Some Applications Of Fractional Calculus To Viscoelasic Fluid Mechanics

This master dissertation consists of three parts. In chapter 1, some preliminary knowledge about the theory of fractional calculus and two kinds of special functions is concisely introduced. In section§1.1, we introduce the development history and recent applied fields of fractional calculus . and give the definitions of Ricmann-Liouville fractional integral operator ₀D_t^-α, differential operator ₀D_t^βand Caputo fractioanl differential operator ₀^CD_t^λ. Besides, some important properties and formulae are given, too. In section§1.2 the definitions, main properties and formulae of Mittag-Leffler funtion and H-Fox function are presented. The two functions are powerful tool in course of solving fractional equations and will be used in the subsequent chapter.Two concrete applications of fractional calculus to viscoelastic fluid are discussed in chapter 2 and chapter 3.In chapter 2, fractioanl calculus is applied to the constitutive equation of second-order fluid, and governing equation for flows of a generalized second-order fluid through a rectangular conduit is established in section§2.2 as followsIn section§2.3, using double finite Fourier transform, temporal Fourier transform and Laplace transform method, we obtain exact solutions for flows through a rectangular conduit under four different conditions. Finally, in section§2.4, we prove that solutions for Newtonian fluid betwwen two infinite parallel plates appear as limiting cases of our solutions. In chapter 3, fractional calculus is applied to the constitutive relationship of Oldroyd-B model fluid, and by introducing inertia force and establishing non-inertial coordinate system governing equations for flows pat an accelerated horizontal plate in a rotating fluid are established in section§3.2 as followsIn section§3.3, by means of dimensionless method and Laplace method we obtain the solutions of velocity and shear stress for above flows. In section§3.4, we made a comparison among the influence of parameters in Oldroyd-B modelλ,θand in fractional differential operatorα,βon the velocity profiles through numerical method in graphic form. The influence of inertia force is also illustrated.