A new constitutive equation and its performance in contraction flows

A new constitutive equation for incompressible materials is obtained by ass uming that the stress tensor is an isotropic function of two kinematic quan tities, namely, the rate-of-strain tensor and the relative-rate-of-rotation tensor. A representation theorem is employed to obtain the most general sy mmetric form of this function. The arising coefficients are assumed to be f unctions of the second invariants of the two tensors only. Because the seco nd invariant of the relative-rate-of-rotation tensor is an indicator of the flow strength for several flows of engineering interest, the equation is t hus sensitive to it. Forms of these functions are proposed, which lend to t he constitutive equation the capability of fitting closely and independentl y data for shear viscosity, first normal stress coefficient, second normal stress coefficient, and extensional viscosity. This constitutive equation i s used in conjunction with the equations of mass and momentum conservation to obtain the partial differential equations that govern the axisymmetric f low through a 4 : 1 abrupt contraction. These differential equations are in tegrated using the finite volume method to obtain velocity, stress and now- type fields. The effect on flow pattern of parameters related to normal str esses and extensional viscosity is investigated. It is observed that the vo rtex size increases when the level of extensional viscosity is increased, w hile it mildly decreases when the parameter related to normal stress coeffi cients is increased. Moreover, the stress power is highly sensitive to the normal stress parameter. (C) 1999 Elsevier Science B.V. All rights reserved .