One-dimensional Model for Fluids of Third-grade in Tubes with Constant Radius

Our goal in this paper is to extend previous results obtained for Newtonian and secondgrade fluids to third-grade fluids in the case of an axisymmetric, straight, rigid and impermeable tube with constant cross-section using a one-dimensional hierarchical model based on the Cosserat theory related to fluid dynamics. In this way we can reduce the full threedimensional system of equations for the axisymmetric unsteady motion of a non-Newtonian incompressible third-grade fluid to a system of equations depending on time and on a single spatial variable. Some numerical simulations for the volume flow rate and the the wall shear stress are presented.

Specific shear-dependent viscoelastic third-grade fluid model

A specific modified constitutive equation for a third-grade fluid is proposed so that the model be suitable for applications where shear-thinning or shear-thickening may occur. For that, we use the Cosserat theory approach reducing the exact three-dimensional equations to a system depending only on time and on a single spatial variable. This one-dimensional system is obtained by integrating the linear momentum equation over the cross-section of the tube, taking a velocity field approximation provided by the Cosserat theory. From this reduced system, we obtain the unsteady equations for the wall shear stress and mean pressure gradient depending on the volume flow rate, Womersley number, viscoelastic coefficient and flow index over a finite section of the tube geometry with constant circular cross-section.

Into the looking glass: Literacy acquisition and mirror invariance in preschool and first-grade children

At what point in reading development does literacy impact object recognition and orientation processing? Is it specific to mirror images? To answer these questions, forty-six 5- to 7-year-old preschoolers and first graders performed two same–different tasks differing in the matching criterion-orientation-based versus shape-based (orientation independent)-on geometric shapes and letters. On orientation-based judgments, first graders out- performed preschoolers who had the strongest difficulty with mirrored pairs. On shape-based judgments, first graders were slower for mirrored than identical pairs, and even slower than preschoolers. This mirror cost emerged with letter knowledge. Only first graders presented worse shape-based judgments for mirrored and rotated pairs of reversible (e.g., b-d; b-q) than nonreversible (e.g., e-ә) letters, indicating readers’ difficulty in ignoring orientation contrasts relevant to letters.