Some exact solutions of the third‐order vector partial differential equation describing nonsteady flows of incompressible second‐order fluids are presented. Although periodic simple shearing flow is also discussed, the main emphasis here is on helical flows between moving coaxial cylinders, particularly periodic flows of the Couette and Poiseuille type. The calculations suggest that, for such periodic helical flows, measurements of the radial thrusts on the bounding cylindrical walls supply practicable methods for the determination of normal‐stress coefficients.
REFERENCES
1.
2.
This is a rough explanation in words of Noll’s [
W.
Noll
, Arch. Rat. Mech. Anal.
2
, 197
(1958
)] mathematical concept of fluidity.3.
Since we are here concerned with incompressible materials only, it is understood that whenever in our text we use the word fluid we mean incompressible fluid. For a survey of the implications of gradually fading memory in the theory of compressible simple fluids see the article by B. D. Coleman and W. Noll in Proceedings of the International Symposium on Second‐Order Effects in Elasticity, Plasticity, and Fluid Dynamics, Haifa, 1962 (Jerusalem Academic Press, Jerusalem, 1964).
4.
5.
6.
7.
H. Markovitz and B. D. Coleman, Adv. Appl. Mech. to be published.
8.
L.
Boltzmann
, Sitzber. Kaiserlich. Akad. Wiss. (Wien), Math‐Naturwiss. Kl.
70
, Sect. II
, 275
(1874
).9.
10.
Normal‐stress measurements have been made by Markovitz and Brown [
H.
Markovitz
and D.
Brown
, Trans. Soc. Rheol.
7
, 137
(1963
);H. Markovitz, in Proceedings of the Fourth International Congress of Rheology, 1963 (John Wiley & Sons, Inc., New York, 1964),] on steady flows of solutions of polyisobutylene in cetane. Their work confirms the inequality for such solutions.
11.
For a discussion of Eq. (1.10) see the survey of B. Gross [B. Gross, Mathematical Structure of the Theories of Viscoelasticity (Hermann & Cie, Paris, 1953)].
In his book J. D. Ferry [J. D. Ferry, Viscoelastic Properties of Polymers (John Wiley & Sons, Inc., New York, 1961)] attributes Eq. (1.11) to H. Fujita.
12.
See Ferry’s book (Ref. 11).
13.
See also Coleman and Noll (Ref. 4), Ting [
T. W.
Ting
, Arch. Rat. Mech. Anal.
14
, 1
(1963
)], and Markovitz and Coleman (Ref. 7).14.
Compare the solution of Markovitz and Coleman which describes a situation related to, but different from, that which we discuss here in Eqs. (2.6)–(2.11).
15.
The work of Ting (Ref. 13) on initial value problems rests heavily on the assumption that ρ, and γ all have the same sign, an assumption contradicting (1.9).
16.
C. Truesdell has advised us that he has also studied problems involving standing waves in second‐order fluids and will publish on them shortly.
17.
Coleman and Noll (Ref. 4), considering the case (i.e., nonsteady Couette flows), obtained Eq. (3.7), remarked on its linearity, and suggested that its theory be studied.
18.
Ting (Ref. 13), considering the case (i.e., non‐steady Poiseuille flows), obtained Eq. (3.8) and studied initial‐value problems for it assuming that ρ, and γ all have the same sign.
19.
20.
Cf. Eqs. (4.11), (4.21), (4.30), (4.35), and (5.14b) of Ref. 6.
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© 1964 The American Institute of Physics.
1964
The American Institute of Physics
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