Comments on the paper ?Remarks on the stability of viscometric flow? by M. Ahrens, D. D. Joseph, M. Renardy, and Y. Renardy

Abstract

In the above-mentioned paper [1], the aim of which was "to work out some consequences of the 'short memory approximation' and to give a critical assessment of it", a number of questions were posed. In the present short communication a few comments, which may be useful for further scientific discussion in this area, are addressed to some of these questions.The first critical remark (or problem) was formulated as follows: "It thus appears that the short memory hypothesis is inconsistent unless the fluid has a different memory for perturbations than it has for the shear flow itself. We do not know if and how this could be made precise". The idea of different relaxation times is one possible way of understanding the short memory approximation. Another way is to construct a constitutive model with the following properties,N~ 2 without distinguishing between the relaxation time of the basic flow and its perturbations by assuming that the relaxation time 2 is a decreasing function of the shear rate × in some range of (probably high) ~. This idea could be demonstrated by a modification of the upper convected Maxwell fluid:(S and D are the stress and the strain rate tensors) with deformation-dependent viscosity i/ and relaxation time 4. 38 It is assumed in a simple shear flow that 2 and ~/are functions of ~ (in general, functions of the invariants of D) and they can be approximated for high shear rates by the equations 2 = 20 , (3 a) ~/= ~/0 , (3 b)with constant exponents m, n ~ R+ (n < 1) and material constants 20, ~/0, n0. Making use of eq. (2) leads to the following expressions:We 2 = 4 (1 -m -n) 2 i----n "For any given small value of 2 z, We 2 can be made larger than 16 by suitable choice of the exponents m and n.The second problem was formulated as follows: "The upper and lower convected Maxwell models satisfy, at high enough shear rates, the condition but it was recently shown [2, 3] that a change of type does not occur in those models. Thus any existing