Incompressible Flow

Abstract

Starting from conservation of mass, momentum, and energy, the equations for incompressible flow are derived using the divergence-free condition ∇ • V↦ = 0, which implies that there is no compression or expansion in the flow field. The equation for conservation of mass becomes

$$ \rho t + \overrightarrow V \, \cdot \,\nabla \rho = 0, $$

(18.1)

indicating that the (possibly spatially varying) density is advected along streamlines of the flow. The Navier-Stokes equations for viscous incompressible flow are

$$ {u_t} + \vec{V} \cdot \nabla u + \frac{{px}}{\rho } = \frac{{{{\left( {2\mu {u_x}} \right)}_x} + {{\left( {\mu \left( {{u_{{_y}}} + {v_x}} \right)} \right)}_y} + {{\left( {\mu \left( {{u_z} + {w_x}} \right)} \right)}_z}}}{\rho },\, $$

(18.2)

$$ {v_t} + \vec{V} \cdot {\nabla_V} + \frac{{{p_y}}}{\rho } = \frac{{{{\left( {\mu \left( {{u_y} + {v_x}} \right)} \right)}_x} + {{\left( {2\mu {v_y}} \right)}_y} + {{\left( {\mu \left( {{v_z} + {w_y}} \right)} \right)}_z}}}{\rho } + g, $$

(18.3)

$$ {w_t} + \vec{V} \cdot {\nabla_w} + \frac{{{p_z}}}{\rho } = \frac{{{{\left( {\mu \left( {{u_z} + {w_x}} \right)} \right)}_x} + {{\left( {\mu \left( {{v_z} + {w_y}} \right)} \right)}_y} + {{\left( {2\mu {w_z}} \right)}_z}}}{\rho }, $$

(18.4)

where μ is the viscosity and g is the acceleration of gravity. These equations are more conveniently written in condensed notation as a row vector

$$ {\vec{V}_t} + \left( {\vec{V} \cdot \nabla } \right)\vec{V} + \frac{{{\nabla_p}}}{\rho } = \frac{{{{\left( {\nabla \cdot \tau } \right)}^T}}}{\rho } + \vec{g}, $$

(18.5)

where “T” is the transpose operator, g↦ = <0, g, 0>, and τ is the viscous stress tensor

$$ \tau = \left( {\begin{array}{*{20}{c}} {2{u_x}} & {{u_y} + {v_x}} & {{u_z} + w_x} \\ {{u_y} + {v_x}} & {2{v_y}} & {{v_z} + {w_{\partial }}} \\ {{u_z} + {w_x}} & {{v_z} + {w_y}} & {2{w_z}} \\ \end{array} } \right), $$

(18.6)

which can be expressed in a more compact form as

$$ \tau = \mu \left( {\begin{array}{*{20}{c}} {{\nabla_u}} \\ {{\nabla_v}} \\ {{\nabla_w}} \\ \end{array} } \right) + \mu {\left( {\begin{array}{*{20}{c}} {{\nabla_u}} \\ {{\nabla_v}} \\ {{\nabla_w}} \\ \end{array} } \right)^{{\rm T}}}. $$

(18.7)