Convective diffusion to a slowly rotating spherical electrode; Basic model for Re → O, Pe → ∞

1983 ◽  
Vol 48 (6) ◽  
pp. 1571-1578 ◽  
Author(s):  
Ondřej Wein
Keyword(s):  
Boundary Layer ◽  
Flow Regime ◽  
Peclet Number ◽  
Convective Diffusion ◽  
Creeping Flow ◽  
Péclet Number ◽  
Spherical Electrode ◽  
Basic Model ◽  
Boundary Layer Solution ◽  
Layer Solution

Theory has been formulated of a convective rotating spherical electrode in the creeping flow regime (Re → 0). The currently available boundary layer solution for Pe → ∞ has been confronted with an improved similarity description applicable in the whole range of the Peclet number.

Convective diffusion toward the sphere in laminar flow around the sphere

1979 ◽  
Vol 44 (4) ◽  
pp. 1218-1238
Author(s):  
Arnošt Kimla ◽  
Jiří Míčka
Keyword(s):  
Laminar Flow ◽  
Velocity Profile ◽  
Peclet Number ◽  
Convective Diffusion ◽  
Péclet Number ◽  
Stability Of The Solution

The problem of convective diffusion toward the sphere in laminar flow around the sphere is solved by combination of the analytical and net methods for the region of Peclet number λ ≥ 1. The problem was also studied for very small values λ. Stability of the solution has been proved in relation to changes of the velocity profile.

NUMERICAL STUDY OF MIXED BIOCONVECTION IN POROUS MEDIA SATURATED WITH NANOFLUID CONTAINING OXYTACTIC MICROORGANISMS

2013 ◽  
Vol 13 (04) ◽  
pp. 1350067 ◽  
Author(s):  
O. ANWAR BÉG ◽  
V. R. PRASAD ◽  
B. VASU
Keyword(s):  
Boundary Layer ◽  
Peclet Number ◽  
Mechanical Design ◽  
Numerical Study ◽  
Lewis Number ◽  
Nanoparticle Concentration ◽  
Péclet Number ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Oxytactic Microorganisms

A mathematical model has been developed for steady-state boundary layer flow of a nanofluid past an impermeable vertical flat wall in a porous medium saturated with a water-based dilute nanofluid containing oxytactic microorganisms. The nanoparticles were distributed sufficiently to permit bioconvection. The product of chemotaxis constant and maximum cell swimming speed was assumed invariant. Using appropriate transformations, the partial differential conservation equations were non-dimensionalised to yield a quartet of coupled, non-linear ordinary differential equations for momentum, energy, nanoparticle concentration and dimensionless motile microorganism density, with appropriate boundary conditions. The dominant parameters emerging in the normalised model included the bioconvection Lewis number, bioconvection Peclet number, Lewis number, buoyancy ratio parameter, Brownian motion parameter, thermophoresis parameter, local Darcy-Rayleigh number and the local Peclet number. An implicit numerical solution to the well-posed two-point non-linear boundary value problem is developed using the well-tested and highly efficient Keller box method. Computations are validated with the Nakamura tridiagonal implicit finite difference method, demonstrating excellent agreement. Nanoparticle concentration and temperature were found to be generally enhanced through the boundary layer with increasing bioconvection Lewis number, whereas dimensionless motile microorganism density was only increased closer to the wall. Temperature, nanoparticle concentration and dimensionless motile microorganism density were all greatly increased with a rise in Peclet number. Temperature and dimensionless motile microorganism density were reduced with increasing buoyancy parameter, whereas nanoparticle concentration was increased. The present study found applications in the fluid mechanical design of microbial fuel cell and bioconvection nanotechnological devices.

