How Fluid Mechanics Function - Essay Example

April 09, 2008 Basic Principals of Fluid Mechanics A fluid is a substance that can flow Therefore the term fluid includes liquid and gases. Fundamental Principals Fluid thrust and pressure: Thrust of a liquid is defined as the total forced acting on the surface of solid with which the liquid is in contact. Thrust has the same units as force. Thrust per unit area is called the pressure of the liquid at that point Laws of liquid pressure (i) The pressure at a point within a liquid is directly proportional to the depth of the point from the free surface of the liquid or the height of the liquid column above the point H2 p H1 P is proportional to H (ii) The pressure at a point is directly proportional to the density of the liquid P is proportional to d (iii) The pressure and thrust of a liquid at rest are always perpendicular to the surface in contact with it. (iv) The pressure at any point within a liquid is same in all directions According to Pascal law, “liquids transmit pressure equally in all directions’ (v) The pressure at a point within a liquid does not depend on the volume of the liquid in the vessel but depends only on the heights of the liquid above the point. (vi) The pressure at a point is independent of the shape of the vessel. Archimedes Principle When a body is immersed in liquid, it takes up the space which was earlier occupied by the liquid. The volume of the liquid displaced is equal to the volume of the solid. The weight of the liquid displaced is equal to the upthrust experienced by the solid due to the liquid. This is the upthrust due to which the body immersed in the liquid appears lighter. The apparent loss in weight is equal to the upthrust on the body. This is called the Archimedes principle. Thus Archimedes’ principle states that when a body is immersed partially or completely in a liquid, it experiences an upthrust which is equal to the weight of the liquid displaced by it. This principle applies not only to the liquids, but it applies equally well to the gases also. Principle of floatation (i) When a body is immersed in a liquid, it experiences a n upward thrust according to Archimedes principle. The body will sink or float on the surface of the liquid depending on it s density. There are three possibilities (ii) If the density of the body is more than the density of the liquid, the body sinks. In this case, the weight of the body is more than the upward thrust. (iii) If the density of the body is equal to the density of the liquid, the body will remain in equilibrium anywhere within the liquid. The weight of the body is equal to the upward thrust exerted by the liquid. In this case the weight of the body is equal to the weight of the displaced liquid. (iv) If the density of the body is less than the density of the liquid, the body, when completely immersed in liquid, experiences an upward thrust more than its weight, the body moves up to the surface. In the equilibrium position the body floats. A part of the body is above the surface of the liquid and the other part is immersed in the liquid. Bernoulli Principle Application of the principle of conservation of energy to the fluid flow leads to an important equation which was derived by Daniell Bernoulli in 1738. It state that the total energy of a small amount of an incompressible non viscous liquid flowing without friction from one point to another, in a streamlined flow , remains constant throughout the displacement. Consider an element of liquid of mass ‘m’, density ‘d’. Its total energy ‘E’ is sum equal of its kinetic energy, potential energy and pressure energy. ½ mv2 + mgh + m p/d = constant ……. (1) Dividing throughout by ‘m’ i.e. considering total energy per unit mass ½ v2 + gh + p/d = constant …. (2) Considering total energy per unit volume of the liquid ½ d v2 +dgh + p = constant …….. (3) Dividing equation 3 by ‘g’ throughout, v2/2g + h + p/dg = Constant ……………(4) It can be seen that all the three terms have the dimensions of length. The term v2/2g is called velocity head, h is called gravitational head and p/dg is called pressure head. The names have been given to these terms accordance with the cause of their origin. Thus Bernoulli’s equation or Bernoulli’s principle can be stated in an alternatives form as follows. It states that in case of a small amount of an incompressible, non viscous fluid, flowing from one point to another in a streamlined flow, the sum total of velocity head, gravitational head and the pressure head is a constant quantity Equations 1 and 2 give various forms of Bernoulli’s equation. Application of Archimedes Principle Equilibrium of floating bodies For a floating body in equilibrium, the following two conditions must be satisfied: 1. The weight of the body must be equal to the weight of the liquid displaced (i.e. upthrust). 2. The centre of gravity G of the body and the centre of buoyancy B of the displaced liquid must lie in the same vertical line a. The design of a small boat A solid body of density greater than that of water sinks while a body of density lowers than that of water floats. If an iron nail is placed on a surface of water, it sinks. This is because the density of iron is greater than that of water, so the weight of the nail is more than the upthrust of water on it. On the other hand, boats are also made of wood and iron but they do not sink. This is because the boat is hollow and empty space contains air which makes the average density of the boat less than that of water. Therefore, even with a small part of it submerged into water, the weight of the water displaced becomes equal to the total weight of the boat and the boat floats. If extra cargo is loaded on the boat, its weight