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## Q. 10.14.9

For an incompressible viscous fluid moving under a conservative body force, show that the circulation $I_{c}$ round a circuit c moving with the fluid is not constant in general. Deduce that $I_c$ is constant if and only if curl w = ▽ξ, for some ξ.

## Verified Solution

For the given flow, the Navier-Stokes equation is given by (10.14.6) with $\textbf{b}=-\triangledown\chi$.
From (6.6.4), we recall that the rate of change of circulation round a material circuit is given by

$\frac{DI_{c}}{Dt}=\int_{C}\frac{Dv}{Dt}.d\textbf{x}$           (6.6.4)

$\frac{DI_{c}}{Dt}=\oint_{c}\frac{D\textbf{v}}{Dt}.d\textbf{x}$                        (10.14.68)

Substituting for Dv/Dt from (10.14.6) with $\textbf{b}=-\triangledown\chi$ in (10.14.68), we get

$v\triangledown^2\textbf{v}-\frac{1}{\rho}\triangledown\rho+\textbf{b}=\frac{D\textbf{v}}{Dt}$           (10.14.6)

$\frac{DI_{c}}{Dt}=v\oint_{c}\triangledown^2\textbf{v}.d\textbf{x}$                    (10.14.69)

Since the fluid is viscous, $v\neq 0$ and consequently $DI_{c}/Dt\neq 0\ when\ \triangledown^2\textbf{v}\neq 0.\ Thus,\ I_{c}$ is not constant in general.
Since $\triangledown^2\textbf{v}=-curl\ \textbf{w}$ for an incompressible fluid, expression (10.14.69) can be rewritten as

$\frac{DI_{c}}{Dt}=-v\oint_{c}curl\ \textbf{w}.d\textbf{x}$                    (10.14.70)

Evidently, $I_{c}=constant$ if and only if curl w = ▽ξ for some scalar ξ.
(Thus, Kelvin’s circulation theorem is not generally valid for viscous fluids.)