Fluid Mechanics Chapter 4 PDF

Chapter 4 Fluid Mechanics  A fluid is any substance that can not maintain its own shape; in other words, it is a substance with no rigidity. This definition includes liquids, such as water; gases, such as air; some mixture of liquids and solids, such as crèmes  The study of fluids dates...

Chapter 4 Fluid Mechanics  A fluid is any substance that can not maintain its own shape; in other words, it is a substance with no rigidity. This definition includes liquids, such as water; gases, such as air; some mixture of liquids and solids, such as crèmes  The study of fluids dates back to some of the earliest discoveries in physics. Many of the principles we examine in this chapter are associated with great scientists of the past, such as Archimedes (300BC), Pascal (17th century), Bernoulli (18th century), and Stoke (19th century). However, the study of fluid flow remains an active area of research.  Volume: is the space occupied by a body. Its unit is (length3) e.g. m3,cm3,inch3…. Density: is the mass per unit volume. ρ= Mass Volume Its unit is kg/m3, gm/cm3, ….  Compressible Fluids: They have no definite volume; that is, they expand to fill their container. Gases are example of this type. Incompressible Fluids: They have definite volume, and so definite density. They take the shape of their container. Liquids are example of this type. Unless otherwise stated, we deal in this chapter with incompressible fluids (liquids).  Specific Gravity: Specific gravity of a substance is given by the equation: Specific Gravity = Mass of the substance Mass of equal volume of water  Here water is taken as a reference for determining the specific gravity of a substance. Specific gravity is a unit less quantity and is very useful in qualitative analysis such as urine analysis. The specific gravity of a given substance equals numerically its density.  The Pressure: pressure is defined as "the force per unit area" Pressure (P) = Force Area Units: Newoton/meter2, Dyne/centimeter2 ,…. If we use the SI system of units, the unit of the pressure would be Newton/ meter2 , which is called Pascal (symbol Pa) in the honor of Baliase Pascal 1 Pa =1 N/m2  The pressure of a liquid depends on its height only: Consider a liquid put in an evacuated tube we will now try to find the pressure Pressure= Force = Weight of liquid (1) Area A Weight of liquid = Mass × g (2) Mass=density × volume=ρ × V= ρ × h × A (3) If we substitute (2) & (3) in (1) we can write: Pressure of the liquid= ρ × h × A × g = ρ × h × g A  Re-arranging we can write: Pliquid= ρ.g. h Since ρ and g are constants, we see that the pressure of a liquid varies with its height.  Example1: A tank is filled with water to a depth of 1.5 m. What is the pressure at the bottom of the tank due to water a lone? (g=9.8 m/sec2, ρwater=1×103 kg/m3) Solution: Pwater= ρ.g. h=1×103×9.8×1.5=1.5×104 N/m2 =1.5×104 Pa  Example2: A nurse administers medication in a saline solution to a patient by infusion into a vein in the patient’s arm. The density of the solution is 1.0 ×103 kg/m3 and the pressure inside the vein is 2.5×103 Pa. How high above insertion point must the container be hung so that there is sufficient pressure to force the fluid into the patient  Solution: The container must be hung high enough that the pressure due to the liquid in the tube and container is at least as great as the pressure inside the vein. Pliquid = ρ g h = 1.0 × 103× 9.8 × h This pressure must be greater than 2.4×103 Pa, so: 1.0 ×103 × 9.8 × h > 2.5×103 h> 0.25m (25 cm)  The atmospheric pressure: Is the pressure exerted by the air at any object in the earth. This pressure varies with the height of an object from the surface of the earth. When we lifted to higher latitudes above the surface of the earth, the atmospheric pressure decreases. Other units for measuring the pressure: Since the pressure for fluids varies with their height, we may use fluids for measuring the pressure. Here are examples of such units: mm or cm mercury(Hg): Here the mercury is used as a reference for measuring the pressure  1 mm Hg=1.33×102 Pa Atmosphere: The pressure exerted by the air at the sea level is taken to be 1 atmosphere (1 atm ). 1 atmosphere=101.3×103 Pa  Example: Calculate the pressure at a depth 30 m below the sea level. Given ρ sea water=1030 kg/m3 Solution: P=Patm + ρ.g. h =101.3×103 + 1030 ×9.8× 30= 404120 Pa  Gauge Pressure: Is the pressure in excess of atmospheric pressure.  Pascal's Principle: "The pressure applied at one point in an enclosed fluid is transmitted undiminished to every part of the fluid and to the walls of the container". Pascal's principle holds for gases as well as liquids, with some minor modifications due to the change in the volume of a gas when the pressure is changed.  Pascal's principle in practice: Hydraulic lift: Pa= Pb Fa = Fb Aa Ab  Example: A piston is fitted into an 3 cm2 area cylinder which is connected to a larger cylinder of 24 cm2 area. If a 50 kg women puts all her weight on the handle of the smaller piston, how much weight can be lifted in the larger one?.  Fa = Fb Aa A b m×g = Fb Aa Ab 50×9.8 = Fb 3 24 Fb = 50 ×9.8×24 = 3920 N 3 Medical Application: measuring human blood pressure: The heart is a large muscle, responsible for pumping blood to all parts of the body. It works like two pumps: the blood returns from the body through the veins to the right side of the heart, which pumps the blood to the lungs. The lungs remove the carbon dioxide from the blood and add oxygen. The left side of the heart receives the oxygenated blood from the lungs and pumps it to the body by way of the arteries.  Two pressures in the heart's action are of particular interest: the systolic pressure, when the heart is contracted, and the diastolic pressure, when the heart is relaxed between beats.  The commonly used method for measuring the blood pressure is by a device called sphygmomanometer. A non elastic cuff that has as inflatable bag within it is placed around the upper arm, roughly at the same vertical level as the heart. The cuff is connected directly to some pressure gauge such as a manometer.  When the cut cuff is inflated, the tissue in the arm is compressed; if sufficient pressure is applied, the flow of arterial blood in the arm stops. The pressure in the tissues in the arm is the same as the pressure in the cuff, and is also the same as the pressure in the artery. Pascal's principle holds for the system composed of cuff, arm, artery, and manometer.  After the blood flow has been cut of, the pressure in the cuff is reduced by releasing some of the air. At some point the maximum arterial pressure slightly exceeds the pressure in the surrounding tissue and cuff, allowing the blood to resume flowing. The acceleration of the blood through the arteries gives rise to a characteristic sound, which can be identified by means of a stethoscope.  When this sound occurs, the manometer's reading indicates the maximum, or systolic pressure. As the pressure in the cuff falls further, a second change in the sound is heard, characteristic of he drop below diastolic pressure.  A typical value of the systolic pressure for a healthy person is 120 mm Hg, and for diastolic pressure is 80 mm Hg. These two values are written in the form 120/80 mm Hg.  Fluids in motion Continuity Equation: Consider an incompressible fluid flowing steadily through a tube from a region of cross-sectional area A1 to a region of area A2. Because the fluid is incompressible, the same a mount of it leaves each region per unit time:  mass1=mass2 => Density×Volume1= Density× Volume2 => ρ× V1= ρ× V2 => ρ×A1×v1×Δt= ρ×A2×v2×Δt => A1×v1= A2×v2 => A×v =Constant {continuity equation}  Example: An incompressible fluid moves with speed 4m/sec in a cylinder of cross-sectional area 25 cm2. The cylinder feeds into smaller cylinder of cross-sectional area 10 cm2. What is the velocity of water in the smaller pipe? Solution:  A1×v1= A2×v2 25×4=10×v2 v2 = 100 = 10 m/sec 10