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Abstract
The insertion of an elastic rod or fiber into a confining cavity is studied. Such an insertion is a feature of a variety of problems, including packing and unpacking of DNA in viral capsids and the insertion of catheters during surgery. We consider a simplified geometry in which the container is a smooth (frictionless) circular cylinder of radius a. The fiber is pushed through a hole in the curved surface of the cylinder and is then assumed to stay in a cross-sectional plane perpendicular to the cylinder axis. A solution is found for the fiber shape in which most of the fiber lies against the curved interior surface of the cylinder, apart from the final end section of the fiber, of length 2.0888a, which crosses the interior of the cylinder before ending at the opposite side, which it meets at an angle 1.15 rad to the normal. The force required to push the fiber into the cylinder is EI/2a^{2}, where E is the fiber's Young's modulus and I its cross-sectional moment of inertia. The shape of the final end section of the fiber is confirmed by experiment.
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Affiliation(s)
- J D Sherwood
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
| | - S Ghosal
- Department of Mechanical Engineering and Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
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Abstract
The electrical repulsion between two charged solid surfaces separated by an electrolyte is studied as a function of the permittivity ϵs of the solid in the limit in which potentials are small, and the gap between the plane solid surfaces is small compared to the Debye length κ-1 within the electrolyte. The solid surfaces are uniformly charged in a central region |x|< L outside which they are uncharged. When ϵs = 0, ions from the charge cloud between the charged surfaces spill out into regions of length O(κ-1) beyond x = ± L, thereby reducing the pressure between the surfaces from that predicted by Derjaguin-Landau-Verwey-Overbeek theory for infinite, uniformly charged surfaces. When ϵs>0, ions spill out over much larger O(L) regions, thereby reducing still further both the electrical potential between the solid surfaces and the repulsive force between them. However, this reduction becomes smaller as κL becomes large.
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Affiliation(s)
- J D Sherwood
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, U.K
| | - S Ghosal
- Department of Mechanical Engineering & Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, United States
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Sherwood JD, Mao M, Ghosal S. Electrically generated eddies at an eightfold stagnation point within a nanopore. Phys Fluids (1994) 2014; 26:112004. [PMID: 25489206 PMCID: PMC4247373 DOI: 10.1063/1.4901984] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/15/2014] [Accepted: 11/04/2014] [Indexed: 05/24/2023]
Abstract
Electrically generated flows around a thin dielectric plate pierced by a cylindrical hole are computed numerically. The geometry represents that of a single nanopore in a membrane. When the membrane is uncharged, flow is due solely to induced charge electroosmosis, and eddies are generated by the high fields at the corners of the nanopore. These eddies meet at stagnation points. If the geometry is chosen correctly, the stagnation points merge to form a single stagnation point at which four streamlines cross at a point and eight eddies meet.
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Affiliation(s)
- J D Sherwood
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge , Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - M Mao
- Department of Mechanical Engineering, Northwestern University , 2145 Sheridan Road, Evanston, Illinois 60208, United States
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Abstract
A theoretical model of electroosmosis through a circular pore of radius a that traverses a membrane of thickness h is investigated. Both the cylindrical surface of the pore and the outer surfaces of the membrane are charged. When h ≫ a, end effects are negligible, and the results of full numerical computations of electroosmosis in an infinite pore agree with theory. When h = 0, end effects dominate, and computations again agree with analysis. For intermediate values of h/a, an approximate analysis that combines these two limiting cases captures the main features of computational results when the Debye length κ(-1) is small compared with the pore radius a. However, the approximate analysis fails when κ(-1) ≫ a, when the charge cloud due to the charged cylindrical walls of the pore spills out of the ends of the pore, and the electroosmotic flow is reduced. When this spilling out is included in the analysis, agreement with computation is restored.
