A new constitutive equation for incompressible materials is obtained by assuming that the stress tensor is an isotropic function of two kinematic quantities, namely, the rate-of-strain tensor and the relative-rate-of-rotation tensor. A representation theorem is employed to obtain the most general symmetric form of this function. The arising coefficients are assumed to be functions of the second invariants of the two tensors only. Because the second invariant of the relative-rate-of-rotation tensor is an indicator of the flow strength for several flows of engineering interest, the equation is thus sensitive to it. Forms of these functions are proposed, which lend to the constitutive equation the capability of fitting closely and independently data for shear viscosity, first normal stress coefficient, second normal stress coefficient, and extensional viscosity. This constitutive equation is used in conjunction with the equations of mass and momentum conservation to obtain the partial differential equations that govern the axisymmetric flow through a 4 : 1 abrupt contraction. These differential equations are integrated using the finite volume method to obtain velocity, stress and now-type fields. The effect on flow pattern of parameters related to normal stresses and extensional viscosity is investigated. It is observed that the vortex size increases when the level of extensional viscosity is increased, while it mildly decreases when the parameter related to normal stress coefficients is increased. Moreover, the stress power is highly sensitive to the normal stress parameter.