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Chapter 10

Q. 10.14.5

For a nonsteady flow of an incompressible viscous fluid under conservative body force with curl\ \textbf{w}=\triangledownξ for some scalar function ξ show that the Navier-Stokes equation becomes

\frac{\partial\textbf{v}}{\partial t}+\textbf{w}\times\textbf{v}=-\triangledown H^*                    (10.14.39)

where

\textbf{H}^*=\frac{p}{\rho}+\frac{1}{2}v^2+\chi+vξ                (10.14.40)

Deduce the following.
(i) If the flow is of potential kind, then \textbf{H}^*+(\partial\phi/\partial t)=f(t),\ where\ f(t) is an arbitrary function of t.
(ii) If the flow is steady and \textbf{v}\times\textbf{w}\neq 0,then\ H^* is constant along stream lines and vortex lines.
(iii) If the flow is steady and \textbf{v}\times \textbf{w}=\textbf{0},\ then\ H^* is constant everywhere in the fluid.

Step-by-Step

Verified Solution

For an incompressible viscous fluid moving under a conservative body force, the Navier-Stokes equation is given by (10.14.9).
Since curl\ \textbf{w}=-\triangledown^2\textbf{v}, the given condition, curl\ \textbf{w}=\triangledownξ,\ yields\ \triangledown^2\textbf{v}=-\triangledown curl\ \textbf{w}=\triangledownξ.
Using this, equation (10.14.9) becomes (10.14.39), with H* defined by (10.14.40).
We observe that equation (10.14.39) is strikingly similar to equation (10.8.1). Indeed, in the absence of viscous effects, function H* defined by (10.14.40) reduces to Bernoulli’s function H defined by (10.8.2). Following the steps that led to Bernoulli’s equations (10.8.4), (10.8.7) and (10.8.8) from (10.8.1), we arrive at the desired results starting from the equation (10.14.39).

v\triangledown^2\textbf{v}-\triangledown\left(\frac{p}{\rho}+\chi+\frac{1}{2}v^2\right)=\frac{\partial \textbf{v}}{\partial t}+\textbf{w}\times\textbf{v}                 (10.14.9)

\frac{\partial\textbf{v}}{\partial t}+\textbf{w}\times \textbf{v}=-\triangledown H              (10.8.1)

H=P+\chi+\frac{1}{2}v^2           (10.8.2)

H=\frac{\partial\phi}{\partial t}=f(t)           (10.8.4)

P+\frac{1}{2}v^2+\chi=constant          (10.8.7)