The velocity field $v$ represents a steady helical flow, where the fluid rotates around the $z$-axis while simultaneously translating along it. By transforming the Cartesian expression into cylindrical coordinates, we find that the radial velocity component is zero, which is why the mass flux through the cylinder walls is null. The mass flux through the horizontal disc depends purely on the uniform vertical velocity $v_0$, while the flux through the vertical $\phi$-plane depends on the rotational "swirl" component, which increases linearly with the distance from the axis.
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title Visualizing the Dynamics of Helical Fluid Flow
Resulmation: Visualize the velocity field of particles flowing inside a 3D cube
: Track a small fluid element (a cube) as it moves through the given velocity field
: Visualize the rotational aspect of the flow
IllustraDemo: Visualizing Mass Flux in Helical Flows
Ex-Demo: The Mechanics of Incompressible Helical Flow
Narr-graphic: The Dynamics and Geometry of Helical Fluid Flow
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