Unidirectional flows of fractional Jeffreys' fluids: Thermodynamic constraints and subordination

A class of initial-boundary value problems governing the velocity distribution of unidirectional flows of viscoelastic fluids is studied. The generalized fractional Jeffreys' constitutive model is used to describe the viscoelastic properties. Thermodynamic constraints on the parameters of the model are derived from the monotonicity of the corresponding relaxation function. Based on these constraints, a subordination principle for the considered class of problems is established. It gives an integral representation of the solution in terms of a probability density function and the solution of a related wave equation. Explicit representation of the probability density function is derived from the solution of the Stokes' first problem. Numerical verification of the obtained analytical results is provided.