Differences in Normal Stress in Aqueous Solutions of Turbulent Drag Reducing Additives

Abstract

WHEN a fluid is in motion the normal stresses acting on the faces of a small fluid element which is instantaneously in the form of a cube are not in general equal. For example, in a Newtonian liquid the tensile stresses τxx, τyy, τzz acting on the cube faces perpendicular to orthogonal axes x, y, z are given by where p is pressure, μ is viscosity, and u, v w are the velocity components in the x, y, z directions. In the special case of flow through a long straight tube of uniform cross-section, however, u does not vary with x, and v and w are zero, so that then τxx = τyy = τzz = −p for a Newtonian liquid. For non-Newtonian liquids, on the other hand, such an equality may no longer obtain, even for this restricted type of flow. Thus for axial flow in the annular passage between two concentric tubes, of inner arid outer radius r1 and r2, respectively, the radial normal stress τrr may differ from the circumferential stress τθθ. For radial equilibrium or Thus measurements of the pressure difference can give an indication of the average magnitude of the normal stress difference τrr − τθθ.