Electromechanics and electrorheology of fluid flow with internal micro-particle electrorotation

Abstract

The negative electrorheological responses of two dimensional Couette and Poiseuille flows with internal micro-particle electrorotation are modeled and analyzed via a set of "fully continuum mechanical modeling field equations" formulated in this thesis. By combining the theories of particle electromechanics and continuum anti-symmetric/couple stresses, general governing equations are presented to describe the physical aspects of mass conservation, linear momentum balance, angular momentum balance, and electro-quasi-static field of the negative electrorheological fluid flow. A "rotating coffee cup model" is also developed for the first time to derive the retarding polarization relaxation equation with its accompanying equilibrium retarding polarization in order to characterize the non-equilibrium motion effects of the continuum spin velocity, co, continuum linear velocity, v, and micro-particle rotation speed, n, on the polarization responses as well as the electrical body torque inputs in the negative electrorheological flow field. Using the general assumptions of steady, incompressible, fully developed, and two dimensional flows, we reduce and simplify the full general governing equations in the zero spin viscosity and the finite spin viscosity small spin velocity limits for both Couette and Poiseuille flow geometries. In the zero spin viscosity limit, expressions for the spin velocity and effective viscosity of Couette flow as well as the spin velocity, linear velocity, and two dimensional volume flow rate of Poiseuille flow are derived in terms of the applied direct current electric field strength, shear rate (for Couette flow), driving pressure gradient (for Poiseuille flow), and spatial coordinate by solving the simplified continuum linear and angular momentum equations with the linear flow velocity being subjected to the no-slip boundary condition. As for the finite spin viscosity small spin velocity limit, analytical solutions to the spin velocity, linear velocity, and effective viscosity of Couette flow as well as solutions to the spin velocity, linear velocity, and two dimensional volume flow rate of Poiseuille flow are obtained and expressed in terms of the applied direct current electric field strength, boundary condition selection parameter (p), spin viscosity, and driving shear rate (for Couette flow) or pressure gradient (for Poiseuille flow) by solving a set of differential equations coupling the linear and angular momentum balances of the negative electrorheological fluid flow subjected to the no-slip and co=0.5/Vxv (with 0 , 1) boundary conditions. After obtaining the solutions in the respective zero spin viscosity and finite spin viscosity small spin velocity limits, series of parametric studies are then performed on these solutions via varying the pertinent physical parameters involved in several parametric regimes of interest so as to illustrate the negative electrorheological behavior and fluid flow response due to internal micro-particle electrorotation. Modeling results in the two limits generally show that with a direct current electric field applied perpendicularly to the flow direction, the spin velocity is increased and the effective viscosity is decreased as compared to the zero electric field values of the electrorheological fluid flow in Couette geometries at a given driving shear rate. It is also found that with a constant driving pressure gradient, the internal micro-particle electrorotation induces increased continuum fluid spin velocity, linear flow velocity, and two-dimensional volume flow rate on the macroscopic level in Poiseuille flow geometries when a direct current electric field perpendicular to the direction of flow is applied. Results of the Couette effective viscosity and Poiseuille volume flow rate obtained from our present continuum mechanical formulation are further compared to the experimental measurements as well as modeling results from single particle dynamics based two-phase volume averaged effective medium analysis found in current literature. With the "rotating coffee cup" fluid polarization model, the present zero spin viscosity continuum solutions to the effective viscosity and volume flow rate agree with the theoretical solutions obtained from single particle dynamics analysis. The zero spin viscosity solutions to the Couette effective viscosity also fall closer to the experimental measurements reported in current literature for low to moderate direct current electric field strengths. Moreover, the present continuum mechanical formulation in the finite spin viscosity small spin velocity limit is more capable of accurately capturing the negative electrorheological flow responses in the low shear rate and low driving pressure gradient flow regimes characterized by the respective Couette effective viscosity and Poiseuille volume flow rate. These finite spin viscosity small spin velocity results agree better with previous experimental measurements reported in the literature and bring the theoretical modeling of the negative electrorheological flow phenomenon due to internal micro-particle electrorotation closer to physical reality-both of which were generally not possible in previous literature. This important improvement in modeling the negative electrorheological response considered in this thesis is due to our proposed "rotating coffee cup model," which is likely the first model to treat the continuum spin velocity and the micro-particle rotation speed as separate physical variables. Using the finite spin viscosity small spin velocity analysis, we also derive for the first time a characteristic length scale determined by the balances between the electrical body torque input and the angular momentum conversion between the linear and spin velocity fields, which can be used to explain why the present continuum zero spin viscosity solutions are very much similar to those obtained from single particle dynamics based two-phase volume averaged effective medium analysis found in current literature. Future work includes a more advanced modeling of the polarization relaxation processes in the negative electrorheological fluid flow, the full non-linear analysis of finite spin viscosity effects on the angular momentum balances within the electrorheological flow field without the restriction of the small spin velocity limit, and the search of possible applications of our proposed continuum mechanical modeling field equations theory in the research areas of micro/nano-fluidics, biofluid dynamics, and engineering torque-shear rate control systems.