Navier-Stokes Equation
     

Using rate of stress and rate of strain tensors, it can be shown that the components of a viscous force in a non-rotating frame are given by

(1)

where is the dynamic viscosity and is the second viscosity coefficient (Tritton 1988). For an incompressible fluid, is constant the term drops out. Taking to be constant in space, this leaves
\begin{displaymath} {{\mathbf{F}}_{\rm viscous}\over V} = \eta\nabla^2 {\mathbf{u}}. \end{displaymath} (2)

There are two additional forces, pressure force
\begin{displaymath} {{\mathbf{F}}_{\rm\href{pressure}}\over V} = -\nabla P \end{displaymath} (3)

and a body force
\begin{displaymath} \mathcal{F} = {{\mathbf{F}}_{\rm body}\over V}. \end{displaymath} (4)

Adding these three forces and equating them to Newton's law (fluid) yields
\begin{displaymath} \rho {\partial {\mathbf{u}}\over\partial t} +\rho {\mathbf{u... ...mathbf{u}} = -\nabla P+\eta\nabla^2 {\mathbf{u}} +\mathcal{F} \end{displaymath} (5)


\begin{displaymath} {\partial {\mathbf{u}}\over\partial t} + {\mathbf{u}}\cdot\n... ...P\over\rho} +\nu\nabla^2 {\mathbf{u}}+ {\mathcal{F}\over\rho}, \end{displaymath} (6)

where the kinematic viscosity is defined by
\begin{displaymath} \nu \equiv {\eta\over\rho}. \end{displaymath} (7)

This is the Navier-Stokes Equation. When combined with the continuity equation of fluid flow, it yields 4 equations in 4 unknowns ($\rho$ and ${\mathbf{u}}$). Because of its vector form, however, it is impossible to solve exactly, so approximations are commonly made. The equation satisfies three conservation laws. Mass conservation is included implicitly through the continuity equation,
\begin{displaymath} {\partial\rho\over\partial t} = -\nabla\cdot\Phi_{\rm mass} ... ... = -{\mathbf{u}}\cdot\nabla\rho -\rho\nabla\cdot {\mathbf{u}}. \end{displaymath} (8)

For a constant density liquid,
\begin{displaymath} {\partial\rho\over\partial t} = -\rho\nabla\cdot {\mathbf{u}}. \end{displaymath} (9)

Momentum conservation gives
$\displaystyle {\hbox{momentum}\over\hbox{mass}}$ $\textstyle =$ $\displaystyle \hbox{[mass]}+\hbox{[\href{pressure}]}$  
  $\textstyle \phantom{=}$ $\displaystyle +\hbox{[body force]}+\hbox{[viscosity]}$ (10)


\begin{displaymath} {\partial{\mathbf{u}}\over\partial t} = ({\mathbf{u}}\cdot\n... ...r\rho}\nabla p+{\mathcal{F}\over\rho}+\nu\nabla^2{\mathbf{u}}. \end{displaymath} (11)

energy conservation follows from
\begin{displaymath} {\partial s\over\partial t} = -{\mathbf{u}}\cdot\nabla s + {Q\over T}, \end{displaymath} (12)

where s is the entropy per unit mass.


The irrotational Navier-Stokes Equation is given in Cartesian coordinates by

\begin{displaymath} {\partial u\over\partial x} + {\partial v\over\partial y} + {\partial w\over\partial z} = 0 \end{displaymath} (13)

and
$\quad\rho\left({{\partial u\over\partial t} +u{\partial u\over\partial x}+v{\partial u\over\partial y} +w{\partial u\over\partial z}}\right)$
$ = - {\partial p\over\partial x}+\eta\left({{\partial^2 u\over\partial x^2}+{\partial^2 u\over\partial y^2} +{\partial^2 u\over\partial z^2}}\right)+F_x\quad$ (14)
$\quad\rho\left({{\partial v\over\partial t} +u{\partial v\over\partial x} +v{\partial v\over\partial y} +w{\partial v\over\partial z}}\right)$
$ = - {\partial p\over\partial y} + \eta\left({{\partial^2 v\over\partial x^2} +... ...artial^2 v\over\partial y^2} + {\partial^2 v\over\partial z^2}}\right)+F_y\quad$ (15)
$\quad\rho\left({{\partial w\over\partial t} +u{\partial w\over\partial x} +v{\partial w\over\partial y} +w{\partial w\over\partial z}}\right)$
$ = - {\partial p\over\partial z} +\eta\left({{\partial^2 w\over\partial x^2} + ... ...rtial^2 w\over\partial y^2} + {\partial^2 w\over\partial z^2}}\right)+F_z.\quad$ (16)


In cylindrical coordinates,


\begin{displaymath} {\partial u_r\over\partial r} + {u_r\over r} + {1\over r} {\... ...l u_\phi\over\partial\phi} + {\partial u_z\over\partial z} = 0 \end{displaymath} (17)

