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Author

Philip Geoffrey Saffman

Other affiliations: University of Cambridge
Bio: Philip Geoffrey Saffman is an academic researcher from California Institute of Technology. The author has contributed to research in topics: Vortex & Vorticity. The author has an hindex of 49, co-authored 86 publications receiving 15759 citations. Previous affiliations of Philip Geoffrey Saffman include University of Cambridge.


Papers
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Journal ArticleDOI
TL;DR: In this article, it was shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamline moving in the direction opposite to V.
Abstract: It is shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to the streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamlines moving in the direction opposite to V. Here, a denotes the radius of the sphere, κ the magnitude of the velocity gradient, and μ and v the viscosity and kinematic viscosity, respectively. The relevance of the result to the observations by Segree & Silberberg (1962) of small spheres in Poiseuille flow is discussed briefly. Comments are also made about the problem of a sphere in a parabolic velocity profile and the functional dependence of the lift upon the parameters is obtained.

2,912 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that a flow is possible in which equally spaced fingers advance steadily at very slow speeds, such that behind the tips of the advancing fingers the widths of the two columns of fluid are equal.
Abstract: When a viscous fluid filling the voids in a porous medium is driven forwards by the pressure of another driving fluid, the interface between them is liable to be unstable if the driving fluid is the less viscous of the two. This condition occurs in oil fields. To describe the normal modes of small disturbances from a plane interface and their rate of growth, it is necessary to know, or to assume one knows, the conditions which must be satisfied at the interface. The simplest assumption, that the fluids remain completely separated along a definite interface, leads to formulae which are analogous to known expressions developed by scientists working in the oil industry, and also analogous to expressions representing the instability of accelerated interfaces between fluids of different densities. In the latter case the instability develops into round-ended fingers of less dense fluid penetrating into the more dense one. Experiments in which a viscous fluid confined between closely spaced parallel sheets of glass, a Hele-Shaw cell, is driven out by a less viscous one reveal a similar state. The motion in a Hele-Shaw cell is mathematically analogous to two-dimensional flow in a porous medium. Analysis which assumes continuity of pressure through the interface shows that a flow is possible in which equally spaced fingers advance steadily. The ratio λ = (width of finger)/(spacing of fingers) appears as the parameter in a singly infinite set of such motions, all of which appear equally possible. Experiments in which various fluids were forced into a narrow Hele-Shaw cell showed that single fingers can be produced, and that unless the flow is very slow λ = (width of finger)/(width of channel) is close to , so that behind the tips of the advancing fingers the widths of the two columns of fluid are equal. When λ = 1/2 the calculated form of the fingers is very close to that which is registered photographically in the Hele-Shaw cell, but at very slow speeds where the measured value of λ increased from 1/2 to the limit 1.0 as the speed decreased to zero, there were considerable differences. Assuming that these might be due to surface tension, experiments were made in which a fluid of small viscosity, air or water, displaced a much more viscous oil. It is to be expected in that case that λ would be a function of μU/T only, where μ is the viscosity, U the speed of advance and T the interfacial tension. This was verified using air as the less viscous fluid penetrating two oils of viscosities 0.30 and 4.5 poises.

2,634 citations

Journal ArticleDOI
TL;DR: It is suggested that for a realistic situation translational diffusion should be about four times faster in relation to rotational diffusion than in the isotropic case.
Abstract: Brownian motion (diffusion) of particles in membranes occurs in a highly anisotropic environment. For such particles a translational mobility (independent of velocity) can be defined if the viscosity of the liquid embedding the membrane is taken into account. The results of a model calculation are presented. They suggest that for a realistic situation translational diffusion should be about four times faster in relation to rotational diffusion than in the isotropic case.

1,611 citations

Journal ArticleDOI
TL;DR: In this paper, a theory of collisions between small drops in a turbulent fluid which takes into account collisions between equal drops was proposed, and it was shown that the collision rate due to the spatial variations of turbulent velocity is N = 1.30(r_1 + r_2)^2(n_1n_2)(e | v)^(1/2), valid for r_1|r_2 between one and two.
Abstract: This paper proposes a theory of collisions between small drops in a turbulent fluid which takes into account collisions between equal drops. The drops considered are much smaller than the small eddies of the turbulence and so the collision rates depend only on the dimensions of the drops, the rate of energy dissipation e and the kinematic viscosity v. Reasons are given for believing that the collision rate due to the spatial variations of turbulent velocity is shown to be N = 1.30(r_1 + r_2)^2(n_1n_2)(e | v)^(1/2), valid for r_1|r_2between one and two. A numerical integration has been performed using this expression to show how an initially uniform distribution will change because of collisions. An approximate calculation is then made to take account also of collisions which occur between drops of different inertia because of the action of gravity and the turbulent accelerations. The results are applied to the case of small drops in atmospheric clouds to test the importance of turbulence in initiating rainfall. Estimates of e are made for typical conditions and these are used to calculate the initial rates of collision, the change in mean properties and the rate of production of large drops. It is concluded that the effects of turbulence in clouds of the layer type should be small, but that moderate amounts of turbulence in cumulus clouds could be effective in broadening the drop size distribution in nearly uniform clouds where only the spatial variations of velocity are important. In heterogeneous clouds the collision rates are increased, and the effects due to the inertia of the drop soon become predominant. The effect of turbulence in causing collisions between unequal drops becomes comparable with that of gravity when e is about 2000 cm^2 sec^(−3).

