scholarly journals Axisymmetric spreading of surfactant from a point source

2017 ◽  
Vol 832 ◽  
pp. 777-792 ◽  
Author(s):  
Shreyas Mandre
Keyword(s):  
Fluid Velocity ◽  
Radial Distance ◽  
Domain Size ◽  
Fluid Interface ◽  
Viscous Forces ◽  
Constant Rate ◽  
Adsorbed Phase ◽  
Self Similar ◽  
Marangoni Stress ◽  
Dissolved Phase

Guided by computation, we theoretically calculate the steady flow driven by the Marangoni stress due to a surfactant introduced on a fluid interface at a constant rate. Two separate extreme cases, where the surfactant dynamics is dominated by the adsorbed phase or the dissolved phase, are considered. We focus on the case where the size of the surfactant source is much smaller than the size of the fluid domain, and the resulting Marangoni stress overwhelms the viscous forces so that the flow is strongest in a boundary layer close to the interface. We derive the resulting flow in a region much larger than the surfactant source but smaller than the domain size by approximating it with a self-similar profile. The radially outward component of fluid velocity decays with the radial distance $r$ as $r^{-3/5}$ when the surfactant spreads in an adsorbed phase, and as $r^{-1}$ when it spreads in a dissolved phase. Universal flow profiles that are independent of the system parameters emerge in both the cases. Three hydrodynamic signatures are identified to distinguish between the two cases and verify the applicability of our analysis with successive stringent tests.

Local implications for self-similar turbulent plume models

2007 ◽  
Vol 575 ◽  
pp. 257-265 ◽  
Author(s):  
M. M. SCASE ◽  
C. P. CAULFIELD ◽  
P. F. LINDEN ◽  
S. B. DALZIEL
Keyword(s):  
Radial Velocity ◽  
Phase Boundary ◽  
Fluid Model ◽  
Fluid Velocity ◽  
Ambient Fluid ◽  
Self Similar ◽  
The Difference ◽  
Two Fluid Model ◽  
Natural Way ◽  
Total Fluid

The local implications of the well-known flux conservation equations of Morton et al. (Proc. R. Soc. Lond. A, vol. 234, 1956, p.1) for plumes and jets are considered. Given the vertical velocity distributions of a model plume or jet, the divergence-free radial velocity distributions are calculated. It is shown that in general the velocity of the plume boundary is not described by the local total fluid velocity in this way. A two-fluid model tracking the evolution of both ‘plume’ and ‘ambient’ fluid is proposed which resolves this apparent inconsistency and also provides a way of explicitly describing the mixing process within a model plume. The plume boundary acts as a phase boundary across which ambient fluid is entrained, and the plume boundary moves at the velocity of the plume fluid. The difference between the plume-fluid radial velocity and the total fluid velocity quantifies in a natural way the purely horizontal entrainment flux of ambient fluid into the plume across the phase boundary at the plume edge.

The magneto-hydrodynamics of a rotating fluid and the earth’s dynamo problem

1963 ◽  
Vol 274 (1357) ◽  
pp. 274-283 ◽  
Keyword(s):  
Magnetic Force ◽  
Necessary Conditions ◽  
Fluid Velocity ◽  
Rotating Fluid ◽  
Rigid Boundary ◽  
Viscous Forces ◽  
Magnetic Forces ◽  
The Earth ◽  
The Magnetic Field

This paper discusses a rotating, incompressible fluid enclosed within a rigid boundary which is a surface of revolution. It is shown that if viscous forces are negligible, then, in the presence of magnetic fields, the fluid can execute slow, steady relative motions only if the magnetic force satisfies a constraint. In cylindrical polar co-ordinates this constraint can be written that is, the couple exerted by the magnetic forces on any cylinder of fluid coaxial with the axis of rotation must vanish. Furthermore, subject to certain restrictions on the shape of the container (which, for example, are fulfilled by a sphere but not by a cylinder), it is shown that if the field satisfies the above condition then the fluid velocity is completely determined by the instantaneous value of the magnetic field (together with that of the density if buoyancy forces are important). This velocity is such that the necessary conditions on the field will continue to be satisfied. An algorithm for the determination of the velocity is given and its application to the earth ’s dynamo problem is indicated.