$\rho\left({{\partial u_r\over\partial t} + u_r {\partial u_r\over\partial r} + ... ..._r\over\partial z} - {{u_\phi}^2\over r}}\right)= - {\partial p\over\partial r}$
$\mathop{+}\eta\left({{\partial^2 u_r\over\partial r^2} + {1\over r} {\partial u... ...over\partial z^2} - {2\over r^2} {\partial u_\phi\over\partial\phi}}\right)+F_r$
$\quad$ (18)
$\rho\left({{\partial u_\phi\over\partial t} + u_r {\partial u_\phi\over\partial... ...tial u_\phi\over\partial z}}\right)= - {1\over r} {\partial p\over\partial\phi}$
$ \mathop{+}\eta\left({{\partial^2 u\phi\over\partial r^2} + {1\over r} {\partia... ...over\partial z^2} + {2\over r^2} {\partial u_r\over\partial\phi}}\right)+F_\phi$

(19)
$\rho\left({{\partial u_z\over\partial t} + u_r {\partial u_z\over\partial r} + ... ...r r}{\partial u_z\over\partial\phi} + u_z {\partial u_z\over\partial z}}\right)$
$ = - {\partial p\over\partial z} +\eta\left({{\partial^2 u_z\over\partial r^2} ... ...^2 u_z\over\partial\phi^2}+ {\partial^2 u_z\over\partial z^2}}\right)+F_z.\quad$

(20)


In spherical coordinates,


\begin{displaymath} {\partial u_r\over\partial r} + {2u_r\over r} + {1\over r} {... ... + {1\over r\sin\theta} {\partial u_\phi\over\partial\phi} = 0 \end{displaymath} (21)

$\quad\rho\left({{\partial u_r\over\partial t} + u_r {\partial u_r\over\partial ... ...partial u_r\over\partial\phi} - {u_\theta^2\over r} - {u_\phi^2\over r}}\right)$
$ = - {\partial p\over\partial r} +\eta\left({{\partial^2 u_r\over\partial r^2} ... ...tial\theta^2} + {\cot\theta\over r^2} {\partial u_r\over\partial\theta}}\right.$
$\left.{\mathop{+} {1\over r^2\sin^2\theta} {\partial^2 u_r\over\partial\phi^2} ... ...er r^2} - {2\over r^2\sin\theta} {\partial u_\phi\over\partial\phi}}\right)+F_r$

(22)
$\rho\left({{\partial u_\theta\over\partial\theta} + u_r {\partial u_\theta\over... ...eta} {\partial u_\theta\over\partial\phi} - {u_\phi^2\cot\theta\over r}}\right)$
$ = - {1\over r} {\partial p\over\partial\theta} +\eta\left({{\partial^2 u_\thet... ...theta^2} + {\cot\theta\over r^2} {\partial u_\theta\over\partial\theta}}\right.$
$\left.{\mathop{+} {1\over r^2\sin^2\theta} {\partial^2 u_\theta\over\partial\ph... ...ta\over r^2\sin\theta} {\partial u_\phi\over\partial\phi}}\right)+F_\theta\quad$ (23)
$\quad\rho\left({{\partial u_\phi\over\partial t} + u_r {\partial u_\phi\over\pa... ...u_\phi\over r} + {u_\theta\over r} {\partial u_\phi\over\partial\theta}}\right.$
$\left.{ + {u_\theta u_\phi\cot\theta\over r} + {u_\phi\over r\sin\theta} {\partial u_\phi\over\partial\phi}}\right)\quad$
$ = - {1\over r\sin\theta} {\partial p\over\partial\phi} +\eta\left({{\partial^2... ...^2\sin^2\theta} + {1\over r^2} {\partial^2 u_\phi\over\partial\theta^2}}\right.$
$\left.{\mathop{+} {\cot\theta\over r^2} {\partial u_\phi\over\partial\theta} + ... ...+ {2\cot\theta\over r^2\sin\theta} {\partial u_\theta\over\partial\phi}}\right)$
$ \mathop{+}F_\phi.\quad$ (24)


The Navier-Stokes equation with no body force

\begin{displaymath} \rho {\partial {\mathbf{u}}\over \partial t} +\rho {\mathbf{u}}\cdot\nabla {\mathbf{u}} = -\nabla P+\eta\nabla^2 {\mathbf{u}} \end{displaymath} (25)

can be put into dimensionless form with the definitions
$\displaystyle x'$ $\textstyle \equiv$ $\displaystyle {x\over L}$ (26)
$\displaystyle y'$ $\textstyle \equiv$ $\displaystyle {y\over L}$ (27)
$\displaystyle z'$ $\textstyle \equiv$ $\displaystyle {z\over L}$ (28)
$\displaystyle {\mathbf{u}}'$ $\textstyle \equiv$ $\displaystyle {{\mathbf{u}}\over U}$ (29)
$\displaystyle P'$ $\textstyle =$ $\displaystyle {P\over\rho U^2}$ (30)
$\displaystyle \nabla'$ $\textstyle \equiv$ $\displaystyle \hat {\mathbf{x}} {\partial \over\partial x'} +\hat{\mathbf{y}} {\partial\over\partial y'} +\hat{\mathbf{z}} {\partial\over\partial z'} = L\nabla$ (31)
$\displaystyle t'$ $\textstyle \equiv$ $\displaystyle t {U\over L},$ (32)

