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By Uriel Frisch, Joseph B. Keller, George C. Papanicolaou, Olivier Pironneau

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In addition, one defines the second invariant II(M) of a symmetric tensor M through II(M) = 1 1 1 tr M2 = M : M = ||M||2 . 11) In fact, the non-negative square root of the invariant II(M) turns out to be more useful. And, in connection with this matter, we introduce the kinematic invariant or the second invariant K(A) of the Rivlin-Ericksen tensor A. 4 Conservation of Energy 43 1 K(A) = [II(A)]1/2 = √ ||A|| ≥ 0. 12) When deemed necessary, K(A(v)), where A(v) is derived from the velocity field v, is expressed as K(v).

2. A uniquely defined pressure imposes a restriction on the trace of the extra stress tensor in every deformation. 3. Just because the density of an incompressible fluid is not affected by pressure, it is not true that pressure has no effect on rheological properties. In fact, there exists a fully developed theory to incorporate this dependence [2]. We shall now turn to Lagrangian mechanics to demonstrate its relevance to rheology and its use in establishing the above three claims. 1 The Meaning of Pressure In Lagrangian mechanics, the forces acting on a rigid body are split into two classes: those forces which arise due to externally imposed constraints and the rest, usually described as given forces.

Secondly, the critical value τc may be quite different from the yield stress τy in a Bingham fluid; hence, whether a flow exists with or without wall slip depends on the relative strengths of these two properties. 1), the velocity dependent wall shear stress is given by f (uw )uw = Duw D is a constant and the exponent s > 0. 2) ˜ the non-dimensional pressure drop per unit length; next Sc is the and denote by G, critical yield stress number and, finally, Sn is called the slip number. Of course, U is the characteristic velocity and H is the length scale.

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