1) Lamb’s form of N-S equation:-
The
N-S equation of motion for a viscous incompressible fluid is
$\frac{D\overrightarrow{q}}{Dt}=\overrightarrow{F}-\frac{\nabla
p}{\rho }+\upsilon {{\nabla }^{2}}\overrightarrow{q}$
or, $\frac{\partial
\vec{q}}{\partial t}+\left( \vec{q}.\nabla
\right)\vec{q}=\vec{F}-\frac{\nabla p}{\rho }+\upsilon {{\nabla
}^{2}}\overrightarrow{q}$……(1)
Where $\vec{q}=(u,v,w)$
$\vec{F}=(X,Y,Z)$
$\nabla
={{(}^{\partial }}{{/}_{\partial x}}{{,}^{\partial }}{{/}_{\partial
y}}{{,}^{\partial }}{{/}_{\partial z}})$
r =
Pressure
u= Kinematic viscosity
$or,\frac{\partial
\vec{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}}
\right)-\overrightarrow{q}\times \left( \nabla \times \overrightarrow{q}
\right)=\vec{F}-\frac{\nabla p}{\rho }+\upsilon {{\nabla
}^{2}}\overrightarrow{q}$ [∵ By Lagrange’s vector identity]
or, , $\frac{\partial \vec{q}}{\partial t}+\nabla \left(
\frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left(
\widetilde{w} \right)=\vec{F}-\frac{\nabla p}{\rho }+\upsilon {{\nabla
}^{2}}\overrightarrow{q}$
where $\widetilde{w}=\nabla \times
\overrightarrow{q}$ as vorticity vector
Or, \[\frac{\partial
\vec{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}}
\right)+\widetilde{w}\times \overrightarrow{q}=\vec{F}-\frac{\nabla p}{\rho
}-\upsilon {{\nabla }^{2}}\overrightarrow{q}\] which
is known as Lamb’s form of N-S equation
2) If external forces form
a conservative field of force then F=-ÑW,
where W is potential
function.
The lamb’s form of N-S
equation now becomes
\[or,\frac{\partial \vec{q}}{\partial
t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}}
\right)-\overrightarrow{q}\times \widetilde{w}=-\nabla \Omega -\frac{\nabla
p}{\rho }-\upsilon {{\nabla }^{2}}\overrightarrow{q}\]
\[or,\frac{\partial \vec{q}}{\partial
t}-\overrightarrow{q}\times \widetilde{w}=-\nabla \left(
\frac{1}{2}{{\overrightarrow{q}}^{2}}+\Omega +\frac{p}{\rho } \right)-\upsilon
{{\nabla }^{2}}\overrightarrow{q}\]$$
But,
\[{{\nabla }^{2}}\overrightarrow{q}=\nabla
(\nabla .\overrightarrow{q})-\nabla \times \left( \nabla \times
\overrightarrow{q} \right)=0-\nabla \times \left( \nabla \times
\overrightarrow{q} \right)=-\nabla \times \widetilde{w}\]for incompressible
fluid.
The above equation reduces to
\[\frac{\partial \vec{q}}{\partial t}-\overrightarrow{q}\times
\widetilde{w}=-\nabla \left( \frac{p}{\rho }+\Omega
+\frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\upsilon \nabla \times
\widetilde{w}\]which is another form of Lamb’s form of N-S equation under the
external force is conservative.
3) When there exist a functional relation between pressure and
density, the N-S equation of motion for compressible viscous fluid is
$\frac{D\overrightarrow{q}}{Dt}=\overrightarrow{F}-\int{\frac{\partial
\rho }{\rho }}+\upsilon {{\nabla }^{2}}\overrightarrow{q}+\frac{\upsilon
}{3}\nabla (\nabla .\overrightarrow{q})$….(1)
Where $\vec{q}=(u,v,w)$
$\vec{F}=(X,Y,Z)$
$\nabla
={{(}^{\partial }}{{/}_{\partial x}}{{,}^{\partial }}{{/}_{\partial
y}}{{,}^{\partial }}{{/}_{\partial z}})$
$\frac{D\vec{q}}{Dt}=\frac{\partial
\vec{q}}{\partial t}+\left( \vec{q}.\nabla
\right)\vec{q}$
$=\frac{\partial
\overrightarrow{q}}{\partial t}+\nabla \left(
\frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left(
\nabla \times \overrightarrow{q} \right)$
[∵$\left(
\overrightarrow{q}.\nabla
\right)\overrightarrow{q}=\nabla \left(
\frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left(
\nabla \times \overrightarrow{q} \right)$] \[and\nabla
\times \left( \nabla \times \overrightarrow{q} \right)=\nabla \left( \nabla
.\overrightarrow{q} \right)-{{\nabla }^{2}}\overrightarrow{q}\Leftrightarrow
{{\nabla }^{2}}\overrightarrow{q}=\nabla \left( \nabla .\overrightarrow{q}
\right)-\nabla \times \left( \nabla \times \overrightarrow{q} \right)\]
Then the equarion (1) reduces to
$\frac{\partial \overrightarrow{q}}{\partial t}+\nabla \left(
\frac{1}{2}{{\overrightarrow{q}}^{2}} \right)-\overrightarrow{q}\times \left(
\nabla \times \overrightarrow{q} \right)=\overrightarrow{F}-\int{\frac{\partial
\rho }{\rho }}+\upsilon \left\{ \nabla \left( \nabla .\overrightarrow{q}
\right)-\nabla \times \left( \nabla \times \overrightarrow{q} \right)
\right\}+\frac{\upsilon }{3}\nabla (\nabla .\overrightarrow{q})$
$or,\frac{\partial
\overrightarrow{q}}{\partial t}+\nabla \left( \frac{1}{2}{{\overrightarrow{q}}^{2}}
\right)-\overrightarrow{q}\times \left( \nabla \times \overrightarrow{q}
\right)=\overrightarrow{F}-\int{\frac{\partial \rho }{\rho }}-\upsilon \nabla
\times \left( \nabla \times \overrightarrow{q} \right)+\frac{4\upsilon }{3}\nabla
(\nabla .\overrightarrow{q})$
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