By Michael Růžička (auth.)

This is the 1st booklet to provide a version, according to rational mechanics of electrorheological fluids, that takes under consideration the complicated interactions among the electromagnetic fields and the relocating liquid. a number of constitutive kinfolk for the Cauchy rigidity tensor are mentioned. the most a part of the booklet is dedicated to a mathematical research of a version owning shear-dependent viscosities, proving the life and area of expertise of susceptible and robust recommendations for the regular and the unsteady case. The PDS structures investigated own so-called non-standard progress stipulations. life effects for elliptic platforms with non-standard progress stipulations and with a nontrivial nonlinear r.h.s. and the 1st ever effects for parabolic platforms with a non-standard development stipulations are given for the 1st time. Written for complex graduate scholars, in addition to for researchers within the box, the dialogue of either the modeling and the math is self-contained.

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**Additional resources for Electrorheological Fluids: Modeling and Mathematical Theory**

**Example text**

While they are compressible if the electric field E changes. This situation is similar to that in a viscous fluid which can undergo only isochoric motions if the temperature is constant, while it can sustain motions that are not isochoric due to changes in the temperature, which is at the heart of the celebrated Oberbeck-Boussinesq approximation, which was recently re-examined in Rajagopal, R~i~ka, Srinivasa [106]. We here will follow the approach outlined in [106] for treating internal constraints.

43) pv - div T + V ¢ = pf + [V E ] P . 43) we obtain an equation for ¢, which reads --A¢ -t- -1V p. L T-div (divT). 44). 17). 45) where ¢, 7], T and P are functions of E, D and 0. 45) reduces to D + kL:~ > 0. 3), where (~q are functions of 0 only, because p is determined through E. 6), where ¢ has to be omitted. 3. 17). 49) by changing the sign of D, and choosing tr D = 0. 30)1-4, where ¢(IEI 2) has to be replaced by - ¢ . 51) can be re-written as 17 0131 ---~0 , a15 = - ~ a2 5 1 , a12 0132 -> 0, = - ~ a13 1 , a51 _> 0.

We assume that there is no external body force and that the electric field and the velocity are given by E = E2 0 , v = . 31). Firstly, we observe that the extra stress S is a function of x2 only. 21) 0a¢ = 0. From the last equation follows that ¢ = ¢(xl, x2) and from the first two equations we conclude a~¢ = 0, ala2¢ = 0, and therefore ¢ is an affine function of xl given by ¢(xl, x2) -- - A x l + k(x2). 22) The constant A can be interpreted as the pressure drop, which maintains the flow. 21) and integrating we obtain $12 = - A x 2 + Co, Co = const.