increases so it sinks further till the weight of the water displaced by it is again equal to the weight of the loaded boat. If cargo is unloaded, the boat will rise in water. Water of different seas have different densities therefore when a boat sails from the sea of higher water density to the sea of lower water density or from the sea water to the river water, it sinks further because according to the law of floatation now a greater volume of water is required to be displaced. It is also dangerous for a boat to be too lightly loaded. Then its centre of gravity will be higher and it may be blown over on its side by the strong winds. When cargo is not loaded, the bottom of the boat is loaded with sand or stones called the ballast, so that the centre of gravity is lowered and the metacentre lies above the centre of gravity (Meta centre it is the point where the new vertical line passing through the centre of buoyancy in the displaced position meets the original vertical line passing through the centre of gravity of the floating body). This makes the equilibrium of the boat stable. b. Floatation of submarines A submarine is a fish shaped water tight body which can travel under water. Its engine works on electric batteries or nuclear energy. It can be made to diver into the water or rise up to the surface of water as and when desired. The reason is that the submarines are provided with floatation (or ballast) tanks if a submarine is to dive, the ballast tanks are filled with water so that the average density of the submachine becomes greater than the density of sea water and the submarine dives into the water. If the submarine is to rise, the water from the ballast tank is forced out into the sea by allowing the compressed air to enter the tank. This makes the average density of the submarine less than that of sea water so that the weight of water displaced by it becomes greater than the weight of submarine and hence the submarine rises up to the surface of the water. Applications of Bernoulli Principle (i) Aerofoil (Dynamic lift): this is a device shaped so that the relative motion between it and the fluid produces a force perpendicular to the flow. Examples of aerofoil are aircraft wings, turbine blades and propellers. The shape of the aerofoil is such that fluids flows faster over the top surface than over the bottom, i.e. the streamlines are closer above than below the aerofoil. By Bernoulli, it follows that the pressure underneath is increased and that above reduced. Wings of an airplane are made tapering. The upper surface is made convex and the lower surface is made concave. Due to this shape of the wings, the air currents at the top have a large velocity then at the bottom. Consequently the pressure above the surface of the wing is less as compared to the lower surface of the wing. This difference of pressure is helpful in giving a vertical lift to the plane. A resultant upward force is thus created normal to the flow and it is this surface which provides most of the ‘lift’ of an aircraft. Its value increases with the angle between the wing and the airflow (called the ‘angle of attack’); until at a certain angle the flows separate from the upper surface. Lift is lost almost completely, drag increases sharply, the flow downstream becomes very turbulent and the aircraft stalls. (ii) Spinning ball: If a tennis ball cut or a gold ball sliced’ it spins as it travels though the air and experiences a sideways force which causes it to curve in flight. This is due to air being dragged round by the spinning ball, thereby increasing the air flow on one side and decreasing it on the other. A pressure difference is thus created. The swing of a fast moving cricket ball can also be explained on the concepts of Bernoulli’s equation. The bowler imparts rotational motion to the fast moving ball. As air moves across the ball, the direction of motion of a particle on the surface of ball coincides with the direction of motion of air while the case is opposite. Thus relative velocity of air with respect to ball is different for the two diametrically opposite points. This in a accordance with the concepts of Bernoulli’s equation creates a difference of pressure, as a result of which the ball turns towards the region of low pressure. When a fast moving ball changes its direction of motion, it causes a swing. (iii) Working of carburetors in a petrol engine can also be explained on the basis of Bernoulli’s equation. Air is sucked in by the engine; it rushes past the nozzle of petrol tube. A decrease in pressure in front of the nozzle takes place as a result which petrol comes out. (iv) In strong storms roofs of the houses are blown off, leaving the other structures unaffected. As a wind blows with a high speed it creates a low pressure above the roof as a result of which it is blown off. Pressure below the roof surface is normal atmospheric pressure. This difference of pressure exerts a force, on roof in upward direction. Hence it is blown off. (v) Two ships moving parallel to each other have a tendency to move closer. The liquid in between the two ships has a greater velocity than that on the other two sides. Greater velocities result in a low pressure. As a result of this, the two ships get pressed closer to each other. (vi) It is always dangerous to stand very close to the track of the fast moving railway train. In that case, the air in between man and train moves faster than air behind the man. This result in a decrease in pressure in between man and train. Therefore a man will receive a push towards the train Works Cited Singh S B (2002), Fundamentals of Physics, Macmillan limited, New Delhi Goyal R P & Tripathi S P (1999), Fluid Mechanics, Selina publishers Read More