer of aerosols with axial diffusion in narrow rectangular channels

The problem of mass transfer of aerosols with axial, as well as radial, diffusion in laminar flow in a narrow rectangular channel is studied. Two cases are investigated. The first case is where all particles enter the channel inlet and none form within the channel; and the second, where no particles enter the channel, and “formation in flight” occurs within the channel. For each case, analyses are made for both slug and Poiseuille flows. The first twenty modes of the eigenvalues, the eigenfunctions, and the coefficients of series expansion are obtained for several diffusion Péclet numbers, Pe. The first twelve of them are presented for Pe=1, 5, 10, 100, and ∞. Asymptotic expressions for the eigenvalues and the eigenfunctions are also given. The effects of axial diffusion on the local particle concentration, the bulk concentration, the Sherwood number, and the fraction of aerosols arriving at any cross-section of the channel are studied for various diffusion Péclet numbers. It was found that, for diffusion with or without formation in flight, the effect of axial diffusion may be neglected at an axial distance from the channel inlet greater than one and a half times that of the channel height for 1<Pe<100.     

Rigorous upscaling of the reactive flow with finite kinetics and under dominant Péclet number

We consider the evolution of a reactive soluble substance introduced into the Poiseuille flow in a slit channel. The reactive transport happens in presence of dominant Péclet and Damköhler numbers. We suppose Péclet numbers corresponding to Taylor’s dispersion regime. The two main results of the paper are the following. First, using the anisotropic perturbation technique, we derive rigorously an effective model for the enhanced diffusion. It contains memory effects and contributions to the effective diffusion and effective advection velocity, due to the flow and chemistry reaction regime. Error estimates for the approximation of the physical solution by the upscaled one are presented in the energy norms. Presence of an initial time boundary layer allows only a global error estimate in L ² with respect to space and time. We use the Laplace’s transform in time to get optimal estimates. Second, we explicit the retardation and memory effects of the adsorption/desorption reactions on the dispersive characteristics and show their importance. The chemistry influences directly the characteristic diffusion width.     

Thermal Instability in a Horizontal Porous Channel with Horizontal Through Flow and Symmetric Wall Heat Fluxes

The linear thermoconvective instability of the basic parallel flow in a plane and horizontal porous channel is investigated. The boundary walls are assumed to be impermeable and subject to symmetric and uniform heat fluxes. The wall heat fluxes produce either a net heating or a net cooling of the fluid saturated porous medium. A horizontal mass flow rate is externally impressed leading to a stationary basic state with a temperature gradient inclined to the vertical. A region of possibly unstable thermal stratification exists either in the lower half-channel (boundary heating), or in the upper half-channel (boundary cooling). The convective instability of the basic flow is governed by the Rayleigh number and by the Péclet number. In the case of boundary heating, the thermal instability arises when the Rayleigh number exceeds its critical value, that depends on the Péclet number. The change of the critical Rayleigh number as a function of the Péclet number is determined numerically for arbitrary normal modes oblique to the basic flow direction. The most dangerous modes are the longitudinal rolls, with a wave vector perpendicular to the basic velocity. There exists a minimum value of the Péclet number, 19.1971, below which no linear instability is detected.     

On the Finite Increment Calculus method for stabilizing advection‐diffusion equations,analysis and computation of the stabilization parameter

In this work we are concerned with the finite increment calculus (FIC) method. The method has been developed for efficient approximation of advection‐diffusion equations with high Péclet numbers. Since the natural application of FIC is within the framework of the FEM, we consider the BVP in a weak sense on finite dimensional spaces. Here we provide a result on existence and uniqueness of the solution as well as an error analysis. Also we propose a choice of the stabilization parameter. We test the method on some troublesome 2D problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Studying the Accuracy and Applicability of the Finite Difference Scheme for Solving the Diffusion–Convection Problem at Large Grid Péclet Numbers

Journal of Applied Mechanics and Technical Physics - The work is devoted to studying a finite difference scheme for solving the diffusion–convection problem at large grid Péclet numbers....     

Heat transfer by Hagen-Poiseuille flow in the thermal development region with axial conduction

The heat transfer in the region of circular pipes close to the beginning of the heating section is investigated for low-Péclet-number flows with fully developed laminar velocity profile. Axial heat conduction is included and its effect on the temperature distribution is studied not only for the region downstream of the start of heating but also for that upstream. The energy equation is solved numerically by a finite difference method. Results are presented graphically for various Péclet numbers between 1 and 50. The boundary conditions are uniform wall temperature and uniform wall heat flux with step change at a certain cross-section. For the latter case, also some results for the region near the end of the heating section are reported. The solutions are applicable for the corresponding mass transfer situations where axial diffusion is important if the temperature is replaced by the concentration andPe byReSc.     

Stationary Transverse Diffusion in a Horizontal Porous Medium Boundary-Layer Flow with Concentration-Dependent Fluid Density and Viscosity

Stable transport of high-concentrated solute is considered in horizontal boundary-layer flows above a wall of constant concentration. Mixing is accomplished by advection and molecular diffusion only. The utilized boundary-layer approximation allows to investigate the exclusive influence of gravity on vertical diffusion. The hydrodynamic dispersion mechanism was disregarded in the present study which confines its applicabilty to flows with small molecular Péclet numbers. A linear variability of both the fluid's density and viscosity with changing concentration is taken into account as well as the complete set of mass-fraction based balance equations. Steady-state concentration and velocity distributions above the horizontal wall have been obtained using the series truncation method which recently had proven successful to solve the corresponding problem using the Boussinesq assumption. The impact of the latter on these distributions is discussed by what has been additionally-facilitated by the existence of an exact analytical solution for the simpler Boussinesq case. Whereas no density variability influence exists with use of the Boussinesq assumption the complete system of mass-fraction based equations predicts opposing effects of density and viscosity differences between oncoming and near-wall fluids on concentration distributions. Larger density differences narrow the transition zone between both fluids, larger viscosity differences widen it. Thus, a compensation of both effects can be observed for individual fluids and for certain regions of the flow field.     

