Renardy

Renardy, M., 1997. Imposing no boundary condition at outflow why does it work Int. J. Numer. Methods Fluids TA, 413-417. 

Proc of Specialists Meeting held at Les Renardiers May 1982 on Erosion Corrosion of Steels in High Temperature Water and Wet Steam, Eds P. Berge and E. Khan, Electricite de France Paris (1983)... 

Planck H., General aspects in the use of medical textiles for implantation in Planck H., Dauner M., Renardy M. (eds) Medical Textiles for Implantation, Spinger-Verlag, Heidelberg, 1991,1-16. 

Joseph Renardy Fundamentals of Two-Fluid Dynamics Part F Mathematical Theory and Applications... 

The first example, due to Renardy [17,29], deals with a special nonlinear situation, namely that of a small perturbation of a uniform flow v = ( /, 0,0) transverse to a strip, as shown on Figure 1. 

The analysis of the corner singularity is delicate. We refer to the recent works of Hinch [36] and Renardy [37,38], who have contructed a matched asymptotic expansion for the steady solution to a Maxwell fluid flow near the corner. 

We now turn to local existence of solutions for Maxwell-type models. The situation is much trickier here since these models can display Hadamard instabilities (see Section 2.1), and no general results seem to be known so far. One has, in any case, to restrict initial data to Hadamard stable ones. A possible way to overcome the difficulty is to consider models satisfying an eUipticity condition, which will imply well-posedness. This approach was followed by Renardy [41], whose results are briefly described below. 

Under hypothesis (15) and under some smothness conditions on the (bounded) domain of the flow, and on the functions Aijki and, Renardy [41] proves the local existence and the uniqueness of a Hj fl solution of equations (3) and (14), provided that the initial data Vo and To are smooth, and satisfy a compatibility condition at time t = 0. 

In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes. 

Remark 4.5 Corollary 4.1 obviously implies the stability of the rest state (the zero solution) for e not to close to 1. Using a different approach, Renardy [42] has recently removed this restriction and showed the Liapunov stability of the rest state for all e s, 0 < e < 1. This fact is not known for Maxwell-type models (e = 1). 

Remark 4.7 In the case e = 1 (Maxwell models), system (16)-(17) is not always of evolution type. (See section 2.1.) Indeed, Renardy et al. [49] have constructed initial data in the hyperbolic domain, with steep gradients, such that the velocity and the stress develop singularities in their first space derivatives in finite time. The idea is to reduce the system, by a clever change of variables, to a degenerate system of three nonlinear hyperbolic equations. 

A good situation to first at look is the one of linear hyperbolic equations or systems. Actually the implication (S3) ==> (S2) has been proven in [59] for certain hyperbolic systems in one space variable. A more general (and simpler) proof heis been given by Renardy [60]. On the other hand, Renardy [61] has constructed a simple example (namely the wave equation Wtt = + u v + with periodic boundary conditions), showing... 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

In a recent paper [70] Renardy has investigated the nonlinear stability of flows of Jeffreys-type fluids at low Weissenberg numbers. More precisely, assuming the existence of a steady flow (v, r), he proves that this flow is linearly and Liapunov stable provided the spectrum of the linearized operator lies entirely in the open plane 3 A < 0 and that the following quantity is sufficiently small... 

Renardy and Renardy [66,73] have investigated the stability of plane Couette flows for Maxwell-type models involving the derivative (2). The flow lies between parallel plates at a = 0 euid x = 1, which are moving in the j/-direction with velocities 1, such as in Figure 6. 

D.D. Joseph, M. Renardy and J.-C. Saut, Hyperbolicity and change of type in the flow of viscoelastic fluids. Arch. Rat. Mech. Anal., 87 (1985) 213-251. 

M. Renardy, A well-posed boundary value problem for supercritical flow of viscoelastic fluids of Maxwell-type, in Nonlinear Evolution Equations That Change Type, B.L. Keyfitz and M. Shearer (eds.), IMA Volumes in Mathematics and its Applications 27, Springer-Verlag, Berlin, 1991, 181-191. 

M. Renardy, An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions, J. Non-Newtonian Fluid Mech., 36 (1990) 419-425. 

M. Renardy, Existence of steady flows of viscoelaistic fluids of Jeffreys-type with traction boundary conditions, Diff. Int. Eq., 2 (1989) 431-437. 

M. Renardy, Existence of steady flows for Maxwell fluids with traction boundary conditions on open boundaries, Z. Angew. Math. Mech. 75 (1995) 153-155. 

M. Renardy, Local existence of the Dirichlet initial boundary value problem for incompressible hypoelcistic materials, SIAM J. Math. Anal., 21 (1990) 1369-1385. 

M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific and Technical, Burnt Mill, Harlow, 1987. 

M. Renardy and Y. Renardy, Pattern selection in the Benard problem for a viscoelastic fluid, Z. Angew. Math. Mech., 43 (1992) 154-180. 

M. Renardy, On the type of certain Co semi-groups. Comm. Part. Diff. Eq., 18 (1993) 1299-1307. 

M. Renardy, On the linear stability of hyperbolic partial differential equations and viscoelastic flows, Z. Angew. Math. Phys., 45 (1994) 854-865. 

M. Renardy, A rigorous stability proof for plane Couette flow of an uppe-convected Maxwell fluid at zero Reynolds number, Eur. J. Mech. B, 11 (1992) 511-516. 

M. Renardy and Y. Renardy, Stability of shear flows of viscoelastic fluids under perturbations perpendicular to the plane of flow, J. Non-Newtonian Fluid Mech., 32 (1989) 145-155. 

Senecio renardi Senecio retrorsus Senecio rhombifoiius... 

Several snake species contributed the venom used by the Scythians, including the steppe viper Vipera ursinii renardi, the Caucasus viper Vipera kasnakovi, the European adder Vipera berus, and the long-nosed or sand viper Vipera ammodytes transcaucasiana. In ancient India, one of the most feared poisons was derived from the rotting flesh and venom of the white-headed Purple Snake, described by the natural historian Aelian (third century ad). His detailed description suggests that the Purple Snake was the rare. 

Walters, K., Overview of macroscopic viscoelastic flow, in Viscoelasticity and Rheology, Lodge, A.S., Renardy, M., and Nobel, J.A., Eds., Academic Press, London, 1985, p. 47. 

Seneciphylline N-oxide Senecio carthamoides Greene (84, 85) Senecio eremophilus Richards (84, 85) Senecio fremonti Torr. and Gray (83) Senecio ilicifolius Thunb. (117, 118) Senecio jacobaea L. (80, 99-101) Senecio longilobua Benth. (85, 120, 121) Senecio platyphyllua D.C. (116) Senecio pterophorus D.C. (117, 118) Senecio renardi Winkl. (81) Senecio vulgaris L. (84, 85) Erechtites quadridentata D.C. (117) CO. 120 ... 

---