Hydrodynamics of Viscous Fluid

Abstract

In Chap. 5, the equation of continuity and the Navier-Stokes equations, which are momentum conservation equations, were derived as the fundamental equations to describe fluid motion.

$$ \frac{\partial \rho }{\partial t}+\frac{\partial \rho u}{\partial x}+\frac{\partial \rho v}{\partial y}+\frac{\partial \rho w}{\partial z}=0 $$

$$ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\frac{1}{\rho}\left(\frac{\partial {\tau}_{xx}}{\partial x}+\frac{\partial {\tau}_{yx}}{\partial y}+\frac{\partial {\tau}_{zx}}{\partial z}\right)+\frac{F_x}{\rho } $$

$$ \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial y}+\frac{1}{\rho}\left(\frac{\partial {\tau}_{xy}}{\partial x}+\frac{\partial {\tau}_{yy}}{\partial y}+\frac{\partial {\tau}_{zy}}{\partial z}\right)+\frac{F_y}{\rho } $$

$$ \frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial z}+\frac{1}{\rho}\left(\frac{\partial {\tau}_{xz}}{\partial x}+\frac{\partial {\tau}_{yz}}{\partial y}+\frac{\partial {\tau}_{zz}}{\partial z}\right)+\frac{F_z}{\rho } $$

Of these equations, the Navier-Stokes equations are nonlinear partial differential equations. It is generally difficult to exactly obtain solutions for nonlinear partial differential equations; however, since exact solutions can be obtained under special conditions, some examples of such solutions are described below.