The solution of heat-transfer problems by the Wiener—Hopf technique II. Trailing edge of a hot film

1974 ◽  
Vol 337 (1610) ◽  
pp. 395-412 ◽  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Half Plane ◽  
Dimensionless Temperature ◽  
Fluid Temperature ◽  
Trailing Edge ◽  
Boundary Layer Solution ◽  
Leading Term ◽  
Hot Film ◽  
Layer Solution

An incompressible fluid of constant thermal diffusivity k , flows with velocity u = Sy in the x -direction, where S is a scaling factor for the velocity gradient at the wall y = 0. The region — L ≤ x ≤ 0 is occupied by a heated film of temperature T 1 , the rest of the wall being insulated. Far from the film the fluid temperature is T 0 < T 1 . The finite heated film is approximated by a semi-infinite half-plane x < 0 by assuming that the boundary-layer solution is valid somewhere on the finite region upstream of the trailing edge. Exact solutions in terms of Fourier inverse integrals are obtained by using the Wiener-Hopf technique for the dimensionless temperature distribution on the half-plane x > 0 and the heat transfer from the heated film. An asymptotic expansion is made in inverse powers of x and the coefficient of the leading term is used to calculate the exact value of the total heat-transfer as a function of the length L . It is shown that the boundary layer solution differs from the exact solution by a term of order L -1/3 for large L . An expansion in powers of x for the heat transfer upstream of the trailing edge is also found. Application of the theory, together with that of Springer & Pedley (1973), to hot films used in experiments are discussed for the range of values of L(S/K) ½ , up to 20.

scholarly journals The dynamics of miscible viscous fingering from onset to shutdown

2018 ◽  
Vol 837 ◽  
pp. 520-545 ◽  
Author(s):  
Japinder S. Nijjer ◽  
Duncan R. Hewitt ◽  
Jerome A. Neufeld
Keyword(s):  
Numerical Simulations ◽  
Flow Regime ◽  
Peclet Number ◽  
Viscous Fingering ◽  
Linear Stability Theory ◽  
Late Time ◽  
Exchange Flow ◽  
Mixing Rate ◽  
Péclet Number ◽  
Single Finger

We examine the full ‘life cycle’ of miscible viscous fingering from onset to shutdown with the aid of high-resolution numerical simulations. We study the injection of one fluid into a planar two-dimensional porous medium containing another, more viscous fluid. We find that the dynamics are distinguished by three regimes: an early-time linearly unstable regime, an intermediate-time nonlinear regime and a late-time single-finger exchange-flow regime. In the first regime, the flow can be linearly unstable to perturbations that grow exponentially. We identify, using linear stability theory and numerical simulations, a critical Péclet number below which the flow remains stable for all times. In the second regime, the flow is dominated by the nonlinear coalescence of fingers which form a mixing zone in which we observe that the convective mixing rate, characterized by a convective Nusselt number, exhibits power-law growth. In this second regime we derive a model for the transversely averaged concentration which shows good agreement with our numerical experiments and extends previous empirical models. Finally, we identify a new final exchange-flow regime in which a pair of counter-propagating diffusive fingers slow exponentially. We derive an analytic solution for this single-finger state which agrees well with numerical simulations. We demonstrate that the flow always evolves to this regime, irrespective of the viscosity ratio and Péclet number, in contrast to previous suggestions.

Boundary-layer solution of the diffusion equation with shrinking

1988 ◽  
Vol 37 (2) ◽  
pp. 79-83 ◽  
Author(s):  
Pascual Viollaz ◽  
Constantino Suarez
Keyword(s):  
Boundary Layer ◽  
Diffusion Equation ◽  
Boundary Layer Solution ◽  
Layer Solution

Characterization of stationary mixing patterns in a three-dimensional open Stokes flow: spectral properties, localization and mixing regimes

2009 ◽  
Vol 639 ◽  
pp. 291-341 ◽  
Author(s):  
M. GIONA ◽  
S. CERBELLI ◽  
F. GAROFALO
Keyword(s):  
Finite Length ◽  
Peclet Number ◽  
Scaling Laws ◽  
Eigenvalue Problems ◽  
Operating Conditions ◽  
Creeping Flow ◽  
Scaling Analysis ◽  
Dominant Eigenvalue ◽  
Péclet Number ◽  
The Real

