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Chapter 10

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.

Step-by-Step

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.)