1,304 citations

Journal ArticleDOI
TL;DR: In this paper, the structure of the intense-vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the range Re, = 35-170.
Abstract: The structure of the intense-vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the range Re, = 35-170. In accordance with previous investigators this vorticity is found to be organized in coherent, cylindrical or ribbon-like, vortices (‘worms’). A statistical study suggests that they are simply especially intense features of the background, O(o’), vorticity. Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow. An interesting observation is that the Reynolds number y/v, based on the circulation of the intense vortices, increases monotonically with ReA, raising the question of the stability of the structures in the limit of Re, --z co. Conversely, the average rate of stretching of these vortices increases only slowly with their peak vorticity, suggesting that self-stretching is not important in their evolution. One- and two-dimensional statistics of vorticity and strain are presented; they are non-Gaussian and the behaviour of their tails depends strongly on the Reynolds number. There is no evidence of convergence to a limiting distribution in this range of Re,, even though the energy spectra and the energy dissipation rate show good asymptotic properties in the higher-Reynolds-number cases. Evidence is presented to show that worms are natural features of the flow and that they do not depend on the particular forcing scheme.

965 citations


Cited by
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Journal ArticleDOI
TL;DR: In this paper, the authors present a review of the applicability and applicability of numerical predictions of turbulent flow, and advocate that computational economy, range of applicability, and physical realism are best served by turbulence models in which the magnitudes of two turbulence quantities, the turbulence kinetic energy k and its dissipation rate ϵ, are calculated from transport equations solved simultaneously with those governing the mean flow behaviour.

11,866 citations

Book
01 Jan 1984
TL;DR: In this paper, the authors describe a transition from Laminar boundary layer flow to Turbulent Boundary Layer flow with change of phase Mass Transfer Convection in Porous Media.
Abstract: Fundamental Principles Laminar Boundary Layer Flow Laminar Duct Flow External Natural Convection Internal Natural Convection Transition to Turbulence Turbulent Boundary Layer Flow Turbulent Duct Flow Free Turbulent Flows Convection with Change of Phase Mass Transfer Convection in Porous Media.

4,067 citations

Journal ArticleDOI
12 May 1978-Science
TL;DR: The force required to separate two cells is shown to be greater than the expected electrical forces between cells, and of the same order of magnitude as the forces required to pull gangliosides and perhaps some integral membrane proteins out of the cell membrane.
Abstract: A theoretical framework is proposed for the analysis of adhesion between cells or of cells to surfaces when the adhesion is mediated by reversible bonds between specific molecules such as antigen and antibody, lectin and carbohydrate, or enzyme and substrate. From a knowledge of the reaction rates for reactants in solution and of their diffusion constants both in solution and on membranes, it is possible to estimate reaction rates for membrane-bound reactants. Two models are developed for predicting the rate of bond formation between cells and are compared with experiments. The force required to separate two cells is shown to be greater than the expected electrical forces between cells, and of the same order of magnitude as the forces required to pull gangliosides and perhaps some integral membrane proteins out of the cell membrane.

4,058 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamline moving in the direction opposite to V.
Abstract: It is shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to the streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamlines moving in the direction opposite to V. Here, a denotes the radius of the sphere, κ the magnitude of the velocity gradient, and μ and v the viscosity and kinematic viscosity, respectively. The relevance of the result to the observations by Segree & Silberberg (1962) of small spheres in Poiseuille flow is discussed briefly. Comments are also made about the problem of a sphere in a parabolic velocity profile and the functional dependence of the lift upon the parameters is obtained.

2,912 citations

Journal ArticleDOI
TL;DR: In this paper, a two-equation turbulence model is proposed that is shown to be quite accurate for attached boundary layers in adverse pressure gradient, compressible boundary layers, and free shear flows.
Abstract: A comprehensive and critical review of closure approximations for two-equation turbulence models has been made. Particular attention has focused on the scale-determining equation in an attempt to find the optimum choice of dependent variable and closure approximations. Using a combination of singular perturbation methods and numerical computations, this paper demonstrates that: 1) conventional A:-e and A>w formulations generally are inaccurate for boundary layers in adverse pressure gradient; 2) using "wall functions'' tends to mask the shortcomings of such models; and 3) a more suitable choice of dependent variables exists that is much more accurate for adverse pressure gradient. Based on the analysis, a two-equation turbulence model is postulated that is shown to be quite accurate for attached boundary layers in adverse pressure gradient, compressible boundary layers, and free shear flows. With no viscous damping of the model's closure coefficients and without the aid of wall functions, the model equations can be integrated through the viscous sublayer. Surface boundary conditions are presented that permit accurate predictions for flow over rough surfaces and for flows with surface mass addition.

2,783 citations