Hydrodynamic forces radius

Consider the forces acting on an arbitrary sphere S enclosing the soft sphere at its center. We take the radius of S tends to infinity. Since the net electric charge within S is zero, there is no net electric force acting on S and one needs consider only the hydrodynamic force Eh. The equation of motion for S is thus given by... 

In the model of Agarwal and Khakhar [57] the polymer molecules are taken to be bead-rod chains with the hydrodynamic forces concentrated at the beads. The chains may bend about a bead, and a spring force acts to restore the chain to is equilibrium conformation, which is a straight chain. The connecting rods are inextensible. The system is confined to a plane, and the chains diffuse due to Brownian forces resisted by hydrodynamic forces. Hydrodynamic forces resulting from an imposed shear flow deform and orient the molecules. Two chains may react and combine to form a longer chain if the chain ends approach to within the capture radius (a) and if the angle between the chains is less than the critical value (0 ). The reaction is assumed to be very fast (kfj k j ) so that every collision that satisfies the above criteria results in... 

The use of the macro-d5uiamical method previously in connection with the osmotic theory of force [the diffusion of electrolytes and ionic mobilities by Nemst diffusion and the hydrodynamic (Stokes ) radius of molecules according to W. Sutherland and Einstein] is weU known. An alternative, simple formulation of two-component diffusion, Eq. (10), is possible if another form of the thermod5mamic factor, based upon osmotic pressme is applied ... 

Since Nad 1/ p and Ecyi -- the dependence of Ecyi on the particle radius Up is weaker than in the case of negligible molecular and hydrodynamic forces. 

In these expressions, me, Vc, Fd, F, X, a, H, t, 17, p, R, DLVO force, the hydrodynamic force, the scaled distance, the linear radius of the cells, the dimensionless closest half surface-to-surface distance between two cells, the scaled half separation distance between two cells, the time, the viscosity of liquid phase, the hydrodynamic retardation factor, the density of the cells, the gas constant, the scaled DLVO potential, the scaled van der Waals potential, and the Hamaker constant of the system, and i, , and I are dummy variables. Note that the fourth and fifth integral terms on the right-hand side of (25.141) represent, respectively, the contribution to the electrostatic repulsion force when the fixed positive and negative charge in the membrane phase of a cell appears. Equation (25.135) can be rewritten to become... 

Multibody hydrodynamic interactions have generally been ignored in simulations (for reasons of computational cost) with the notable exception of [243,264] for a monolayer system involving a small number of particles. Satoh et al. [266,267] approximate the multibody hydrodynamic forces by assuming additivity of the velocities. This does, however, not guarantee positive definiteness of the mobility matrix (inverse of the resistance matrix), imless a short cutoff radius of the hydrodynamic interactions is used [266,267]. 

The sedimentation velocity Vq of a very dilute suspension of rigid non-interacting particles vith radius a can be determined by equating the gravitational force with the opposing hydrodynamic force as given by Stokes law, i.e. 

Figure 2.9 Hydrodynamic force acting on a hydrophilic sphere of radius R approaching a smooth hydrophilic (diamonds), smooth hydrophobic (circies), and randomiy rough hydrophilic (triangles) wall with 02 = 4% (adapted from. ) Here Fst = 6nr]RU is the Stokes drag. The separation h is defined on top of the surface roughness, as shown in the inset. Simulation resuits (symbols) compared with theoretical curves F/Fst = 1 + 9R/ 8h] (solid), F/Fst = 1 + 9Rf /(8h) with / = f b/h), calculated with Eq. 2.4 (dash-dotted), and F/Fst = 1-1- 9R/ S[h + s]) (dashed). Values of b and s were determined by fitting the simulation data.

The sedimentation velocity v of a very dilute suspension of rigid noninteracting particles with radius a can be determined by equating the gravitational force with the opposing hydrodynamic force eis given by Stokes law in equation (3.46). Equation (3.46) predicts a sedimentation rate for particles with radius 1 pm in a medium with a density difference of 0.2gcm and a viscosity of ImPas (i.e. water at 20 °C) of 4.4 X 10 ms . Such particles will sediment to the bottom of a 0.1m container in about 60 hours. For 10 pm particles, the sedimentation velocity is 4.4 x 10 ms and such particles will sediment to the bottom of a 0.1 m container in about 40 minutes. 

When two colloidal particles come close to each other, they experience two types of force. The first one, the surface forces of intermolecular origin (the disjoining pressure), which are due to the van der Waals, electrostatic, steric, interactions, for example, have been discussed in Sec. VI. The second type represent the hydrodynamic forces which originate from the interplay of the hydrodynamic flows around two moving colloidal particles or two film surfaces. It becomes important when the separation between the particle surfaces is of the order of the particle radius and increases rapidly with the decrease of the gap width. 

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