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Affiliation(s)
- J. D. Sherwood
- Department
of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom
- E-mail:
| | - M. Mao
- Department of Mechanical Engineering and Department of Engineering Sciences
and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States
| | - S. Ghosal
- Department of Mechanical Engineering and Department of Engineering Sciences
and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States
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Affiliation(s)
- J. D. Sherwood
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K
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Abstract
The streaming potential generated by motion of a long drop of viscosity mu(d) = lambdamu in a uniform circular capillary filled with fluid of viscosity mu is investigated by means of a model previously used to study electrophoresis of a charged mercury drop in water. The capillary wall is at potential zeta c relative to the bulk fluid within it, and the surface of the drop is at potential zeta(d). Potentials are assumed to be sufficiently small so that the charge cloud is described by the linearized Poisson-Boltzmann equation, and the Debye length characterizing the thickness of the charge cloud is assumed to be thin compared with the gap h(0) between the drop and the capillary wall. Ions in the external fluid are not allowed to discharge at the surface of the drop, and the wall of the capillary has a nonzero surface conductivity sigma c. The drop is assumed to be sufficiently long so that end effects can be neglected. Recirculation of fluid within the drop gives rise to an enhanced streaming current when zeta(d) is nonzero, leading to an anomalously high streaming potential. This vanishes as the drop viscosity becomes large. If V is the velocity of the drop and gamma is the coefficient of interfacial tension between the two fluids, then the capillary number is Ca = mu V/gamma, and the gap varies as h(0)planck'sCa(2/3). When Ca is small, the gap h(0) is small and electrical conduction along the narrow gap is dominated by the surface conductivity sigma(c) of the capillary wall, which is constant. The electrical current convected by flowing fluid is proportional to Ca, as is the change in streaming potential caused by the presence of the drop. If sigma(c) = 0, then the electrical conductance of the gap depends on its width h(0) and on the bulk fluid conductivity sigma and becomes small as h(0) approximately equal to Ca(2/3) --> 0. The streaming potential required to cancel the O(Ca) convection current therefore varies as Ca(1/3). If sigma(c) = 0 and the drop is rigid (lambda --> infinity), then the change in streaming potential over and above that expected due to the change in pressure gradient is proportional to the difference in potentials zeta(c)-zeta(d).
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Affiliation(s)
- J D Sherwood
- Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge, U.K.
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Abstract
The primary electroviscous effect in a nondilute suspension of charged spherical particles is studied by means of cell models. The governing equations are derived, and then analytic results are obtained by restricting attention to the limit of thin double layers, small Hartmann and Peclet numbers, and small potentials. Previous work has assumed that the velocity at the outer boundary of the cell is identical to the imposed flow, as proposed by Simha (J. Appl. Phys. 1952, 23, 1020). Results with this boundary condition are compared against those predicted when the tangential shear stress on the outer boundary is assumed to be unperturbed, as proposed by Happel (J. Appl. Phys. 1957, 28, 1288). Both the hydrodynamic and electroviscous contributions to the effective viscosity are smaller with the Happel boundary condition, showing that such cell models offer a range of predictions and should be used with caution.
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Affiliation(s)
- J D Sherwood
- Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, United Kingdom.
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Headen TF, Clarke SM, Perdigon A, Meeten GH, Sherwood JD, Aston M. Filtration of deformable emulsion droplets. J Colloid Interface Sci 2006; 304:562-5. [PMID: 17022995 DOI: 10.1016/j.jcis.2006.09.014] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2006] [Revised: 09/04/2006] [Accepted: 09/09/2006] [Indexed: 11/20/2022]
Abstract
Oil-in-water (o/w) emulsions of different droplet size were filtered on membranes of various pore sizes to investigate the growth and behaviour of o/w filter cakes. The cake desorptivity S and the filter membrane resistance R were measured at various filtration pressures P. The variation of S with P shows that filter cake oil droplets of radius a are effectively rigid for P << gamma/a and fully deformable for P >> gamma/a, where gamma is the oil-water interfacial tension. For the largest P, when S became P-independent, the filter cake remained water-permeable as expected from theory.
