A fractional-order Maxwell model for non-Newtonian fluids

This work considers an extension of the fractional-order Maxwell arrangement to incorporate a relaxation process with non-Newtonian viscosity behavior. The resulting model becomes a fractional-order nonlinear differential equation with stable solution converging asymptotically to a unique equilibrium point. Expressions for the corresponding storage and loss moduli as function of strain frequency and amplitude are computed via a first-harmonic analysis of the differential equation. Some distinctive features and their relationship to the classical and fractional-order linear Maxwell models are discussed. Three examples are used to illustrate the ability of the fractional-order Maxwell model to describe experimental data.

[1]  M. A. Rao.  Rheology of Fluid and Semisolid Foods: Principles and Applications , 2011 .

[2]  Christopher W. Macosko,et al.  Rheology: Principles, Measurements, and Applications , 1994 .

[3]  Hongguang Sun,et al.  An equivalence between generalized Maxwell model and fractional Zener model , 2016 .

[4]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[5]  S. Das,et al.  Forced spreading and rheology of starch gel: Viscoelastic modeling with fractional calculus , 2012 .

[6]  R. Metzler,et al.  Generalized viscoelastic models: their fractional equations with solutions , 1995 .

[7]  A. Pirrotta,et al.  Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results , 2011 .

[8]  Da-Wen Sun,et al.  Computational fluid dynamics (CFD) ¿ an effective and efficient design and analysis tool for the food industry: A review , 2006 .

[9]  C. Gallegos,et al.  Rheology of food, cosmetics and pharmaceuticals , 1999 .

[10]  E. J. Vernon‐Carter,et al.  Designing W1/O/W2 double emulsions stabilized by protein–polysaccharide complexes for producing edible films: Rheological, mechanical and water vapour properties , 2011 .

[11]  P. Saramito,et al.  Efficient simulation of nonlinear viscoelastic fluid flows , 1995 .

[12]  Xiaoyun Jiang,et al.  Parameter estimation for the generalized fractional element network Zener model based on the Bayesian method , 2015 .

[13]  A. Graciaa,et al.  Stability of water/crude oil emulsions based on interfacial dilatational rheology. , 2006, Journal of colloid and interface science.

[14]  N. Heymans,et al.  Fractal rheological models and fractional differential equations for viscoelastic behavior , 1994 .

[15]  R. Bagley,et al.  A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity , 1983 .

[16]  R. Christensen.  Theory of viscoelasticity : an introduction , 1971 .

[17]  Lina Zhang,et al.  Rheological behavior of Aeromonas gum in aqueous solutions , 2006 .

[18]  M. Ghoul,et al.  The structural characteristics and rheological properties of Lebanese locust bean gum , 2014 .

[19]  A. Hanyga.  Fractional-order relaxation laws in non-linear viscoelasticity , 2007 .

[20]  G. W. Blair.  The role of psychophysics in rheology , 1947 .

[21]  Relaxation modulus in the fitting of polycarbonate and poly(vinyl chloride) viscoelastic polymers by a fractional Maxwell model , 2002 .

[22]  G. McKinley,et al.  Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  R. Dupaix,et al.  Exploring the mechanical behavior of degrading swine neural tissue at low strain rates via the fractional Zener constitutive model. , 2014, Journal of the mechanical behavior of biomedical materials.

[24]  R. Metzler,et al.  Relaxation in filled polymers: A fractional calculus approach , 1995 .

[25]  Guillermo Fernández-Anaya,et al.  Lyapunov functions for a class of nonlinear systems using Caputo derivative , 2017, Commun. Nonlinear Sci. Numer. Simul..

[26]  M. P. Gonçalves,et al.  Modelling the rheological behaviour of galactomannan aqueous solutions , 2005 .

[27]  C. Friedrich.  Relaxation and retardation functions of the Maxwell model with fractional derivatives , 1991 .

[28]  Gareth H. McKinley,et al.  A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids , 2014 .

[29]  Helmut Schiessel,et al.  Hierarchical analogues to fractional relaxation equations , 1993 .

[30]  J. Ferry.  Viscoelastic properties of polymers , 1961 .