where U and L are a characteristic velocity and a characteristic length. Then
$\quad \rho {\partial (U{\bf u'})\over\partial\left({{L\over U} t'}\right)} +\rho U{\bf u'}\cdot {1\over L}\nabla'U{\bf u'}$
$ = - {1\over L}\nabla'(\rho U^2 P') + \eta {1\over L^2}\nabla'^2 (U{\bf u'}).\quad$ (33)
Assuming constant $\rho$ and multiplying both sides by ${L/(\rho U^2 )}$ gives
\begin{displaymath} {\partial {\bf u'}\over\partial t'} + {\bf u'}\cdot\nabla'{\... ...a'^2 {\bf u'} =\nabla'P' + {1\over{\rm Re}}\nabla'^2 {\bf u'}, \end{displaymath} (34)

where Re is the Reynolds number. Pressure is a parameter fixed by the observer, so it follows that the only other force is inertia force.
\begin{displaymath} {\rm Re} = {F_{\rm inertia}\over F_{\rm viscous}}. \end{displaymath} (35)


For irrotational, incompressible flow with ${\mathbf{F}} = \mathbf{0}$, the Navier-Stokes equation then simplifies to

\begin{displaymath} \rho {\partial {\mathbf{u}}\over\partial t} +\rho {\mathbf{u... ...dot\nabla {\mathbf{u}} = -\nabla P+\eta\nabla^2 {\mathbf{u}}. \end{displaymath} (36)

For low Reynolds number, the inertia term is smaller than the viscous term and can therefore be ignored, leaving the equation of creeping motion
\begin{displaymath} \rho {\partial {\mathbf{u}}\over\partial t} +\nabla P = \eta\nabla^2 {\mathbf{u}}. \end{displaymath} (37)

In this regime, viscous interactions have an influence over large distances from an obstacle. For low Reynolds number flow at low pressure, the Navier-Stokes equation becomes a diffusion equation
\begin{displaymath} \rho {\partial {\mathbf{u}}\over\partial t} = \eta\nabla^2 {\mathbf{u}}. \end{displaymath} (38)

For high Reynolds number flow, the viscous force is small compared to the inertia force, so it can be neglected, leaving Euler's equation of inviscid motion
\begin{displaymath} \rho {\partial {\mathbf{u}}\over\partial t} +\rho {\mathbf{u}}\cdot\nabla {\mathbf{u}} = -\nabla P. \end{displaymath} (39)

In the absence of a pressure force,
\begin{displaymath} \rho {\partial {\mathbf{u}}\over\partial t} +\rho {\mathbf{u}}\cdot\nabla {\mathbf{u}} = \eta\nabla^2 {\mathbf{u}}, \end{displaymath} (40)

which can be written as
\begin{displaymath} {\partial {\mathbf{u}}\over\partial t} -\nabla\times [{\math... ...imes {\mathbf{u}})] =\nu\nabla^2 (\nabla\times {\mathbf{u}}). \end{displaymath} (41)

For steady incompressible flow,

\begin{displaymath} {\partial {\mathbf{u}}\over\partial t} = \mathbf{0}. \end{displaymath} (42)

At low Reynolds number,
\begin{displaymath} \nabla P = \eta\nabla^2 {\mathbf{u}}. \end{displaymath} (43)

At low Reynolds number and low pressure
\begin{displaymath} \nabla^2 {\mathbf{u}} = \mathbf{0}. \end{displaymath} (44)

At high Reynolds number
\begin{displaymath} \rho {\mathbf{u}}\cdot\nabla {\mathbf{u}} = -\nabla P. \end{displaymath} (45)

For small pressure forces,
\begin{displaymath} \rho {\mathbf{u}}\cdot\nabla {\mathbf{u}} = \eta\nabla^2 {\mathbf{u}}, \end{displaymath} (46)

which can be written as
\begin{displaymath} -\nabla\times [{\mathbf{u}}\times (\nabla\times {\mathbf{u}})] =\nu\nabla^2 (\nabla\times {\mathbf{u}}). \end{displaymath} (47)

References

Clay Mathematics Institute. "Navier-Stokes Equations." http://www.claymath.org/prize_problems/navier_stokes.htm.

Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 20000821820702 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Providence, RI: Amer. Math. Soc., 2000.

Tritton, D. J. Physical Fluid Dynamics. New York: Van Nostrand Reinhold Co., pp. 52-53 and 59-60, 1977.


© 1996-2000 Eric W. Weisstein and Wolfram Research, Inc.