Total Mass Flux Through Cylindrical Surfaces

To accurately calculate and interpret mass flux, converting the velocity field to the appropriate coordinate system (e.g., cylindrical) is crucial, especially when dealing with surfaces defined in that system. The mass flux value itself then provides a quantitative measure of the net rate of mass flow across a surface, with a positive flux indicating outflow, negative indicating inflow, and zero signifying flow along the surface rather than through it.

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✍️Mathematical Proof

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The Importance of Coordinate System Conversion

The first and most critical step was converting the Cartesian velocity field into cylindrical coordinates. This was essential because the surfaces given in the problem were defined cylindrically. Failing to do this would have made the calculations much more complex, if not impossible. The simplified form of the velocity field, $v=\frac{v_0}{L}\left(\rho e_\phi+L e_z\right)$, revealed the true nature of the flow: a combination of rotational motion ( $e_\phi$ ) and a constant vertical flow ( $e_z$ ).

Physical Interpretation of Mass Flux

The value of the mass flux, $\Phi$, represents the net rate of mass flow across a given surface. A positive flux means fluid is flowing out of the volume, a negative flux means fluid is flowing in, and a zero flux means there is no net flow across the surface.

Part a) Mass Flux Through the Disc (Top/Bottom): The non-zero result, $\Phi=\pi \rho_0 v_0 r_0^2$, indicates that there is a net flow of fluid through the disc. This makes sense physically because the velocity field has a constant upward component ( $e_z$ ), meaning fluid is consistently moving in the positive z-direction through the surface.
Part b) Mass Flux Through the Cylinder Wall: The result of zero, $\Phi=0$, confirms that there is no net mass flow through the side walls of the cylinder. This is a direct consequence of the velocity field's rotational component, which is perfectly parallel to the surface normal of the cylinder wall. Since $v \cdot n=0$, no fluid passes through the surface, only along it.
Part c) Mass Flux Through the $\phi$-Surface: The result, $\Phi=\frac{\rho_0 v_0 z_0 r_0^2}{2 L}$, indicates a net flow through the specified slice. This happens because the velocity field has a tangential component ( $e_\phi$ ) and the surface's normal vector is also in the tangential direction, leading to a non-zero dot product.

✍️Mathematical Proof

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🧄Proof and Derivation-2

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