Local Differential Quadrature Method In Flow Problems Of Viscous And Plastic Fluid In Duct

Flow problems of viscous and plastic fluid in duct are very important in the field of physics, mechanics, engineering, etc. The mathematical form can be formulated as mixed elliptic variational inequalities with the second kind. Ordinarily the numerical methods of these problems are finite difference method(FDM) and finite element method(FEM). Currently, meshless method is a new and highly effective approach for solving partial differential equations. It becomes increasely popular because of its no need of mesh property. However, it is seldom used in numerical solution of variational inequalities. Differential quadrature method(DQM) and local differential quadrature method(LDQ) are a kind of meshless methods. In this paper, we construct the couple method of Uzawa and LDQ to solve the flow problems of viscous and plastic fluid in duct.The main work is as follows:First, we introduce the basic knowledge of DQM and LDQ. The derivative of high order at a mesh point and error estimation are derived in detail with Lagrange interpolation. Effectiveness of DQM method is testified by piecewise functions. Then we discuss the effect of total number of nodes and node distribution on computational errors of DQM and LDQ methods through numerical experiments. Second, we discuss the flow problems of viscous and plastic fluid in duct by couple method Uzawa iteration and LDQ. Applying LDQ method, we construct the numerical method for the mixed and elliptic variational inequalities of the second kind. The velocity of fluid in a duct is obtained. Through two numerical experiments, we prove the effectiveness of such couple method. Also, we present the numerical accuracy of LDQ method. Finally, we compare the computational efficiency of our method with the dual method of Uzawa iteration and FEM.