Verification of the Divergence Theorem for a Rotating Fluid Flow

The mass flux out of the closed cylindrical surface is zero, a result confirmed by both the application of the Divergence Theorem and the direct summation of the fluxes through the individual surfaces. The Divergence Theorem revealed that the volume integral of the mass flux is zero because the velocity field $v$ has a zero divergence ( $\nabla \cdot v=0$ ), indicating that the fluid is incompressible and mass-conserving within the volume. This zero net flux was verified by the surface calculations, where the flux out of the top cap ( $\Phi_1= +\rho_0 v_0 \pi r_0^2$) was perfectly balanced by the flux into the bottom cap ($\Phi_2=-\rho_0 v_0 \pi r_0^2$), while the flux through the curved side wall ($\Phi_3$) was also zero due to the specific anti-symmetric nature of the velocity field components on that surface.

🎬Narrated Video

https://youtu.be/H2ndWQfc-qw

🏗️Structural clarification of Poof and Derivation

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🗒️Downloadable Files - Recursive updates

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