State the principle on which each of the following is based. (a) the equation of continuity, Av = constant (b) the Bernoulli equation

conservation of mass

conservation of energy

(a) The equation of continuity, Av = constant, is based on the principle of conservation of mass. This principle states that the mass of a fluid flowing through a pipe remains constant, assuming the fluid is incompressible and there is no leakage.

To derive the equation of continuity, we consider a fluid flowing through a pipe with varying cross-sectional areas. According to the principle of conservation of mass, the mass flow rate (m) of the fluid should remain constant.

The mass flow rate (m) is given by the product of the fluid density (ρ), velocity (v), and cross-sectional area (A) through which the fluid is flowing:
m = ρAv

Since the mass flow rate remains constant, the product of velocity and cross-sectional area should also remain constant. This leads to the equation of continuity, Av = constant. It states that as the cross-sectional area decreases, the fluid velocity must increase to maintain a constant mass flow rate.

(b) The Bernoulli equation is based on the principle of conservation of energy along a streamline in a fluid flow. It is derived from the application of the principles of conservation of mass and conservation of energy.

The Bernoulli equation relates the pressure, velocity, and elevation of a fluid along a streamline. It is given by the equation:

P + 1/2ρv² + ρgh = constant

where P is the pressure, ρ is the density of the fluid, v is the velocity, g is the acceleration due to gravity, and h is the elevation (height) of the fluid above a reference point.

This equation states that the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline when there is no energy loss due to friction or other external factors.

The Bernoulli equation is commonly used to analyze fluid flow problems, such as in pipes, airfoils, and fluid dynamics, by relating changes in pressure, velocity, and elevation to each other.