Question

please help 7. Find the mass flux of a fluid of a constant density \( \rho \), viscosity \( v \) and pressure gradient \( G \) passing through a cross-section of a cylinder pipe of radius \( R \), with the fluid

Answer 1

Final Answer:

The mass flux
`(\( \dot{m} \))` of the fluid through the cross-section of the cylindrical pipe is given by the formula
`\( \dot{m} = \rho \cdot v \cdot A \)`, where
`\( A \)` is the cross-sectional area of the pipe. For a cylindrical pipe,
`\( A = \pi R^2 \)`. Therefore,
`\( \dot{m} = \pi \rho v R^2 G \).`

Step-by-step explanation:

In fluid dynamics, the mass flux represents the rate at which mass flows through a unit area. The formula
`\( \dot{m} = \rho \cdot v \cdot A \)`expresses this relationship, where
`\( \rho \)`is the fluid density,
`\( v \)` is the velocity of the fluid, and
`\( A \)` is the cross-sectional area through which the fluid is flowing.

For a cylindrical pipe, the cross-sectional area
`\( A \)` is given by
`\( \pi R^2 \)`, where
`\( R \)` is the radius of the pipe. Substituting this into the mass flux formula, we get
`\( \dot{m} = \pi \rho v R^2 \).` Additionally, the pressure gradient
`\( G \)` influences the mass flux, so we multiply the formula by
`\( G \)` to incorporate this effect.

Therefore, the final formula for the mass flux
`(\( \dot{m} \))`in a cylindrical pipe with constant density
`\( \rho \)`, viscosity
`\( v \)`, and pressure gradient
`\( G \) is \( \pi \rho v R^2 G \)`. This equation provides a quantitative measure of the mass flow rate through the pipe, taking into account the key factors influencing fluid dynamics in this scenario.