Research And Application Of Heat Transfer And Flow Of Viscoelastic Fluid By Fractional Calculus

The unsteady flow of viscoelastic Non-Newtonian fluids plays a quite applicable role in petroleum, chemical industry, and biological fluid dynamics fields. It is a great outbreat to introduce fractional derivatives into constitutive equations of viscoelastic materials. Considering several fractional constitutive equations of viscoelastic fluids and generalized N-S equation, this paper mainly discuss MHD flow problems of generalized Burgers and Oldroyd-B fluids forced by an accelerating plate, and the impulse fow of fractional second grade fluid. We reveal the flow law of viscoelastic fluids influenced by all the physical parameters and fractional derivatives in order to offer theoretical direction for the carrying application of viscoelastic materials in industrial field.A thermally transient, conductive system takes an important place of research and application in physics and engineering. In Chapter 2, we consider a time-dependent thermal lumped-parameter model with the increasing of Biot numbers in a thermally transient, conductive system. Then the described problem can be approximately modeled by a fractional differential equation, the order of which depends on the Biot number. We take advantage of Laplace transform to get the analytical solution in terms of Mittag-Leffler function.In the research of viscoelastic fluid dynamics Burgers model plays an important practical role in industrial and chemical field. We discuss the magnetohydrodynamic(MHD) flow of an incompressible generalized Burgers’ fluid forced by an infinite accelerating plate in chapter 3. The fractional calculus approach is used both in the motion equation and the constitutive relationship of the Burgers’ fluid. Based on the Mittag-Leffler function and fractional Green’s function, the analytical solution is obtained for the velocity field in integral and series form.In order to research and optimize impluse flow problems of Non-Newtonian fluids in industrial field, we apply a series of integral transforms to deal with impulse flows of viscoelastic fluids in Chapter 4, including the impulse Poiseuille flow of fractional second grade fluid, and the MHD flow of generalized Oldroyd-B fluid due to the boundary condition on central impulse effect. For the former, the analytical solution can be obtained by using finite Fourier sine and Laplace transforms; for the latter, the analytical solution for velocity is obtained in terms of Fox H-function by using the discrete Laplace transform. Moreover, the behavior of many parameters of interest on the velocity fields is shown graphically through different diagrams.From the research data, we conclude that, comparing to the tranditional integer derivative; fractional derivatives and differential equations are more accurate and flexible to solve the problems of heat condutive and flow of viscoelastic fluids.