The Recirculatory Flow Induced by a Laminar Axisymmetric Jet Issuing From a Wall

1987 ◽  
Vol 109 (3) ◽  
pp. 237-241 ◽  
Author(s):  
W. Schneider ◽  
E. Zauner ◽  
H. Bo¨hm
Keyword(s):  
Point Sources ◽  
Creeping Flow ◽  
Outer Flow ◽  
Axisymmetric Jet ◽  
Submerged Jet ◽  
Viscous Forces ◽  
Recirculatory Flow ◽  
Self Similar ◽  
Infinite Wall ◽  
Main Region

The laminar, axisymmetric, submerged jet issuing from a plane, infinite wall perpendicular to the jet axis is considered at very large distance from the nozzle. Based on previous results of an asymptotic analysis, an approximate analytical solution for the complete flow field is obtained. The structure of the far field is discussed by considering various regions the size of which depends strongly on the Reynolds number. The main region is a toroidal eddy in which both inertial and viscous forces are of importance. Closer to the nozzle there is a slender jet flow with slowly varying momentum flux together with a self-similar viscous outer flow. At larger distances, the flow resembles the creeping flow due to point sources of momentum and mass, with the former decaying more rapidly than the latter as infinity is approached. Analytical predictions of the location of the eddy center compare favorably with experimental and numerical results.

Theory of the almost-highest wave: the inner solution

1977 ◽  
Vol 80 (4) ◽  
pp. 721-741 ◽  
Author(s):  
M. S. Longuet-Higgins ◽  
M. J. H. Fox
Keyword(s):  
Free Surface ◽  
Surface Condition ◽  
Radial Distance ◽  
Transcendental Equation ◽  
Vertical Acceleration ◽  
Maximum Slope ◽  
Corner Flow ◽  
Particle Speed ◽  
Damped Oscillation ◽  
Self Similar

This paper investigates the flow near the summit of steep, progressive gravity wave when the crest is still rounded but the flow is approaching Stokes's corner flow. The natural length scale in the neighbourhood of the summit is seen to be l =q2/2g, where g denotes gravity and q is the particle speed at the crest in a reference frame moving with the wave speed. We show that a class of self-similar smooth local flows exists which satisfy the free-surface condition and which tend to Stokes's corner flow when the radial distance r becomes large compared withl. The behaviour of the solution at large values of r/l is shown to depend on the roots of the transcendental equation \[ K \tan h K = \pi/2\surd{3}. \] The two real roots correspond to a damped oscillation of the free surface decaying like (l/r)½. The positive imaginary roots correspond to perturbations vanishing like higher negative powers of r.The complete flow is calculated by transforming the domain onto the interior of a circle in the complex plane and expanding the potential at the surface in a Fourier series. The computation is checked by an independent method, based on approximating the flow by a sequence of dipoles. The profile of the surface is found to intersect its asymptote at large values of r/l. This implies that the maximum slope slightly exceeds 30°. The computed value 30·37° is in close agreement with that obtained by extrapolating the maximum slopes of steep gravity waves, as calculated by previous authors. The vertical acceleration of a particle at the crest is 0·388g. In the far field, however, the acceleration tends to the value ½g corresponding to the Stokes corner flow.

Transient Stresses and Displacement Around a Wellbore Due to Fluid Flow in Transversely Isotropic, Porous Media: II. Finite Reservoirs

1968 ◽  
Vol 8 (01) ◽  
pp. 79-86 ◽  
Author(s):  
M.S. Seth ◽  
K.E. Gray
Keyword(s):  
Porous Media ◽  
Radial Displacement ◽  
Outer Boundary ◽  
Stress Gradient ◽  
Fluid Pressure ◽  
Radial Distance ◽  
Transversely Isotropic ◽  
Vertical Stress ◽  
State Flow ◽  
Constant Rate