Laminar mixed convection on a horizontal plate of finite length in a channel of finite width

The steady two-dimensional laminar mixed-convection flow past a horizontal plate of finite length is analysed for large Péclet numbers, small Prandtl numbers and weak buoyancy effects. The plate is placed in a channel of finite width, with the plane walls of the channel being parallel to the plate. The temperature of the plate is assumed to be constant. The hydrostatic pressure difference across the wake behind the plate is compensated by a perturbation of the inviscid channel flow. This outer flow perturbation affects the temperature distribution in the thermal boundary layer at the plate and the heat transfer rate, respectively. Solutions in closed form are given. The forces acting on the plate due to the potential flow perturbation are also determined.     

Instability of Hadley–Prats Flow with Viscous Heating in a Horizontal Porous Layer

The onset of convective rolls instability in a horizontal porous layer subject to a basic temperature gradient inclined with respect to gravity is investigated. The basic velocity has a linear profile with a non-vanishing mass flow rate, i.e., it is the superposition of a Hadley-type flow and a uniform flow. The influence of the viscous heating contribution on the critical conditions for the onset of the instability is assessed. There are four governing parameters: a horizontal Rayleigh number and a vertical Rayleigh number defining the intensity of the inclined temperature gradient, a Péclet number associated with the basic horizontal flow rate, and a Gebhart number associated with the viscous dissipation effect. The critical wave number and the critical vertical Rayleigh number are evaluated for assigned values of the horizontal Rayleigh number, of the Péclet number, and of the Gebhart number. The linear stability analysis is performed with reference either to transverse or to longitudinal roll disturbances. It is shown that generally the longitudinal rolls represent the preferred mode of instability.     

Mathematical model of hydrate formation from microbial methane in marine sediments

A mathematical model of gas hydrate formation from microbial methane in marine sediments is proposed. An analytical solution of the problem is obtained, determining dimensionless parameters are distinguished, and numerical investigation is performed. In the region of main parameters a critical diagram of hydrate existence is plotted. It is shown that at small sediment accumulation rates microbial methane disperses due to diffusion of dissolved gas which does not reach saturation. If the sediment accumulation is intense, the region of possible hydrate formation falls within the region of stable thermodynamic states only at large depths. Correspondingly, in these cases, the probability of formation of hydrate-containing sediment layers is small. The most probable hydrate formation regime is realized at moderate sediment accumulation rates corresponding to Péclet numbers of the order of unity.     

Influence of the thermal boundary layer on the contact layer between a liquid and a cold plate in a solidification process

In this paper, the influence of both the hydrodynamic and the thermal boundary layer on the solidification process of the flowing liquid on a cold plate is theoretically analyzed. Heat transfer between a frozen layer which is created and a laminar flowing liquid over that layer is considered. The development of the boundary layers and the relation between them on the solidification process are studied. An integral method for the solution of the boundary layer equations was used to obtain approximative solutions. The influence of the Prandtl and Reynolds number on the formation of the solid crust is shown and discussed for time dependent and steady-state solutions.     

Numerical Investigation of Supersonic Flows Around Hyperelliptic Cones

The results of numerical simulation of supersonic flows around hyperelliptic cones with different cross-sections are presented. For solving the problem within the inviscid gas model, the finite volume method based on an integral approximation of the Euler equations is used. The steady-state solution is found using the saturation method. The flow pattern is studied and it is shown that bodies with integral geometric characteristics (midsection area, volume, etc.) similar to those of the elliptic cone but with a more uniform flowfield over most of the lower surface can be constructed.     

Operating regimes of a chemical reactor with longitudinal and transverse mixing in the case of large Péclet numbers

A two-dimensional model of a chemical reactor with longitudinal and transverse mixing is investigated in the case of large Péclet numbers calculated from the effective thermal conductivity in the transverse direction. For this model the existence of at least one steady-state regime has been demonstrated [1], sufficient criteria of its uniqueness have been determined, an asymptotic expansion of the solution has been constructed in the case of small Péclet numbers, and the critical ignition and quenching parameters have been found. In this paper the other limiting case of the model, in which heat is propagated in the transverse direction much more slowly than it is transported by the flow along the reactor (large Péclet numbers), is analyzed in detail. An asymptotic expansion of the solution which closely coincides with the data of numerical calculations is constructed. The critical quenching and ignition conditions of the process are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 120–127, January–February, 1987.     

Detection and Quantification of Unswept Oil Pockets

Unswept oil pockets can be detected and possibly quantified by the natural tracers which diffuse into the flowing water and are transported to the wells. This situation is schematized and addressed mostly by numerical means though an elementary analytical solution could be derived. A dimensional analysis was performed and the parameters listed. A parametric study was made in two dimensions on two particular configurations. A particular emphasis was put on the study of the Péclet number and of the pocket size; the numerical solutions were successfully compared to the analytical one. Other parameters such as the presence of several pockets, their shape, were briefly addressed.