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Department of Physical and Mathematical Sciences Olawumi R. Kaka 3.0 BOUYANCY, ARCHIMEDES' PRINCIPLES , SURFACE TENSION AND HYDRODYNAMICS 3.1 BOUYANCY AND ARCHIMEDES' PRINCIPLES When an object is submerged in a fluid, it appears to weigh less than they do when outside the fluid. For example, a large rock at the bottom of a stream would be easily lifted compare to lifting it from the ground. As the rock breaks through the surface of the water, it becomes heavier. This phenomenon is as a result of upward force called the buoyant force (upthrust) acting on the rock plus the downward gravitational force. The buoyant force occurs because the pressure in fluid increases with depth. Thus, the upward pressure at the bottom surface of a submerged object is greater than the downward pressure at its top surface. Hence, a buoyant force is the upward force exerted by water on any immersed object. To see the effect of this, consider a cylinder of height h whose top and bottom ends have an area A and completely submerged in a fluid of density as shown in figure . Therefore, The pressure exerted by the fluid at the top surface of the cylinder is The downward force due to this pressure on top of the cylinder is The upward force on the bottom of the cylinder is The net force due to the fluid pressure, which is the buoyant force, , acts upward with a magnitude: 1 *2 Where: is the volume of the cylinder Since is the density of the fluid, then is the weight of fluid which takes up a volume equal to the volume of the cylinder. Thus, buoyant force on the cylinder equals the weight of fluid displaced by the cylinder. The mode in which buoyant forces act is summarised is known as Archimedes’s principle; which states the buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by the object. 3.1.1 Forces Acting on a Totally Submerged Object and a Floating Object (Partly Submerged) 3.1.1.1 Totally Submerged Object When an object is totally submerged in a fluid of density , the magnitude of the upward buoyant force is ; Where Vo is the volume of the object If the object has a mass M and density , its weight is The net force acting on the object is = 3.1.1.2 ) Floating Object 2 Consider an object of volume Vo in a static equilibrium floating on a fluid (i.e an object that is partially submerged). In this case, the upward buoyant force is balanced by the downward gravitational force acting on the object. If Vf is the volume of the fluid displaced by the object (the part of the object beneath the fluid level). . The buoyant force has a magnitude; Because the weight of the object is: Since Therefore; ; Example on Buoyancy: A 15.0kg solid gold statue is being raised from a sunken ship. (i) What is the buoyant force when the statue is immersed in a fluid (ii) the force when it is out of the water? Hence, find the tension in the hoisting cable when the statue is (iii) at rest and completely immersed (iv) at rest and out of water. Density of gold is Solution i. The buoyant force when the statue is immersed in a fluid, we use But Therefore, ii. The force when it is out of the water 3 iii. The tension when the object is completely immersed Since the statue is at rest the net external force acting on it is zero Therefore, iv. The tension when the statue is out of water = Statue weight 3.2 SURFACE TENSION You would have noticed this interesting behaviour that can take place at the surfaces of liquids. According to Archimedes' principle, a steel needle should sink in water if its density is greater than density of water. A needle placed carefully on water, however, can be supported by the surface tension because the liquid responds in a way similar to a stretched membrane. Also, water is often used for cleaning, but the surface tension makes it hard for water to penetrate into small crevices or openings, such as are found in clothes. Soap is added to water to reduce the surface tension, so clothes (or whatever else) get much cleaner. The cohesive forces between liquid molecules are responsible for the phenomenon known as surface tension. The cohesive forces between molecules down into a liquid are shared with all neighbouring atoms but those on the surface have no neighbouring atoms above, and thus 4 exhibit stronger attractive forces upon their nearest neighbours on the surface. This enhancement of the intermolecular attractive forces at the surface is called surface tension. 3.3 HYDRODYNAMICS: Description of the properties of a fluid as it moves at each point as a function of time is termed hydrodynamics. 