Energy Conservation and Bernoulli's Equation

Bernoulli's equation states that the sum of fluid pressure, potential energy per unit volume, and kinetic energy per unit volume is constant along a streamline. It applies to incompressible, inviscid fluids in laminar and turbulent flow at steady state. All terms have dimensions of energy per unit volume, making it a powerful tool for calculating pressure at different points when fluid speeds are known.

Applying the work-energy theorem to an incompressible fluid element moving through a non-uniform tube shows that net work done equals the change in kinetic energy. As the fluid element rises, it gains gravitational potential energy. Dividing the total energy change by volume yields Bernoulli's equation, confirming that total energy per unit volume remains constant along a streamline.

Bernoulli's equation requires the fluid to be incompressible, meaning density remains constant throughout the flow. The flow must be laminar and irrotational at steady state. For gases with negligible compressibility effects, density is assumed constant and the equation applies. Since gases are generally compressible, Bernoulli's equation does not apply to them.

Kinetic energy density is the kinetic energy per unit volume of flowing fluid, reflecting how fast the fluid moves. Potential energy density is the potential energy per unit volume, accounting for the fluid's elevation. Both terms have dimensions of energy per unit volume and are essential components of Bernoulli's equation for analyzing fluid behavior.

In a non-uniform tube, the cross-sectional area varies along the flow path. Since the fluid is incompressible, its volume and mass remain constant. When the tube narrows, the same volume of fluid must pass through a smaller area, forcing the fluid to accelerate. Conversely, the fluid decelerates when the tube widens, resulting in different speeds at different locations.

Since the sum of pressure, kinetic energy density, and potential energy density is constant along a streamline, you can calculate pressure at one point if you know the pressure, speed, and elevation at another point. This makes Bernoulli's equation a practical tool for determining pressure variations in flowing fluids when speeds and elevations are known.

A fluid element's kinetic energy changes as it moves through a non-uniform tube because its speed changes. When the tube narrows, the fluid accelerates and kinetic energy increases. When the tube widens, the fluid decelerates and kinetic energy decreases. The work done by surrounding fluid pressure accounts for these kinetic energy changes according to the work-energy theorem.