Electrorheological fluids: modeling and mathematical theory.

*(English)*Zbl 0962.76001
Lecture Notes in Mathematics. 1748. Berlin: Springer. xvi, 161 p. (2000).

In this book the author investigates a model of electrorheological fluids in which the extra part \(S\) of the stress tensor \(-\phi I+S\) is given by \(S=\alpha _{21}((1+|D|^2)^{\frac{p-1}{2}}-1)E\otimes E+\) \((\alpha _{31}+\alpha _{33}|E|^2)(1+|D|^2)^{{\frac{p-2}{2}}}D\)+ \(\alpha _{31}(1+|D|^2)^{{\frac{p-2}{2}}}(DE\otimes E+E\otimes DE)\), where \(D=(\nabla v+\nabla v ^{T})/2\) is the rate of strain tensor, and \(E\) the electric field vector. It is important to note that \(p\) depends on \(|E|^2\), and \(1<p_{\infty}\leq p\leq p_0 <\infty\), for some constants \(p_{\infty}\) and \(p_0\). The total model consists of the quasi-static Maxwell’s equations \(\text{div}E=0,\) \(\text{curl}E=0,\) the linear momentum equation \(\rho_0 v_t +\rho _0 (v\cdot\nabla)v=\) \(-\nabla \phi +\text{div}S+\) \(\rho _0 f+\chi (E\cdot\nabla)E\), and the incompressibility equation \(\text{div}v=0\), where \(v\) is the velocity vector.

First, the author studies steady equations in a bounded domain \(\Omega\subset \mathbb{R}^3\) with boundary conditions \(v=0\) and \(E\cdot n=E_0 \cdot n\) at \(\partial\Omega\). Fortunately, the system is separated into the Maxwell’s equations for \(E\) and the balance equations for \(v\), where \(E\) can be viewed as a parameter. Generally, \(E\) is not constant. So, the parameter \(p\) in the stress-strain law is a given function of the space variable \(x\). Naturally, the corresponding functional setting are the spaces \(L^{p(x)}(\Omega)\) and \(W^{1,p(x)}(\Omega)\), the so-called generalized Lebesgue and generalized Sobolev spaces, respectively. Particularly, for given \(p(x)\), the norm in \(L^{p(x)}(\Omega)\) is defined by \(\|f\|_{p(x)}=\) \(\text{inf}\{\lambda >0:|f/\lambda |_{p(x)}\leq 1\},\) \(|f|_{p(x)}=\int_{\Omega}|f(x)|^{p(x)}dx\).

The existence of weak solutions is proved if \(p_{\infty}>9/5\). The solution is unique if data are small enough. The existence result is accomplished by adapting the theory of monotone operators to the framework of generalized Sobolev spaces. The existence of strong solutions is proved by another approach. Approximate solutions are constructed, and their second derivatives are estimated locally by the difference quotient method uniformly with respect to the approximation parameter.

In the final part, the study of unsteady flows is performed for the simplified stress-strain law \(S= \alpha _{31}(1+|E|^2)(1+|D|^2)^{{\frac{p-2}{2}}}D\). The main results are the global existence of weak (for \(p_{\infty}\geq 2\)) and strong solutions, and the uniqueness of strong solutions.

First, the author studies steady equations in a bounded domain \(\Omega\subset \mathbb{R}^3\) with boundary conditions \(v=0\) and \(E\cdot n=E_0 \cdot n\) at \(\partial\Omega\). Fortunately, the system is separated into the Maxwell’s equations for \(E\) and the balance equations for \(v\), where \(E\) can be viewed as a parameter. Generally, \(E\) is not constant. So, the parameter \(p\) in the stress-strain law is a given function of the space variable \(x\). Naturally, the corresponding functional setting are the spaces \(L^{p(x)}(\Omega)\) and \(W^{1,p(x)}(\Omega)\), the so-called generalized Lebesgue and generalized Sobolev spaces, respectively. Particularly, for given \(p(x)\), the norm in \(L^{p(x)}(\Omega)\) is defined by \(\|f\|_{p(x)}=\) \(\text{inf}\{\lambda >0:|f/\lambda |_{p(x)}\leq 1\},\) \(|f|_{p(x)}=\int_{\Omega}|f(x)|^{p(x)}dx\).

The existence of weak solutions is proved if \(p_{\infty}>9/5\). The solution is unique if data are small enough. The existence result is accomplished by adapting the theory of monotone operators to the framework of generalized Sobolev spaces. The existence of strong solutions is proved by another approach. Approximate solutions are constructed, and their second derivatives are estimated locally by the difference quotient method uniformly with respect to the approximation parameter.

In the final part, the study of unsteady flows is performed for the simplified stress-strain law \(S= \alpha _{31}(1+|E|^2)(1+|D|^2)^{{\frac{p-2}{2}}}D\). The main results are the global existence of weak (for \(p_{\infty}\geq 2\)) and strong solutions, and the uniqueness of strong solutions.

Reviewer: Vladimir Shelukhin (Novosibirsk)

##### MSC:

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

76A05 | Non-Newtonian fluids |

35Q35 | PDEs in connection with fluid mechanics |

76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |

76W05 | Magnetohydrodynamics and electrohydrodynamics |