Fluid Mechanics: Bernoullis Theorem - Assignment Example

Fluid Mechanics: Bernoulli’s Theorem Fluid Mechanics: Bernoulli’s Theorem Daniel Bernoulli developed the theory of fluid mechanism by studying the flow of fluid through various diameters of pipes. He investigated that the pressure of the fluid changes as the diameter of the pipe changes. In this way, by varying the diameter of pipe, pressure of the fluid can be altered. Moreover, the percentage of flow of fluid can be restricted or increased by changing the diameter of the pipe. Bernoulli and Euler discovered the relationship between the pressure of the blood and the speed through which it flows out of the heart. Bernoulli evaluated that when he punctured the wall of the pipe containing a fluid. The eight of the fluid rose at the open ended end of the pipe. He further discovered that there is a deep relationship between how much the fluid rose in the pipe and the difference in the pressure between the two ends of the pipe. Very soon, a blood pressure measuring system was introduced at over Europe that used the similar statics as that of the Bernoulli apparatus. This apparatus involves sticking the capillary tube directly into the artery of the patient. This method for measuring blood pressure remained in use for 170 years. An Italian doctor introduced a better method for measuring h blood pressure in 1896 without puncturing the artery. This method is still use today for measuring blood pressure. However, Bernoulli method for measuring blood pressure is used in aviation for measuring the speed of air passing the aircraft. Discovering more about the flow of fluids, Bernoulli studied about the conservation of energy regarding the flow of fluids and evaluated the relationship between the potential and kinetic energy and the connection between the conversation of energy and the flow of fluids. He formerly discovered that as the height of the fluids is increased, it gains potential energy and the kinetic energy of the fluid when it is released depends on the potential energy it gained through height. Dams and water reservoirs and hydroelectric generation plants utilize the similar principle to evaluate the amount of energy being taken as the output of the hydrodynamic energy. In dams and reservoirs sum of forces acting on the water increases its pressure at output, speed of flow and, dynamic pressure and its kinetic energy. While discussing the fluid dynamics, Bernoulli explained that the speed of a non conducting fluid increases as the decrease in pressure or decrease in its potential energy occurs. The principle was first published in the denial Bernoulli book “Hydrodynamica” that was published in 1738. The theory is named after the scientist. Bernoulli’s theorem resulted in the derivation of Bernoulli’s equation that can be altered with respect to the different types of the fluids and types of flow. The simplest form of Bernoulli’s equation is only true for the incompressible fluids. Advanced formulations helped us to evaluate the flow rates of the compressible lows including gaseous flows. Bernoulli derived the equation using conservation of energy principles. According to the law of conservation of energy the sum of energies in the system of the streamline remains constant at all points. This means, the sum of kinetic energy, potential energy and internal energy is constant in the system of fluid flow. An increase in the speed influences the dynamic pressure and kinetic energy by increasing both the two. Bernoulli’s theorem can also be derived from Newton’s second law of motion. The particles of the fluid play an important role in enhancing the density of the liquid and thus influencing the total dynamic pressure of the fluid. Fluids that are heavier (like concrete in liquid form or other such fluids) exert more pressure and require more energy to be pumped vertically as their own weight is exerting an inverse pressure. Fluids move from the region of higher pressure to the region of lower pressure, whenever an escape route is possible. Bernoulli only performed experiments on the incompressible flowing fluids. Thus, the density of the fluid is kept constant. Bernoulli’s equation can be simplified as: Where, is the speed of the flowing fluid, is the acceleration in the flowing fluid due to gravity. It has a constant value of 9.8 m/s2, is the point to which the fluid is elevated with respect to the reference plane, is the pressure of the flowing fluid and is the density of the flowing fluid. Bernoulli did not consider the fluid friction (viscocity) and thus an ideal frictionless fluid is considered. On the other hand, the density of the fluid must remain constant. The equation can be transformed into a more standardized form by multiplying it with density with whole equation: Equation can be simplified as: Here and , here q is the dynamic pressure and h is the hydraulic head of the flowing fluid. It is sum of elevated height of the fluid and the dynamic pressure of the fluid. is the sum of dynamic pressure p and dynamic pressure q. The constant in the derived equation can be equalized to energy head H. in this manner, the equation becomes: The derived equation depicts that as the flow speed is increased the pressure drops. Thus Bernoulli’s equation is limited to zero pressure limits. In most applications regarding the fluid mechanics, density ρ, acceleration due to gravity g  and elevation of the fluid z brought changes so minimum that the is negligible. In this manner, the terms can be ignored. Other terms have much influence with little change in the parametric values. In this way, the equation (a) can easily be simplified as: Or, Static pressure + dynamic pressure = Total pressure In Aerodynamics, L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure." Engineering Connection Bernoulli principle is much related to the civil engineering, particularly with the field of fluid mechanics and hydraulics. On the other hand, it has equal importance in the field of aero space and the construction and working of hydro electric power generation units. Bernoulli equation is utilized to distinguish the design parameters of the wind blade and the design parameters of the wind turbine blades. Thus Bernoulli theorem has variety of applications in nearly all fields of engineering. The Bernoulli principle helps the engineers to plan the flow rate of water through the rivers and irrigation channels. In this manner, engineers can determine the flow of water and plan how fast the water should be flowing through the channel to supply enough water to the irrigating channels. On the other hand, the similar tactic is used to supply enough water with adequate pressure for the power generation in the hydro power generation units. Fluid mechanics, hydrodynamics and aerodynamics are the major areas of civil engineering. Bernoulli’s theorem involves the three areas by evaluating the idealistic conditions for the flow of fluids. It is being used to evaluate the problems regarding fluids. However, recently Bernoulli theorem is used to evaluate the flow of wind through and into the large buildings and skyscrapers. A refined form of Bernoulli theorem is being used in the building to calculate and estimate the flow of wind through the corners and edges of the building and how much the wind will influence the structural strength of the building and what other disadvantages can the flow of wind will have on the building. The buildings are well researched now a day and wind drifting buildings are made to present least possible damage due to the flow of air. Buildings are such built to withstand heavier winds and harsh weather conditions, due to the use of proper knowledge and theorems such as Bernoulli’s theorem. Read More