Dipolar fluids

Travelling waves of density for a fourth-gradient model of fluids

“…This is a necessary step to study dispersive wave or localisation phenomena [11,12]. In fluid dynamics mathematical models that takes into account higher order derivatives of the stretching tensor have been introduced mainly to describe dipolar fluids and turbulence [13,14,15,16]. In the present paper we consider higher gradients of the density.…”

Steady flows of nonlinear bipolar viscous fluids between rotating cylinders

“…(i.4) T\ 2 Also, the pressure distribution required to maintain the flow (1.3) is given by p~lp(r) = \A2r2 -\B2r~2 + 2A-B\nr (1.5) where p is the (constant) fluid density and B= (1.6) 2 ~ A r2 ~ ri Finally, the frictional couple exerted, per unit length, across a cylindrical surface in the fluid of radius r, rx <r <r2, is independent of r and is given by 2nr trg = ~mpQ (1)(2)(3)(4)(5)(6)(7) where tr8 is the tangential stress. In deriving the relations (1.3), (1.5), (1.6), and (1.7), one writes the equilibrium Navier-Stokes equations (based on (1.1)) in cylindrical coordinates (r, 6, z) and looks for solutions, subject to (1.4), of the form vr = r = 0, vg-rd(r), vz = z = 0 (1.8) which can be supported by a pressure distribution p -p(r); in such a situation (1.1) reduces to tre~^^0er6 (1)(2)(3)(4)(5)(6)(7)(8)(9) where In recent work [4] Necas and Shilhavy have examined the foundations of a continuum mechanical theory for the response of a multipolar viscous fluid; the theory that is proposed in [4] is consistent with the second law of thermodynamics, in the form of the Clausius-Duhem inequality, and builds upon earlier work of Toupin [5], Green and Rivlin [6,7], and Bleustein and Green [8]. Bellout, Bloom, and Necas [9] explored several of the consequences of the theory formulated in [4] with emphasis on the isothermal, incompressible, bipolar case that is described below; special consideration was given in [9] to the nature of the velocity profiles predicted for several important cases of laminar flows at low viscosity.…”

“…(where n is given by (1.12), with 1 < p < 2), 14) and V1 = 0' tijkvjvk = ®> ' -1, 2> 3 on dSl X [0, T) (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) (with prescribed initial conditions relative to u and |y) where v is the exterior unit normal to dQ. In (1.13) p is the (constant) density and f is the external body force vector; the second set of boundary conditions is a direct consequence of the principal of virtual work (e.g., Toupin [5]) and expresses the condition that the first moments of the traction vanish on the boundary (similar boundary conditions are present in the work of Bleustein and Green [8]).…”

“…In (1.13) p is the (constant) density and f is the external body force vector; the second set of boundary conditions is a direct consequence of the principal of virtual work (e.g., Toupin [5]) and expresses the condition that the first moments of the traction vanish on the boundary (similar boundary conditions are present in the work of Bleustein and Green [8]). For the initial-boundary value problem (1.13-1.15), with ji as given by (1.12), and \ < p <2, existence of unique solutions has been proved in [20] and the existence of a global compact attractor for the semigroup associated with (1.13-1.15) has been demonstrated in [21]; the analogous results for the case p > 2 (with n1 > 0) may be found in [19] and [22], Our concern in this paper, however, is with a much more mundane problem for the bipolar viscous fluid, namely, the existence of equilibrium solutions of the form (1.8) for the case where Q. is the domain bounded by two concentric circular cylinders of radii r, and r2 that rotate with constant angular velocities Q, and £22.…”

Inertial manifolds of incompressible, nonlinear bipolar viscous fluids

“…The theory of multipolar materials is due to Green and Rivlin [1], [2], who considered the constitutive equations for an elastic, nonviscous material; a model for a bipolar fluid may be found in the paper of Bleustein and Green [3]. Necas and Silhavy [4] developed a thermodynamic theory of constitutive equations for multipolar viscous fluids within the framework of the theory of Green and Rivlin [1], [2]; the general constitutive theory developed in [4] is consistent with the principle of material-frame indifference and the second law of thermodynamics as expressed by the Clausius-Duhem inequality.…”