## Fluid dynamics of viscoelastic liquids.(English)Zbl 0698.76002

Applied Mathematical Sciences, 84. New York etc.: Springer-Verlag. xvii, 755 p. DM 128.00 (1990).
This work relates to various basic problems associated with viscoelastic fluids and is self contained for a reader in that appropriate appendices are attached. A favoured topic is Hadamard instability and loss of type, in that it figures in six of the twenty chapters, which are on the whole rather short.
The style suggests that it is intended as a primer for the intending research worker, but the individual would need to be wary for slovenliness both in language and reason can be found. For example, on page 15, we find this theorem: ‘The eigenvalues of the tensor $$C_t(\tau)(= F_t(\tau)^TF_t(\tau))$$, are real because the tensor is symmetric; they are positive because $$F_t(\tau)$$ and its transpose have the same eigenvalues’. Since a rigid body rotation – which $$F_t(\tau)$$ may represent – has eigenvalues $$e^{\pm iw(t-\tau)}$$ and 1, it is obvious that the conclusion is not reached as suggested. On the same page the position of a ‘fluid particle’ $$X$$ at time $$-\infty <\tau \le t$$ is given by $$\xi = \chi_t(x,\tau): \chi_t(x,t) = x$$, and it is deduced – by an alternative argument – that naturally $d\xi/dt=d \chi_t(x,\tau)/dt=d{\hat \chi}(X,\tau)/dt = 0$ at $$-\infty <\tau <t$$, i.e. that the material time rate of change of the position of a fluid particle $$X$$ at fixed past time $$-\infty <\tau <t$$ with respect to the current time $$t$$ is zero. This is said to show: ‘The path line is invariant to the motion’. The approach to the definition of mathematical quantities such as Gâteaux and Fréchet derivatives and their relationship, which figure in Chapter 16 on fading memory fluid approximation, is cavalier.
On the whole the book would more appropriately have been named aspects of fluid dynamics of viscoelastic liquids, as it represents a rather personal view of the subject by one who has made considerable contributions to its study. Examination of the reference list would indicate a contribution by him and his co-workers in excess of 20 %.
The fluid models treated early in the work are rate models of Maxwell and Voigt types, their equivalent integral forms and generalizations such as Boltzmann, integral Jeffrey and Oldroyd models. However, the complete generalization of rate type models and the representation theorems used to generate their explicit description are not mentioned [cf. C. Truesdell and W. Noll, The nonlinear field theories of mechanics (1965)]. Fluids of the Rivlin-Ericksen type are totally ignored without explanation, though they appear later in the form of approximations to the simple fluids with fading memory introduced by Coleman and Noll (loc. cit.), and their limitations as approximations is stressed there. That they may be considered as independent consistent fluid models as demonstrated by J. E. Dunn and R. L. Fosdick [Arch. Ration. Mech. Anal. 56, 191–252 (1974; Zbl 0324.76001)] and that stability of the rest state of memory fluids to ‘retarded’ perturbations may be meaningfully discussed by their correct use as an approximation [cf. J. Dunwoody J. Rheol. 27, 373–385 (1983; Zbl 0533.76006)] is ignored. Similar omissions can be detected elsewhere. For example, the works of E. Varley [Arch. Ration. Mech. Anal. 19, 215–225 (1965; Zbl 0136.22101)], and J. Dunwoody and N. T. Dunwoody [Int. J. Eng. Sci. 3, No. 3, 417–427 (1965; doi:10.1016/0020-7225(65)90025-X)] on nonlinear acceleration fronts are totally ignored in Chapter 20. The first named had the original derivation of the nonlinear differential equation governing the strength of the discontinuity in acceleration at a wavefront in memory fluids. Indeed Varley’s derivation, unlike that given by Joseph, is for all wavefront geometries. The same applies to the work of the Dunwoodys on a class of nonlinear Maxwell fluids. To restrict the discussion unnecessarily to plane waves leads to the loss of genuinely important conclusions deriving from the geometry of the front.
Though the book would not qualify as scholarship, perhaps the author never aspired to it, it forms the basis of a very good introduction to the subject matter selected. The problems treated cover, in some detail, steady pipe flows, rod climbing and free surface flows, flow past bodies, elongational flows, die swell and wave propagation.

### MSC:

 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics 76A10 Viscoelastic fluids

### Citations:

Zbl 0324.76001; Zbl 0533.76006; Zbl 0136.22101