3.3.1 Characteristics of a Fluid Considering the motion of real fluids, it is very complex and not fully understood. When fluid is in motion, its flow can be characterized as being one of 2 main types. Steady or laminar flow. The flow is steady if the overall flow pattern does not change with time. In this type of flow every element passing through a given point follows the same flow line i.e. the velocity of the fluid at any point remains constant in time. In a laminar flow, each particle of the fluid follows a smooth path, such that the paths of different particles never cross each other. Above a certain critical speed, fluid flow becomes turbulent; turbulent flow is irregular flow characterized by small whirlpool-like regions. The term viscosity is commonly used in the description of fluid flow to characterize the degree of internal friction in the fluid. This internal friction, or viscous force, is associated with the resistance that two adjacent layers of fluid have to moving relative to each other. Figure 1a: Hot gases from a cigarette made visible by smoke particles. The smoke first moves in laminar flow at the bottom and then in turbulent flow above. 5 Figure 1b: Laminar flow around an automobile in a test wind tunnel. Other characteristics involve the streamline the path taken by a fluid particle under steady flow. The velocity of the particle is always tangent to the streamline, as shown in Figure 2. A set of streamlines like the ones shown in Figure 2 form a flow tube. Note that fluid particles cannot flow into or out of the sides of this tube; if they could, then the streamlines would cross each other. along its Figure 2: A particle in laminar flow follows a streamline, and at each point path the particle’s velocity is tangent to the streamline 3.3.2 Properties of an Ideal Fluid Looking at the above, some simplifying assumptions could be said in the model of an ideal fluid as follows: 1. The fluid is non-viscous. In a non-viscous fluid, internal friction is neglected. An object moving through the fluid experiences no viscous force. 2. The flow is steady. In steady (laminar) flow, the velocity of the fluid at each point remains constant. 3. The fluid is incompressible. The density of an incompressible fluid is constant. 6 4. The flow is irrotational. In an irrotational flow, the fluid has no angular momentum about any point. If a small paddle wheel placed anywhere in the fluid does not rotate about the wheel’s center of mass, then the flow is irrotational 3.3.3 Equation of Continuity We now know that the moving fluid doesn’t change as it flows, this leads to an important quantitative relationship called continuity equation. Consider an ideal fluid flowing through a pipe of non-uniform size, as illustrated in Figure 3. The particles in the fluid move along streamlines in steady flow. In a time t, the fluid at the bottom end of the pipe moves a distance, . If A1 is the cross-sectional area in this region, then the mass of fluid contained in the left shaded region in Figure 3 is where is the (non-changing) density of the ideal fluid. Figure 3: A fluid moving with steady flow through a pipe of varying cross-sectional area Similarly, the fluid that moves through the upper end of the pipe in the time t has a mass However, because mass is conserved and because the flow is steady, the mass that crosses A1 in a time t must equal the mass that crosses A2 in the time t. That is, Therefore, And, *3 7 The product Av, which has the dimensions of volume per unit time, is called either the volume flux or the flow rate. Equation *3 represents equation of continuity for an incompressible fluid and it states that the product of the area and the fluid speed at all points along the pipe is a constant 3.3.4 BERNOULLI’S EQUATION The relationship between fluid speed, pressure and elevation was first derived in 1738 by Swiss physicist, Daniel Bernoulli. To derive Bernoulli’s equation, consider the flow of an ideal fluid through a non-uniform pipe in a time t. Figure 4 Figure 4: A fluid in laminar flow through a constricted pipe. The volume of the shaded section on the left is equal to the volume of the shaded section on the right. The force exerted by the fluid in section 1 has a magnitude: 1 The workdone by this force in a time t is = 2 Where V is the volume of section 1 Similarly, the workdone by the fluid in section 2 is the same time t is; = 3 The volume that passes through section 1 in a time t equal the volume that passes through section in the same time.W2 is negative because the fluid force opposes the displacement. 8 Thus, the net workdone by these forces in the time t is 4 Part of this work goes into changing the kinetic energy of the fluid and part goes into changing the gravitational potential energy. If m is the mass that enters one and leaves the other end in a time t, then the change in the kinetic energy of mass m is 5 The change in the gravitational P.E is 6 From equation 4, Total Energy, is 7 If we divide each term by V and recall that this expression reduces to 8 Rearranging, 10 This is Bernoulli’s equation as applied to an ideal fluid. It is often expressed as *4 This expression specifies that, in a laminar flow, the sum of the pressure (P), kinetic energy per unit volume and gravitational potential energy per unit volume has the same value at all points along a streamline. When the fluid is at rest equation *4 becomes 9