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By Javier Jiménez

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Example text

Let us assume for a moment p E (1, 2). 18) ~oo = oo. 13). For a more detailed discussion for purely generalized Newtonian fluids we refer the reader to M~lek, NeSas, Rokyta, Rfi~iSka [70] and M~lek, Rajagopal, Rfi~i~ka [74]. From the mathematical point of view the case when #oo = 0 for p E (1, 2) is more challenging and therefore we will assume a3o = aa2 = c~s0 = 0. 13), respectively, for the material functions a2, a3 and as. 4) it follows that a2o = -a21 (cf. 33)), which is already used in the computation of the limits.

Let ~ = [0, 1] and let p(x) = 2 + x. 36) q-2 where co is independent of q. PROOF : Let bk E (0, 1) be arbitrary. 37) Now, given 0 < ak < b~ < 1, we define bk+l • (0, ak) by (bk--ak) bk+ 1 --a k ,4+2 = 2 k. 2. F U N C T I O N 51 SPACES Such bk+x exists since the function F ( x ) = (bk - ak)~h+~ is continuous and decreasing and satisfies F(0 +) = 2 k+l and F ( a k ) = 1. 39) - and hence (cf. 4o) . 39). 36). Let q > 2 and let ko be such that bko+l < q -- 2 < bko. 40). 39), that ~bko+2/b 12_

It is known t h a t viscometric flows are locally a simple shear flow (cf. Huilgol [45]) . • = const. 1) E = E l e i + E2e2 + E3e3. 3) Si2 = a2EiE2 S13 = a2E1E3 -4- -o~~ E 2 E 3 + 4tc2 Ei E3 a5 ~E1 E3 -4- _~ tc2E2 E3 , ,923 = a2E2E3 -4- -~ where ai are functions of the invariants IEI~,5~ x 2, ~EiE2, ~ ( E 2 + E~). , b u t also in the xl - xa and x2 - x3 planes. The shear stress components S13 and $23 are induced by the electric field component E3, which also contributes to the normal stress T33 • All other components of the e x t r a stress S depend on Ea only through the dependence of the a i ' s on ]El 2.

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