The flow features of inelastic fluids find several industrial applications. Recently, Sacheti et al. have initiated analytical studies to account for the dilatant phenomenon. The authors assumed a linear approximation for the apparent viscosity, and discussed some boundary layer flows. Since it is known that certain applications require the consideration of higher order effects in the rheological models, we have investigated the additive effect of the quadratic term in the constitutive equation with a view to knowing whether such higher order term can have appreciable effects. We assume the constitutive equation of an inelastic fluid as ij 1⁄4 ðI2Þeij 1⁄4 ð 0 þ 1I2 þ 2I 2 þ Þeij, where I2 is the second scalar invariant of the rate of strain tensor, and 0, 1, 2 are the material parameters of the fluid. Here, we extend our previous work to the quadratic approximation for . Since this model has not previously been considered, to the authors’ knowledge, we have derived the governing equations for two-dimensional steady incompressible flow. Using the suffix notations to denote partial derivatives, the continuity equation is, ux þ vy 1⁄4 0, and the momentum equations are