Numerical Solution Study Of Fractional Viscoelastic Fluid Model

With the development and innovation of science and technology,more and more polymers are synthesized and applied,and some polymers,such as paints,plastics,rubbers,etc.,exhibit the properties of viscoelastic fluids,the fractional calculus is widely used in the field of modeling viscoelastic fluids due to its non-local characteristics and memory property,As the study of fractional calculus continues,more and more scholars are interested in the study of fractional calculus operators of variable order.Therefore,this paper focuses on the numerical study of several types of fractional viscoelastic fluid models.First of all,the background and significance of this thesis,the status of domestic and foreign research,and the main research contents are introduced in the first chapter.In the second chapter,the definition of fractional calculus operator,the definition of variable order fractional calculus operator,the discrete Grownwall’s Lemma and other basic symbols and lemmas are given.Then,the generalized fractional Oldroyd-B fluid model and the fractional Maxwell fluid model are solved numerically in Chapter 3,when the order of the fractional derivative operator is constant,the fractional differential operator is discretized by applying the L1-2σ algorithm,and the other terms in the model are approximated numerically by using the Taylor expansion and the central difference quotient formula.The stability and convergence of the difference format are analyzed,and the validity of the above difference format is demonstrated by the numerical analysis of specific models.Finally,the generalized fractional Oldroyd-B fluid model with variable-order fractional differential operator is discussed.The finite difference format of the variableorder fractional model is established by reducing the model to a system of equations of lower order,obtaining the values of σ for each time layer by Newton’s iteration method,and discretizing the variable-order fractional derivative terms by using the L1-2σ algorithm,The stability and convergence of the difference format are verified,and the validity of the above difference format is demonstrated by the numerical analysis results of the specific model.