Theoretical investigations on electrorheological fluids in homogeneous and inhomogeneous electric fields.
(Theoretische Untersuchungen von elektrorheologischen Flüssigkeiten bei homogenen und inhomogenen elektrischen Feldern.)

*(German)*Zbl 0958.76003
Aachen: Shaker Verlag. Darmstadt: TU Darmstadt, Fachbereich Mathematik, iv, 162 S. (2000).

The work deals with the theoretical investigation of electrorheological fluids (ERF) in homogeneous and inhomogeneous electric fields. In particular, the author considers the ERF Rheobay of the Bayer Company. This is a concentrated suspension consisting of a solid and a liquid phase. A characteristic feature of this suspension is the strong influence of electric field on flow properties. Here, this material is modelled within the theory of continuous media as a one-phase homogeneous viscous liquid.

Starting point of the analysis are general balance equations of rational thermodynamics and the equations of electrodynamics of moving media (Maxwell-Minkowski equations). The evaluation of the entropy inequality yieids two material set-ups for Cauchy stress tensor. Two formulations are presented, a linear and a nonlinear one. The latter is kind of a generalized Casson model with fluid stress depending on the electric field strength. This approach agrees well with experimental data. Assuming a homogeneous electric field for both material set-ups, the author determines velocity profiles for a pure shear and pressure flow, respectively, together with the corresponding volume flows. Both set-ups predict a linear velocity profile for a pure shear flow. For the pressure flow, a parabolic profile is predicted by the linear material set-up, while the nonlinear set-up yields a flattened profile with a so-called “plug-zone” depending on the electric field strength. The volume flow based on the nonlinear set-up shows good agreement of calculated and measured values.

Next, the author studies theoretically the influence of a nonhomogeneous electric field on the velocity profile and on the pressure. The field equations are derived for the velocity in the case of a plane incompressible flow for the most general Cauchy stress tensor (compatible with the author’s theory). Concerning an approximation of the first order with respect to a certain small parameter, the author obtains an analytical integral solution for the velocity components. This solution is valid for arbitrary electric fields. The same investigations are also carried out for the nonlinear model. Besides, the author calculates the flow in a rotating viscosimeter. In this case the electric field is not constant. Regarding the linear model, the author obtains a simple closed-form analytical solution for the velocity, which deviates remarkably from the well-known Newtonian solution. Finally, in the case of the nonlinear model, a general integral solution for velocity was given, which allows numerical treatment.

Starting point of the analysis are general balance equations of rational thermodynamics and the equations of electrodynamics of moving media (Maxwell-Minkowski equations). The evaluation of the entropy inequality yieids two material set-ups for Cauchy stress tensor. Two formulations are presented, a linear and a nonlinear one. The latter is kind of a generalized Casson model with fluid stress depending on the electric field strength. This approach agrees well with experimental data. Assuming a homogeneous electric field for both material set-ups, the author determines velocity profiles for a pure shear and pressure flow, respectively, together with the corresponding volume flows. Both set-ups predict a linear velocity profile for a pure shear flow. For the pressure flow, a parabolic profile is predicted by the linear material set-up, while the nonlinear set-up yields a flattened profile with a so-called “plug-zone” depending on the electric field strength. The volume flow based on the nonlinear set-up shows good agreement of calculated and measured values.

Next, the author studies theoretically the influence of a nonhomogeneous electric field on the velocity profile and on the pressure. The field equations are derived for the velocity in the case of a plane incompressible flow for the most general Cauchy stress tensor (compatible with the author’s theory). Concerning an approximation of the first order with respect to a certain small parameter, the author obtains an analytical integral solution for the velocity components. This solution is valid for arbitrary electric fields. The same investigations are also carried out for the nonlinear model. Besides, the author calculates the flow in a rotating viscosimeter. In this case the electric field is not constant. Regarding the linear model, the author obtains a simple closed-form analytical solution for the velocity, which deviates remarkably from the well-known Newtonian solution. Finally, in the case of the nonlinear model, a general integral solution for velocity was given, which allows numerical treatment.

Reviewer: J.Siekmann (Essen)

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76W05 | Magnetohydrodynamics and electrohydrodynamics |

76T20 | Suspensions |