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Nanoscale Charge Nonuniformity on Colloidal Particles
Published in Stoyl P. Stoylov, Maria V. Stoimenova, Molecular and Colloidal Electro-Optics, 2016
The Smoluchowski velocity (USmol) gives the translational velocity due to the average zeta potential of the doublet. Equation 15.6 reveals why it is so difficult to observe charge nonuniformity from translational electrophoresis measurements. The first term completely dominates the second term (which is due to charge nonuniformity), and therefore the second term gets lost in the noise. But for the angular velocity the average zeta potential makes no contribution, regardless of the shape [32]. Thus, the leading order term is due to the charge nonuniformity, as given by Equation 15.7. This gives the angular velocity the sensitivity needed to measure charge nonuniformity at high resolution. It should be noted that for doublets with e perpendicular to E0, Equation 15.7 has been simplified for the case of touching spheres of equal size: 〈Ωx2〉=0.884(εηa)2σζ2NE02
The Dominant Balance and WKB Methods
Published in Alan W. Bush, Perturbation Methods for Engineers and Scientists, 2018
Thus the leading order expression for f is obtained by solving the first order equation (dfdx)2−1x=0,
Theory of the Liquid and Solid Films Rupture
Published in Eli Ruckenstein, Gersh Berim, Wetting Theory, 2018
where C and C′ are the concentrations in the bulks of the film and the bounding fluid, respectively, and subscripts denote differentiation. The stress boundary conditions at the interface z = h(x, t) are highly nonlinear due to the presence of curvature terms and thus only their appropriate forms for the long-wavelength perturbations are reported later. A long-wavelength perturbation is the one for which the wavelength is much larger than the mean film thickness. As has been shown by the linear theories,11,20−23 the dominant growth rate occurs for the long-wavelength disturbances and thus we confine our attention to the long-wavelength perturbations from the onset. This has the advantage of simplifying the hydrodynamic equations without placing any restrictions on the amplitude of the perturbation. The long-wavelength reduction procedure was developed and applied by Benney24 and by Atherton and Homsy20 for the falling films. Williams and Davis16 extended the analysis to a thin wetting film devoid of solutes and derived a nonlinear equation of evolution for the location of the free interface. While the essentials of a formal perturbative analysis may be found in these references and were pursued by us19 for a wetting film (without the Rayleigh-Taylor effect) elsewhere, it suffices to note that the angle Δ in Figure 1 is small for any finite-amplitude, albeit long-wavelength, disturbance. It is in view of this that the curvature effects are negligible in the leading order equations,19,26 as are the transient and the inertia terms in the leading order Navier-Stokes equations. The following simplified Navier-Stokes equations thus describe the hydrodynamics of the thin film subjected to long-wavelength disturbances19 (these are, in effect, the equations in the “lubrication approximation”).
Weakly nonlinear interactions of collective oscillations in a correlated degenerate fluid
Published in Waves in Random and Complex Media, 2022
The WNL analysis predict that the dynamics of the dispersive modes (longitudinal and hydrodynamic) are governed by a KdV equation with a non-local nonlinear forcing term (HS equation with dispersive generalization) and a KdVB equation, respectively. The numerical solutions of these nonlinear equations well agree with the analytical solutions. The analytical solutions derived are the approximated leading-order solutions (Equations (51) and (78)) using multi-time-scale analysis. The higher-order terms in the perturbation analysis introduce only corrections (change in amplitude, velocity and width remain same as obtained in the leading-order approximation). Interestingly, the solvability conditions (Equations (61) and (81)) in case of multi-time-scale analysis are equivalent to the use of conservation laws (specially energy) in adiabatic perturbation theory of solitons to explain the effects of small disturbance (here weak correlation effect) on the initial soliton (leading order) [46–48,51].
A variety of negative-order integrable KdV equations of higher orders
Published in Waves in Random and Complex Media, 2019
First, the leading order and the leading coefficient should be determined. To obtain this, we substitute
A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions
Published in Waves in Random and Complex Media, 2018
First, the leading order and the leading coefficient should be determined. To obtain this, we substitute