Asymptotic relations between shear stresses and normal stresses in general incompressible fluids

It is shown that in the general theory of incompressible simple fluids with fading memory there are, for several types of nonsteady shearing motions, simple universal asymptotic relations between the shear stress S₁₂ and the first normal stress difference N₁ = S₂₂ – S₁₁. The kinematical situations considered include initiation of steady shearing, rest after steady shearing, and sinusoidal oscillation. In, for example, relaxation following cessation of a steady shearing flow with rate of shear κ, there holds, to within an error O(κ⁴): This and the other derived universal relations between N₁ and S₁₂ are either consequences of, or are closely related to a general asymptotic formula B. D. Coleman and W. Noll, Revs. Mod. Phys., 33 , 239 (1961), eq. (6.15)] expressing N₁ as an integral of the product of the shear relaxation modulus and the square of the history of the relative shear strain.