Convected derivatives for differential constitutive equations

• Published in: Journal of non-Newtonian fluid mechanics Vol. 24; no. 3; pp. 331 - 342
• Main Author: Larson, R.C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 1987; Elsevier
• Subjects: Exact sciences and technology; Fluid dynamics; Fundamental areas of phenomenology (including applications); Non-newtonian fluid flows; Physics; Constitutive equation; Non Newtonian fluid; Differential equation; Shear flow
• ISSN: 0377-0257; 1873-2631
• DOI: 10.1016/0377-0257(87)80045-9

To date, no differential constitutive equation has been proposed that agrees with each of four important experimental observations in relaxation after step shear strains: that the stress is often factorable into time and strain-dependent functions, that the strain-dependent function is shear thinning, that the ratio of first normal stress difference to shear stress equals the shear strain—that is, the Lodge-Meissner relationship holds, and that there is a negative second normal stress difference. The Johnson-Segalman model satisfies three of these, but fails to satisfy the Lodge-Meissner relationship, because in step strains the principal stress and strain axes do not rotate together. Using a mathematical technique for forcing co-rotation of stress and strain axes in an arbitrary deformation, we here present an explicit differential constitutive equation that satisfies all four of the above experimental observations.