Bernoulli's Theorem Essay

Bernoulli’s theorem is a special application of the laws of motion and energy. The principle equation describes the pressure measured at any point in a fluid, which can be a gas or a liquid, to the density and the velocity of the specified flow.

The theorem can be explained by the means of imagining a particle in a cylindrical pipe. If the pressure on both sides of the particle in the pipe is equal, the particle will be stationary and in equilibrium. By implementing the second law of motion the particle will accelerate or decelerate if there exists a pressure difference over the particle. The particle’s velocity will increase when it is approaching a low-pressure region and decrease its velocity at a high-pressure region. This principle can also be seen in terms of pressure. If a fluid is slowed down in the pipe the pressure will rise and vice versa.

This principle is applicable to the basic way an aircraft’s wing is able to generate lift (Figure 10). Figure 16: Bernoulli’s principle applied to an airfoil

The equation of Bernoulli’s Principle if given by:

(ρV^2)/2+P+ρgh=Constant [6] Continuity equation

M=ρV_1 A_1=ρV_2 A_2 [7]

With

M=ρVA=Constant [8]

This relationship is known as the condition of continuity. According to R. Von Mises [3] the theorem states that:

Von Mises [3] states that “the forces due to viscosity appear as products of μ and expressions that have the dimensions area times (velocity / length)”. By further investigation, the mathematical analysis of these principles leads to a system of partial derivatives known as the Navier-Stokes equations. These equations are used to describe fluid flow and can be used to solve specific dynamic fluid flow cases. These include; velocities, pressure, temperature, density and can also be used to solve viscous problems of a dynamic fluid flow problems. These partial derivative equations relating to the specific variables are extremely complex and time-consuming to