tailieunhanh - Environmental Fluid Mechanics - Chapter 3

CÁC HÌNH THỨC của các phương trình của MOTION Dòng chảy lỏng có độ nhớt cao là toán học được đại diện bởi các giải pháp của các phương trình chuyển động, dựa trên chuyển động lực trong một thể tích dịch tiểu học. các Các phương trình của chuyển động đối với các dòng nhớt các phương trình Navier-Stokes giới thiệu trong các chương trước. Để thuận tiện, chúng tôi lặp lại những phương trình này ở đây, đối với trường hợp trong đó sự thay đổi độ nhớt là không đáng kể: . | 3 Viscous Flows VARIOUS FORMS OF THE EQUATIONS OF MOTION Viscous flows are mathematically represented by solutions of the equations of motion based on momentum transfer in an elementary fluid volume. The equations of motion for viscous flows are the Navier-Stokes equations introduced in the previous chapter. For convenience we repeat these equations here for cases in which variations in viscosity are negligible mz . . 1 C V V V --V p C pgZ C vV2V dt p where V is the velocity t is time V is the gradient vector p is the density p is the pressure g is gravitational acceleration Z is the elevation with regard to an arbitrary reference and v is the kinematic viscosity. In Appendix 1 tables of Navier-Stokes equations for Cartesian cylindrical and spherical coordinate systems are listed. In Appendix 2 relationships are given between stress components and velocity components as implied by the Navier - Stokes equations. Using Cartesian tensor notation Eq. is represented as dUị dUi 1 dp dZ d2 ui 1Ĩ C H pi - - C v lị where ui represents components of the velocity vector and xi represents the coordinates. This equation incorporates four unknown quantities three components of the velocity vector and the pressure. Along with the continuity equation we thus have a system of four differential equations with four unknowns. The solution of this system subject to appropriate initial and boundary conditions provides the required information about the distribution of the unknown quantities in the domain. The distributions of velocities and pressure depend on the three space coordinates x y and z and the time coordinate t. It should be noted that the Copyright 2001 by Marcel Dekker Inc. All Rights Reserved. order of the differential equation varies with regard to the unknown quantities as well as with regard to the various coordinates. The velocity components contribute terms of first order with regard to time and of both first and second order with regard