This article analyses stationary scalar mixing downstream an open flow Couette device operating in the creeping flow regime. The device consists of two coaxial cylinders of finite length Lz, and radii κ R and R (κ < 1), which can rotate independently. At relatively large values of the aspect ratio α = Lz/R ≫ 1, and of the Péclet number Pe, the stationary response of the system can be accurately described by enforcing the simplifying assumption of negligible axial diffusion. With this approximation, homogenization along the device axis can be described by a family of generalized one-dimensional eigenvalue problems with the radial coordinate as independent variable. A variety of mixing regimes can be observed by varying the geometric and operating parameters. These regimes are characterized by different localization properties of the eigenfunctions and by different scaling laws of the real part of the eigenvalues with the Péclet number. The analysis of this model flow reveals the occurrence of sharp transitions between mixing regimes, e.g. controlled by the geometric parameter κ. The eigenvalue scalings can be theoretically predicted by enforcing eigenfunction localization and simple functional equalities relating the behaviour of the eigenvalues to the functional form of the associated eigenfunctions. Several flow protocols corresponding to different geometric and operating conditions are considered. Among these protocols, the case where the inner and the outer cylinders counter-rotate exhibits a peculiar intermediate scaling regime where the real part of the dominant eigenvalue is independent of Pe over more than two decades of Pe. This case is thoroughly analysed by means of scaling analysis. The practical relevance of the results deriving from spectral analysis for fluid mixing problems in finite-length Couette devices is addressed in detail.

Mixing at the interface between two fluids in porous media: a boundary-layer solution

2007 ◽  
Vol 584 ◽  
pp. 455-472 ◽  
Author(s):  
AMIR PASTER ◽  
GEDEON DAGAN
Keyword(s):  
Boundary Layer ◽  
Sea Water ◽  
Mixing Layer ◽  
Sharp Interface ◽  
Sea Water Intrusion ◽  
Water Intrusion ◽  
Two Fluids ◽  
The Past ◽  
Boundary Layer Solution ◽  
Layer Solution

A lighter fluid (fresh water) flows steadily above a body of a standing heavier one (sea water) in a porous medium. If mixing by transverse pore-scale dispersion is neglected, a sharp interface separates the two fluids. Solutions for interface problems have been derived in the past, particularly for the case of interest here: sea-water intrusion in coastal aquifers. The Péclet number characterizing mixing, Pe = b′/αT where b′ is the aquifer thickness and αT is transverse dispersivity, is generally much larger than unity. Mixing is nevertheless important in a few applications, particularly in the development of a transition layer near the interface and in entrainment of sea water within this layer. The equations of flow and transport in the mixing zone comprise the unknown flux, pressure and concentration fields, which cannot be separated owing to the presence of density in the gravity term. They are nonlinear because of the advective term and the dependence of the dispersion coefficients on flux, the latter making the problem different from that of mixing between streams in laminar viscous flow.The aim of the study is to solve the mixing-layer problem for sea-water intrusion by using a boundary-layer approximation, which was used in the past for the case of uniform flow of the upper fluid, whereas here the two-dimensional flux field is non-uniform. The boundary-layer solution is obtained in a few steps: (i) analytical potential flow solution of the upper fluid above a sharp interface is adopted; (ii) the equations are reformulated with the potential and streamfunction of this flow serving as independent variables; (iii) boundary-layer approximate equations are formulated in terms of these variables; and (iv) simple analytical solutions are obtained by the von Káarmán integral method. The agreement with an existing boundary-layer solution for uniform flow is excellent, and similarly for a solution of a particular case of sea-water intrusion with a variable-density code. The present solution may serve for estimating the thickness of the mixing layer and the rate of sea-water entrainment in applications, as well as a benchmark for more complex problems.

Convective diffusion to a slowly rotating spherical electrode; The effect of axial diffusion in the bulk liquid for Re 10

1985 ◽  
Vol 50 (4) ◽  
pp. 806-827
Author(s):  
Pavel Mitschka ◽  
Ondřej Wein
Keyword(s):  
Mathematical Model ◽  
Convective Diffusion ◽  
Creeping Flow ◽  
Bulk Liquid ◽  
Axially Symmetric ◽  
Flow Conditions ◽  
Transfer Coefficients ◽  
Axial Diffusion ◽  
Spherical Electrode ◽  
Method Of Solution

A complete mathematical model has been solved of the steady axially symmetric convective diffusion toward the surface of a spherical electrode of radius R rotating at an angular velocity Ω under the creeping flow conditions Re ≡ ΩR2ρ/η < 10 and Pe ≡ Ω2R4ρ/(12Dη) > 10 by the method of singular perturbations. For Pe > 300 the effect of axial diffusion has been found entirely negligible; for 10 < Pe < 300 it causes an increase of local transfer coefficients by 1-10%. For Pe < 10 the applied asymptotic method of solution, assuming Pe >> 1 is no longer applicable.

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