Abstract In Part 1 of this work,1 equations of elasticity were formulated for transversely isotropic, axisymmetric, homogeneous, porous media exhibiting pore fluid pressure. Equations of elasticity and the thermal analogy method were used to determine transient horizontal, tangential, and vertical stresses and radial displacement in a semi-infinite cylindrical region when either a constant rate of pressure or a constant rate of flow is maintained at the wellbore. In this paper, the approach presented earlier is extended to finite reservoirs for the cases ofsteady-state flow,constant pressures at the well bore and outer boundary andconstant pressure at the wellbore and no flow at the outer boundary. Results of this work show that radial and tangential stress gradients are high near the wellbore but diminish rapidly away from the well; the vertical stress gradient behaves in the same way but is less severe. Radial stresses are compressive or neutral, whereas tangential and vertical stresses may be tensile, neutral or compressive, depending upon the boundary conditions, the physical properties of the system and the radial distance involved (vertical stresses are always compressive in an unbounded system1). For constant boundary pressures, both radial and tangential stresses increase with time whereas they both decrease for a closed outer boundary and constant pressure at the wellbore. The vertical stress decreases with time for both systems. For steady-state systems, radial displacement may be positive or negative, depending upon the dimensions of the system, the pressure differential and the porosity. Radial displacement may be positive or negative for a closed outer boundary but is positive for constant pressures at both boundaries. INTRODUCTION The importance, utility and complexity of a realistic appraisal of the stress state at and local to a wellbore were indicated in Part 1. In this paper the analytical approach presented earlier is extended to finite, cylindrical reservoir geometry for the cases ofsteady-state flow,constant pressures at wellbore and outer boundary andconstant pressure at the wellbore and no flow at the outer boundary. Other than the outer boundary of the reservoir being finite, the physical system and assumptions pertinent thereto are the same as before. The reader may wish to review the mathematical development through Eq. 49 of Part 1 before proceeding here.

Island Dynamics in Film Aging

1990 ◽  
Vol 202 ◽  
Author(s):  
S. P. Marsh ◽  
M. E. Glicksman
Keyword(s):  
Growth Rate ◽  
Domain Size ◽  
Cube Root ◽  
Global Constraint ◽  
Concentration Gradients ◽  
Average Growth ◽  
Zero Growth ◽  
Self Similar ◽  
The Individual ◽  
Island Dynamics

ABSTRACTA theory is presented which describes the capillary-driven aging of discontinuous thin films on a substrate, where the primary transport mechanism among the domains is two-dimensional diffusion of species over the substrate. This theory employs a statistical dynamics formulation, whereby the average growth rate for each domain size class is determined relative to the critical (zero-growth) domain size. The time dependence of the critical size is determined through a global constraint on the individual fields. The effect of fractional area coverage, Aa, is accounted for through a second global constraint over the distribution of island sizes.This theory yields a self-similar size distribution that is fairly insensitive to Aa. The critical island radius, R*, is found to increase asymptotically as the cube-root of time. The growth rate of R* increases with Aa, which results from the closer proximity of the islands and steeper concentration gradients as Aa increases.

Vortex formation in front of a piston moving through a cylinder

2000 ◽  
Vol 416 ◽  
pp. 1-28 ◽  
Author(s):  
J. J. ALLEN ◽  
M. S. CHONG
Keyword(s):  
Boundary Layer ◽  
Vortex Formation ◽  
Field Measurements ◽  
Inviscid Flow ◽  
Cylinder Wall ◽  
Viscous Effects ◽  
Viscous Forces ◽  
Vorticity Distribution ◽  
Piston Speed ◽  
Self Similar

