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.
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title The Mechanics of Fluid Divergence and Vorticity Dynamics
Resulmation: Visualization of the Velocity Field
: modify the velocity field to see how a "source" (non-zero divergence) would look
: Physical Interpretation - Incompressibility
: see how this equation looks if we add a sink, where density increases as fluid is sucked toward a point
: Calculating the Vorticity (The "Curl")
: See a case where the curl is zero even though the fluid is moving in a circle (like an irrotational vortex)
: the upward flux is perfectly balanced by the downward flux with zero flux through the sides
IllustraDemo: How Divergence and Curl Define Flow
Ex-Demo: The Mechanics of Helical Flow and Fluid Dynamics
Narr-graphic: Divergence and Curl Analysis
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