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Affiliation(s)
- T F Headen
- BP Institute and the Department of Chemistry, University of Cambridge, Madingley Road, Cambridge, UK
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Sherwood JD, Risso F, Collé-Paillot F, Edwards-Lévy F, Lévy MC. Rates of transport through a capsule membrane to attain Donnan equilibrium. J Colloid Interface Sci 2003; 263:202-12. [PMID: 12804903 DOI: 10.1016/s0021-9797(03)00140-1] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Abstract
The swelling of a capsule consisting of salt solution and polyelectrolyte, surrounded by a membrane, is studied. The membrane allows salt and water to pass, but is impermeable to polyelectrolyte molecules. Equilibrium swelling of the capsule is governed by Donnan equilibrium. Transport rates of a salt and water through the membrane are expressed in terms of a Darcy permeability and a salt diffusivity. The governing equations predict that the rate at which equilibrium is attained as the external salt concentration varies is controlled by the timescale for diffusion of salt, rather than by that for Darcy flow. Experiments were performed using capsules with membranes made of covalently linked HSA and alginate. The capsule volume varied with a single relaxation rate when the external salt concentration was changed, as predicted by theory. This constitutes the first step toward a simple method for determining the membrane properties of capsules by measuring rates of change of capsule volume.
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Affiliation(s)
- J D Sherwood
- Schlumberger Cambridge Research, High Cross, Madingley Rd., Cambridge CB3 0DW, UK.
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Abstract
A model for hindered transport of water and ions is used to predict transient flow through a clay membrane caused by an initial difference in the concentration of salt solutions in reservoirs on the two sides of the membrane. Transport is assumed to be controlled by three coefficients, which are analogous to the permeability, salt diffusivity, and salt reflection coefficient of the membrane. Initial fluid motion is caused by osmosis, leading to a buildup of pressure on one side of the membrane. However, the clay forms an imperfect ion exclusion membrane and the final steady state is one of equal concentrations and pressures on the two sides of the membrane. The time-dependent differences in pressure and salt concentration across the membrane are predicted to vary as the sum of two decaying exponentials. When the salt reflection coefficient is small, one time scale governs Darcy flow through the membrane and another the diffusion of salt. Experimental results confirm the analysis. Although the salt concentration in the reservoirs was monitored in the experiments, estimates of the transport coefficients can be obtained by measuring only the pressure change across the membrane. Copyright 2000 Academic Press.
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Affiliation(s)
- JD Sherwood
- Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge, CB3 0EL, United Kingdom
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Sherwood JD, Rubio-Hernández FJ, Ruiz-Reina E. The Primary Electroviscous Effect: Thin Double Layers (akappa>>1) and a Stern Layer. J Colloid Interface Sci 2000; 228:7-13. [PMID: 10882486 DOI: 10.1006/jcis.2000.6935] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
Abstract
The primary electroviscous effect due to the charge clouds surrounding spherical charged particles suspended in an electrolyte was studied by Hinch and Sherwood (J. Fluid Mech. 132, 337 (1983)) in the limit of double layers thin compared to the particle radius a. Here we introduce the effect of a dynamic Stern layer into that analysis, in order to explain the numerical results of Rubio-Hernández et al. (J. Colloid Interface Sci. 206, 334 (1998)) in terms of the ratio of the tangential ionic fluxes within the charge cloud to those within the Stern layer. The predictions of the asymptotic analysis are compared with those of numerical computations. The thickness of the charge cloud is characterized by the Debye length kappa(-1). If akappa>10 the predictions of the asymptotic analysis exhibit the same qualitative behavior as the numerical results, but akappa>1000 is required to achieve quantitative agreement to within 2.5%. Copyright 2000 Academic Press.
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Affiliation(s)
- JD Sherwood
- Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge, CB3 0EL, U.K
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Affiliation(s)
- Wendy Zhang
- Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, and Schlumberger Cambridge Research, Madingley Road, Cambridge CB3 0EL, U.K
| | - H. A. Stone
- Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, and Schlumberger Cambridge Research, Madingley Road, Cambridge CB3 0EL, U.K
| | - J. D. Sherwood
- Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, and Schlumberger Cambridge Research, Madingley Road, Cambridge CB3 0EL, U.K
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Sherwood JD. Nonlinear compaction of an assembly of highly deformable platelike particles. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1994; 50:3834-3840. [PMID: 9962437 DOI: 10.1103/physreve.50.3834] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Sherwood JD. Canadian Colloquium on Computer-Assisted Interpretation of Electrocardiograms. V. Suggested standards for instrumentation in computer processing of electrocardiograms. Can Med Assoc J 1973; 108:1259-61. [PMID: 4574249 PMCID: PMC1941414] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [MESH Headings] [Subscribe] [Scholar Register] [Indexed: 01/11/2023]
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