Bernoulli’s Principle and Newton’s Laws of Motion
Bernoulli’s Principle and Newton’s Laws of Motion
In fluid dynamics, Bernoulli’s principles states that an increase in the speed of the fluid corresponds to a decrease in pressure of the same fluid. Similarly, the decrease in pressure corresponds to a loss in the potential energy of the fluid. The principle is applicable to various types of fluids, which leads to Bernoulli’s equation. There are different types of Bernoulli’s equation depending on the nature of fluid in use. The principle is valid for most compressible and incompressible fluids at low Mach numbers (less than 0.3). For this principle, fluid particles are only subject to pressure and its own weight.
The principle of conservation of energy comes handy in deriving the Bernoulli’s principle. This law of conservation of energy holds that as long as a system is isolated from external factors/interference, the total amount of energy inside the system remains constant despite the energy transformation that also takes place. Therefore, the total sum of mechanical energy for a steady flow of fluid remains the same at all points in a streamline (University of Leeds, 2008). At a constant sum of kinetic and potential energy, an increase in the velocity of the fluid occurs when there is an increase in its kinetic energy and dynamic pressure, and a decrease in both its potential energy and static pressure.
Bernoulli’s principle can also be derived from Newton’s second law of motion. This is possible when a small volume of fluid flows horizontally from a high-pressure region to a region of low pressure resulting to more pressure behind the volume than in front of it (Florida International University, 2009). The pressure difference creates a net force on the volume causing it to accelerate along the streamline.
In the field of aeronautics, Bernoulli’s principle is a useful tool in calculating the lift force on an airfoil. Given the flow behavior of fluid around the airfoil, Bernoulli’s principle can be used to calculate the force required to lift the airfoil upwards. The architecture of the wings of a plane curved upper part and somewhat flat bottom ensures that the air flowing on the curved side is at a high speed than at the flat bottom (Florida International University, 2009). The difference in the speed of fluid flow creates pressure difference; higher pressure at the bottom than at the top, which creates the lift. The lift overcomes the pull of gravity while a thrust produced by the plane overcomes the drag that opposes it.
Bernoulli’s equation given below is based on a venture flume shown, which allows fluid to have different speeds at two different points along the flume. It is the simplest Bernoulli’s equation.
Total Energy per unit volume before = Total Energy per unit volume after
P1+1/2ρv12+ρgh1 = P2+1/2ρv22+ρgh2
Where, P= pressure energy, 1/2ρv2 = kinetic energy per unit volume, ρgh= potential energy per unit volume, point 1= low fluid speed and high internal pressure, point 2= increased fluid speed and decreased internal pressure.
Bernoulli’s equation shown above work on the assumption that the fluid flow is steady, constant density (incompressible fluids), no friction losses and the two points are along a single streamline (not two different streamlines) (University of Leeds, 2008). Newton’s Laws of Motion
Isaac Newton contributed a lot in science by coming up with important laws that laid the foundation for various scientific fields. The greatest contribution was the three Newton’s Laws of motion, which form the foundation of aeronautics. The three Newton’s laws of motion describe the relationship between a body and forces that act on it and how the said forces affect the motion of the body. All the three forces have application in aeronautics and are discussed as in the proceeding paragraphs. Newton’s First Law of Motion
The first law of motion states that a body will remain in its state in a straight line unless external forces are applied to it (National Aeronautics and space Administration [NASA], 2010). The state of the body can be either in motion or at rest. The first law of motion is also known as law of inertia. This law simply implies that when there is no net force from the unbalanced forces acting a body, the body maintains a constant velocity. When the velocity is zero, the body remains at rest. However, upon action of an external force the velocity of the body changes. In this law, prediction of the objects behavior is based on the balance of all the existing forces.
The law is represented by an equation that relates force of gravity/weight of the body (F), the mass of the body (m) and the acceleration of gravity (g). F=ma
In aerodynamics, the first law of motion describes the motion of an airplane whenever the throttle of the engine changes. The plane will be pushed forward by an action of an external force created by throttling of the engine. Newton’s Second Law of Motion
Newton’s second law of motion explores the relationship of force and acceleration of a body. This law states that the acceleration of a body depends upon the resultant force acting on it and its mass. In addition, the acceleration is directly proportional to the net force and inversely proportional to the mass (NASA, 2011). It means that the higher the mass of a body, the higher the force required for the acceleration. Second law of motion explores changes in the momentum (m*v) of the body. Momentum is change in velocity over time, and mathematically it is as follows. Force = m*(V1-V0)/ (t1-t0)
For a body at a constant mass,
Force = mass times acceleration (F=ma)
This second law of motion is applicable when computing the motion of aircraft due to aerodynamic forces, thrust, and aircraft weight. Newton’s Third Law of Motion
Third Newton’s law of motion states that for every action a body there is an equal and opposite reaction (NASA, 2010). Whenever there is interaction between two bodies, there would be a pair forces acting them. For a moving body to stop or slow down, an equal force must be exerted to the body, but in the opposite direction to that of the body.
In aeronautics, the third law helps to explain the generation of lift from an airfoil. For example, on a jet airfoil air is deflected downwards the airfoil, and in response, the wing is pushed upward. The jet engines produce thrust through action-reaction mechanisms. The hot exhaust gases from the engine cause a reaction in the opposite direction and subsequently pushing the jet forward.
Florida International University (2009, Nov 30). Bernoulli’s Principle. Retrieved from http://www.allstar.fiu.edu/aerojava/pic3-2.htm Florida International University (2009, Nov). Bernoulli’s Principle. Retrieved from http://www.allstar.fiu.edu/aerojava/pic3-2.htm National Aeronautics and Space Administration (2010, Sep). Newton’s First Law: Applied to Airplanes. Retrieved from http://www.grc.nasa.gov/WWW/k-12/airplane/newton1.html National Aeronautics and Space Administration (2010, Sep). Newton’s Third Law: Applied to Aerodynamics. Retrieved from http://www.grc.nasa.gov/WWW/k-12/airplane/newton3.html National Aeronautics and Space Administration (2011, Apr). Newton’s Second Law. Retrieved from http://www.grc.nasa.gov/WWW/k-12/airplane/newton2.html University of Leeds (2008, Jan). The Bernoulli’s Equation. Retrieved from http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section3/bernoulli.htm