9. Governing Equations of Fluid Mechanics

The basic equations of continuum mechanics for moving continua are derived from first principals employing the continuum hypothesis, substantial derivative, and material and spatial coordinates. After the continuity equation is deduced from conservation of mass using the Reynolds Transport Theorem, a general balance law is developed and applied to momentum (linear and angular) and energy for nonpolar continua. Then equations of state relevant to ideal and adiabatic gases and constitutive relations relevant to Newtonian fluid flow are introduced involving thermodynamics and Cartesian tensor notation, respectively. In addition the continuity equation in Cartesian coordinates is deduced by a fixed volume big box method as well and then that equation is converted to cylindrical coordinates by direct transformation. The problems examine various aspects of these concepts including incompressibility, stream functions, a big box derivation of the continuity equation in cylindrical coordinates, a transformation of the continuity equation in Cartesian coordinates to spherical coordinates, the constitutive relations for Newtonian fluids, the governing equation for chemical species conservation, the balance of angular momentum for polar continua, the terms in the energy equations for ideal gases, and the Clausius-Duhem inequality.