RHEOLOGICA ACTA AN INTERNATIONAL JOURNAL OF RHEOLOGY Band 5 September 1966 Heft 3 From the Department o/ Chemical and Metallurgical Engineering, University o/Michigan (USA) An Inverse for the Jaumann Derivative and some Applications to the Rheology of Viscoelastic Fluids By J. D. Goddard and Chester Miller (Received June 23, 1966) 1. Introduction As has been pointed out previously by several authors, notably by Oldroyd (1) and Prater (2), there are many "time-like" tensor derivatives which, being equally satisfactory in their invariance properties, are acceptable for formulating rates of change in rheological equations of state. Two such derivatives, which have received considerable application in the theology of fluids, are the convected derivative and the co-rotational or Jaumann derivative (1, 2, 3). While each of these has its particular merits insofar as its physical significance is concerned, the Jaumann derivative is somewhat simpler in its formal mathematical properties, as has been em- phasized by Prater (2), In particular, since it possesses the rather desirable feature of commutativety with the metric tensor, it exhibits essentially all the formal algebraic properties of the ordinary time derivative. Following Oldroyd (1), we shall define here the Jaumann derivative of an absolute tensor field B2~i.2(xJ, t), associated with a material in motion, by ~)t -- Dt [~] where 2J and 2J' represent sums over similar terms and D/Dt denotes the material deriv- ative: DB]~!: 0B:~!: D~ ~- + vmB]ii.], m [2] with xJ and t denoting spatial coordinates and time respectively. Furthermore, v i (xJ, t) denotes in [2] (the contravariant component of) the material velocity vector, while in [1], tgik denotes (the mixed component of) the associated vorticity or rotation-rate tensor: 1 9ik = ~ (vi, k - vk, i) [3l [which here has a sign opposite that used by Oldroyd (it)]. As emphasized by Oldroyd, eq. [1] gives the time rate of change of the ..i. tensor B.k.. as reckoned by an observer fixed in a coordinate frame which moves with the local (linear) velocity v i and rotates rigidly with the local angular velocity, oi, of the material (a co-rotational coordinate frame), where ~ i 1 = y eiJk ~gkj. [4] We recall that Oldroyd (4) has already proposed a so-called "quasi-linear" rheo- logical equation for viscoelastic fluids which involves linear combinations of higher Jau- mann derivatives. These combinations have the general form [ ] 1 + 21"-~-+22 -~ +"" Bij, where 21, 23,... are material constants and where Bij denotes either the (deviatoric) stress tensor Pi j, say, or the deformation-rate tensor Ei j. Such terms can either be derived from an analysis of the microscopic structure of suspensions, or they can simply be assumed as an admissible generalization of a linear viscoelastic model, with terms like ~2 [1 +.~1~-~ + 2 z ~ + ...]Bjj which is the limit of the preceding expression for small velocities and deformation rates. However, as is well known, the latter ex- pression is not the most general viscoelastic operator. From a more general point of view it is desirable to admit a continuous spectrum by introducing the appropriate material "mem- ory" functions in the form of a stress- relaxation modulus or creep compliance. Indeed, some quite general rheological models, which exhibit an arbitrary con- tinuous spectrum in a time parameter in 13