Physics Study Guide/Fluids

Buoyancy

Buoyancy is the force due to pressure differences on the top and bottom of an object under a fluid (gas or liquid).

Net force = buoyant force - force due to gravity on the object

Bernoulli's Principle

Fluid flow is a complex phenomenon. An ideal fluid may be described as:

• The fluid flow is steady i.e its velocity at each point is constant with time.
• The fluid is incompressible. This condition applies well to liquids and in certain circumstances to gases.
• The fluid flow is non-viscous. Internal friction is neglected. An object moving through this fluid does not experience a retarding force. We relax this condition in the discussion of Stokes' Law.
• The fluid flow is irrotational. There is no angular momentum of the fluid about any point. A very small wheel placed at an arbitrary point in the fluid does not rotate about its center. Note that if turbulence is present, the wheel would most likely rotate and its flow is then not irrotational.

As the fluid moves through a pipe of varying cross-section and elevation, the pressure will change along the pipe. The Swiss physicist Daniel Bernoulli (1700-1782) first derived an expression relating the pressure to fluid speed and height. This result is a consequence of conservation of energy and applies to ideal fluids as described above.

Consider an ideal fluid flowing in a pipe of varying cross-section. A fluid in a section of length $\Delta x_{1}$ moves to the section of length $\Delta x_{2}$ in time $\Delta t$. The relation given by Bernoulli is:

$P+{\tfrac {1}{2}}\rho v^{2}+\rho gh={\text{constant}}$

where:

$P$ is pressure at cross-section

$h$ is height of cross-section

$\rho$ is density

$v$ is velocity of fluid at cross-section

In words, the Bernoulli relation may be stated as:

As we move along a streamline the sum of the pressure ($P$), the kinetic energy per unit volume and the potential energy per unit volume remains a constant.

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(To be concluded)