Bipolar coordinates

Fig. P3.16 Bipolar coordinate system. The shaded area denotes the cross section of the fluid, and the constant a, the distance of the pole from the origin. [Reprinted by permission from R. Bird, R. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Volume 1, Fluid Mechanics, Second edition, Wiley, New York, 1987.]...

TABLE P3.16 The Equations of Continuity and Motion in Bipolar Coordinates ( , 0, Q... 

Alternatively the two-roll geometry can be conveniently represented by bipolar coordinates, as suggested by Finston (36). This approach, as well as the FEM, enables the analyses of both equal and unequal roll diameters and frequency of rotation, termed asymmetrical calendering. However, the FEM method provides the most flexibility in dealing with both Newtonian and non-Newtonian fluids and asymmetrical calendering. Chapter 15 covers this method in some detail. 

Unlike the other examples in this section, the equation governing the electrostatics here [i.e., Eq. (53)] is not the linearized Poisson-Boltzmann equation. However, considering interactions outside of thin double layers does have the effect of linearizing the problem. In Eq. (54), n is the fluid viscosity, K is the conductivity, is the zeta potential of the z th surface, and is a bipolar coordinate that is constant on the sphere and wall surfaces. It is this last condition (54), derived by Bike and Prieve [36] as a requirement to satisfy charge conservation, that couples the fluid mechanics with the electrostatics. 

Figure 3 presents the variation of the electrophoretic mobility with the dimensionless distance X when a sphere moves perpendicularly towards a conducting plane. The solid line represents the results from the bipolar coordinate method and the dash curve is the approximate results from the reflection method. A good agreement between the results from the both methods is attained. The electrophoretic velocity of the sphere decreases monotonically with increasing X and is expected to vanish as the particle... 

Once the potential field in the computational domain is available, the EDL force can be determined by integrating the stress over the sphere surface using Eqs. (14) and (15). In the bipolar coordinates, the normal vector of the surface element in Eq. (14) yields ds = sin t] di], where the unit vec-... 

Another simpler and useful form of this equation is obtained by transforming to bipolar coordinates... 

This integral equation can be solved by expansion of the integrand in bipolar coordinates [2, 3], Further improvement to the PY equation can be obtained by analytical fit to simulation studies as described below. 

An extensive discussion and bibliography of further references to conventional wall-effects will be found in Happel and Brenner (H9). To this list should be added the theoretical study by Dean and O Neill (D4) of the rotation of a sphere about an axis parallel to a nearby plane wall in an otherwise unbounded fluid, and a companion study by O Neill (02a) of the translation of a sphere parallel to a plane wall. Bipolar coordinates were employed to obtain exact solutions of the Stokes equations. These studies are particularly interesting in view of the fact that, due to the asymmetry of the flow, the rotating sphere experiences a force parallel to the wall whereas the translating sphere experiences a torque about the sphere center parallel to the wall. According to the remarks made in the second paragraph of Section II,C,3, these cross effects may be expressed in terms of the coupling dyadic in Eqs. (38) and (39) (where 0 refers to the sphere center). The Dean-O Neill in-... 

With regard to wall effects in fluids undergoing net flow, Goldman, Cox, and Brenner (G5e), using bipolar coordinates, obtained an exact solution to the problem of a neutrally buoyant sphere near a single plane wall in a semiinfinite fluid undergoing simple shear. In an unbounded fluid the translational... 

Besides the well-known cartesian system there are a lot of other systems in use in science. These include polar cylindrical coordinates, spherical polar coordinates, ellipsoidal coordinates, parabolic cylindrical coordinates, and cylindrical bipolar coordinates. 

Fig. 2.4. Bipolar coordinates used for the integration over the intersection region of Fig. 2.3.

This is the Percus-Yevick integral equation for y as cited in Section 2.7. Another simpler and useful form of this equation is obtained by transforming to bipolar coordinates (see Section 2.5),... 

This corresponds to a one-particle problem in a noncentral potential. Astute calculations could make use of elliptic or bipolar coordinates. In fact, one may simply use an ordinary expansion into partial... 

In Ref. [376], the two-particle electrocapillary problem was solved in bipolar coordinates without using any superposition approximations. The following expression (power expansion) was obtained... 

On the cylinder surfaces, the boundary conditions in the bipolar coordinates can be expressed as [13]... 

In the bipolar coordinates, Tis constant on the cylinder surface, (o and are related as... 

For the two particles with capillary charges and Q, Eqn. (3.24) is used to compute the interface deformation close to the particles for small or moderate inter-particle separation quY 1). By setting = in Eqn. (3.24) and using the relations of the bipolar coordinates, one obtains [19]... 

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