Rheological Equations of Viscoelastic Liquid

Abstract

Under the assumption that stress is expressible as a polynomial of elastic strain and actual strain rate, a rheological equation optimum to describe the behaviour of viscoelastic liquid is derived as follows:
τ=(p+α₀)1+α₁ε+α₂f+α₃ε²+α₄f²+α₅(εf+fε)+α₆(ε²f+fε²)
+α₇(εf²+f²ε)+α₈(ε²f²+f²ε²)
where p is a hydrostatic pressure, τ, 1, ε and f are stress tensor, metric tensor, strain tensor and strain rate tensor, respectively, and α₀, α₁···α₈ are polynomials of ten invariants of elastic strain and actual strain rate. Assuming that in simple shearing flow the stress tensor has the same principal axes to the elastic strain tensor and two normal stresses in the direction perpendicular to the stream are equal to each other, the above equation is reduced to
τ^ij=(−p+α₀)G^ij+α₁ε^ij+α₇(ε_αⁱf_β^αf^βj+f_αⁱf_β^αε^βj)
Applications of the theory to simple shearing flow and steady flow through pipe are considered, with satisfactory results.