This paper contains the details of an experimental study of the vortex formed in front of a piston as it moves through a cylinder. The mechanism for the formation of this vortex is the removal of the boundary layer forming on the cylinder wall in front of the advancing piston. The trajectory of the vortex core and the vorticity distribution on the developing vortex have been measured for a range of piston velocities. Velocity field measurements indicate that the vortex is essentially an inviscid structure at the Reynolds numbers considered, with viscous effects limited to the immediate corner region. Inviscid flow is defined in this paper as being a region of the flow where inertial forces are significantly larger than viscous forces. Flow visualization and vorticity measurements show that the vortex is composed mainly of material from the boundary layer forming over the cylinder wall. The characteristic dimension of the vortex appears to scale in a self-similar fashion, while it is small in relation to the apparatus length scale. This scaling rate of t0.85+0.7m, where the piston speed is described as a power law Atm, is somewhat faster than the t3/4 scaling predicted by Tabaczynski et al. (1970) and considerably faster than a viscous scaling rate of t1/2. The reason for the structure scaling more rapidly than predicted is the self-induced effect of the secondary vorticity that is generated on the piston face. The vorticity distribution shows a distinct spiral structure that is smoothed by the action of viscosity. The strength of the separated vortex also appears to scale in a self-similar fashion as t2m+1. This rate is the same as suggested from a simple model of the flow that approximates the vorticity being ejected from the corner as being equivalent to the flux of vorticity over a flat plate started from rest. However, the strength of the vorticity on the separated structure is 25% of that suggested by this model, sometimes referred to as the ‘slug’ model. Results show that significant secondary vorticity is generated on the piston face, forming in response to the separating primary vortex. This secondary vorticity grows at the same rate as the primary vorticity and is wrapped around the outside of the primary structure and causes it to advect away from the piston surface.

A MATHEMATICAL MODEL FOR TUMOR CORDS INCORPORATING THE FLOW OF INTERSTITIAL FLUID

2005 ◽  
Vol 15 (11) ◽  
pp. 1735-1777 ◽  
Author(s):  
ALESSANDRO BERTUZZI ◽  
ANTONIO FASANO ◽  
ALBERTO GANDOLFI
Keyword(s):  
Existence And Uniqueness ◽  
Volume Fraction ◽  
Fluid Velocity ◽  
Extracellular Fluid ◽  
Radial Distance ◽  
Viable Cell ◽  
Cell Killing ◽  
Necrotic Region ◽  
A Cell ◽  
Time Existence

This work extends a previous model that described the evolution of a tumor cord (a cylindrical arrangement of tumor cells, generally surrounded by necrosis, growing around a blood vessel of the tumor) under the activity of cell killing agents. In the present model we include the most relevant aspects of the dynamics of extracellular fluid, by computing the longitudinal average of the radial fluid velocity and of the pressure field. We still assume that the volume fraction occupied by the cells always keeps the same constant value everywhere in the cord. The necrotic region is treated as a "fluid reservoir". To improve the modelling of therapeutic treatment, we have subdivided the viable cell population into a proliferating and a quiescent subpopulation. The transitions between the two states are both permitted, and are regulated by rates depending on the local oxygen concentration. For simplicity, the rates of death induced by treatment are assumed to be known functions of the radial distance and time. Existence and uniqueness of the stationary state in the absence of treatment has been shown, as well as the existence and uniqueness of the evolution that arises following a cell killing treatment.

Self-similar states in turbulent mixing layers

2001 ◽  
Vol 446 ◽  
pp. 1-24 ◽  
Author(s):  
ELIAS BALARAS ◽  
UGO PIOMELLI ◽  
JAMES M. WALLACE
Keyword(s):  
Boundary Layer ◽  
Turbulent Mixing ◽  
Random Noise ◽  
Domain Size ◽  
The Self ◽  
Mixing Layers ◽  
Computational Domain ◽  
Mean Flow ◽  
Boundary Layer Turbulence ◽  
Self Similar

Large-eddy simulations of temporally evolving turbulent mixing layers have been carried out. The effect of the initial conditions and the size of the computational box on the turbulent statistics and structures is examined in detail. A series of calculations was initialized using two different realizations of a spatially developing turbulent boundary-layer with their free streams moving in opposite directions. Computations initialized with mean flow plus random perturbations with prescribed moments were also conducted. In all cases, the initial transitional stage, from boundary-layer turbulence or random noise to mixing-layer turbulence, was followed by a self-similar period. The self-similar periods, however, differed considerably: the growth rates and turbulence intensities showed differences, and were affected both by the initial condition and by the computational domain size. In all simulations the presence of quasi-two-dimensional spanwise rollers was clear, together with ‘braid’ regions with quasi-streamwise vortices. The development of these structures, however, was different: if strong rollers were formed early (as in the cases initialized by random noise), a well-organized pattern persisted throughout the self-similar period. The presence of boundary layer turbulence, on the other hand, inhibited the growth of the inviscid instability, and delayed the formation of the roller–braid patterns. Increasing the domain size tended to make the flow more three-dimensional.

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