13.11:
Accelerating Fluids
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When a fluid is in constant acceleration, the pressure and buoyant force equations are modified. Suppose a beaker is placed in an elevator accelerating upward with a constant acceleration, a. In the beaker, assume there is a thin cylinder of height h with an infinitesimal cross-sectional area, ΔS.
The motion of the liquid within this infinitesimal cylinder is considered to obtain the pressure difference. Three vertical forces act on this liquid:
- An upward force due to the liquid present below the bottom surface of the cylinder.
- A downward force due to the liquid above the top surface of the cylinder.
- A downward force is due to weight of the cylindrical element.
Under these three forces, the liquid accelerates upward. Using Newton's second law, the following expression is obtained:
Representing the fluid element's mass in terms of density (⍴) simplifies the equation, and the expression for the pressure difference for an accelerating fluid is obtained.
To obtain the buoyant force, assume a body is dipped inside the same accelerating liquid. It experiences buoyant force and force due to its weight. For simplicity, the body is replaced by an equal volume of the same liquid. From Newton's second law, the buoyant force is expressed in terms of acceleration, and the following expression is obtained:
