Volume Crossing a Surface

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By Twisp [Public domain], via Wikimedia Commons

The amount of fluid that crosses a surface is required in the derivation of the Reynold’s Transport Theorem. One measure of the amount of fluid crossing a surface is the volumetric flow rate.

The differential distance,d, traveled by the fluid in the differential surface element, dS, is obtained by multiplying the normal velocity by a small increment in time, dt.

The differential volume of fluid, dQ, leaving or entering the control volume through the differential surface, dS, is obtained by multiplying the distance, d, by dS.

If the flow is exiting the surface, the differential volume of fluid will have a positive sign.  If the flow is entering the surface, the differential volume of fluid will have a negative sign.  The sign is controlled by the dot product of the velocity vector and the outward unit normal.

The volumetric flow rate leaving or entering the surface is obtained by dividing the previous equation by dt.  The volumetric flow rate is given by the equation

Learning Objectives:

• State how to determine if fluid is entering or leaving a control surface.
• Describe the volumetric flow rate.