Lesson 12: Fluid statics, Continuity equation (Sections ) Chapter 9 Fluids

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1 Lesson : luid statics, Continuity equation (Sections ) Chapter 9 luids States of Matter - Solid, liquid, gas. luids (liquids and gases) do not hold their shapes. In many cases we can think of liquids as being incompressible. Liquids do not change their volume (appreciably) when they are heated. Gases do not have a definite volume or shape. Pressure fluid particle collides with the surface. The change in momentum is caused by an impulse that acts to the right on the particle. By Newton s third law, the particle pushes to the left on the surface. Definition of pressure P av Surprisingly, pressure is a scalar and not a vector. Pressure is measured in N/m which is also called a pascal (Pa). There are a zillion other units: atmosphere, lbs/square inch, torr, bar, etc. Lesson, page

2 Lesson : luid statics, Continuity equation (Sections ) The atmosphere exerts pressure. The pressure of one atmosphere is atm.030 Pascal s Principle change in pressure at any point in a confined fluid is transmitted everywhere throughout the fluid. (p. 36) Pascal s principle is a consequence of the incompressibility of fluids. Pascal s principle is the basis of hydraulics. Hydraulics are the most effective way to transmit a force. Notice that the displacement in the narrow tube is much greater that the displacement in the wide tube. We will return to this observation later. 5 Pa The force on the right () changes the pressure in the fluid. By Pascal s principle, the change in pressure is transmitted throughout the fluid. P P If >, then the >. Modern Marvels: Excavator advertisement: Density is the mass per unit volume. It is defined as Lesson, page

3 Lesson : luid statics, Continuity equation (Sections ) Density is measured in kg/m 3. m V s we descend down into a fluid the amount of fluid above us increases. That additional fluid pushes down and the pressure increases with depth. t depth d the pressure has increased P P gd Measuring Pressure manometer consists of a U-shaped tube containing some mercury. When both sides are open to the atmosphere, the height in both arms are the same. When one side is connected to the pressure to be measured, the heights are different. Lesson, page 3

4 Lesson : luid statics, Continuity equation (Sections ) The pressures at the same height in the same fluid are equal. If they are different, fluid will flow from the high pressure side to the low pressure side. So, the pressures at B and B' are the same, P B P gd C Usually, the manometer is open to the atmosphere. It will measure pressures relative to atmospheric. The gauge pressure is the pressure relative to an atmosphere. P gauge P abs P atm Blood pressure is measured with a sphygmomanometer. The oldest kind of sphygmomanometer consists of a mercury manometer on one side attached to a closed bag the cuff. The cuff is wrapped around the upper arm at the level of the heart and is then pumped up with air. The manometer measures the gauge pressure of the air in the cuff. t first, the pressure in the cuff is higher than the systolic pressure the maximum pressure in the brachial artery that occurs when the heart contracts. The cuff pressure squeezes the artery closed and no blood flows into the forearm. valve on the cuff is then opened to allow air to Lesson, page 4

5 Lesson : luid statics, Continuity equation (Sections ) escape slowly. When the cuff pressure decreases to just below the systolic pressure, a little squirt of blood flows past the constriction in the artery with each heartbeat. The sound of turbulent blood flow past the constriction can be heard through the stethoscope. s air continues to escape from the cuff, the sound of blood flowing through the constriction in the artery continues to be heard. When the pressure in the cuff reaches the diastolic pressure in the artery the minimum pressure that occurs when the heart muscle is relaxed there is no longer a constriction in the artery, so the pulsing sounds cease. The gauge pressures for a healthy heart are nominally around 0 mm Hg (systolic) and 80 mm Hg (diastolic). (p. 333) Buoyant orce When an object is submerged in a fluid, the fluid pushes up on the object. The buoyant force is given by B gd gv rchimedes principle (p. 334) fluid exerts an upward buoyant force on a submerged object equal in magnitude to the weight of the volume of fluid displaced by the object. We still need to use free body diagrams! The force is the force of the fluid above the block pushing down and the force is the force of the fluid below the block pushing up. We have > since the pressure increases with depth. The buoyant force is B By B y The specific gravity is defined as the ratio of the density of the material to the density of water. S.G. water y If S.G. <, the object will float. If S.G. >, the object sinks. rchimedes and the golden crown: The story: We have completed our study of fluids at rest. Now we consider fluids in motion. Lesson, page 5

6 Lesson : luid statics, Continuity equation (Sections ) luid low fluid moving past a surface can exert a viscous force against the surface. This is similar to the frictional force of an object sliding over a surface. We will start by assuming the viscous force to be small. When flow is steady, the velocity at any point is constant in time. The flow may not be the same everywhere. Steady flow is laminar. The streamlines are clearly defined. Turbulence is unsteady fluid flow, not laminar flow. In turbulent flow, swirling vortices appear. The vortices are not stationary and they move with the fluid. The velocity of the fluid flow can change direction and magnitude in an uncontrolled way. s we have done many times this semester, we assume the ideal case first. n ideal fluid is incompressible, undergoes laminar flow, and has no viscosity. The continuity equation Since the fluid is incompressible, the fluid flows faster in the narrow portions of the pipe. Lesson, page 6

7 Lesson : luid statics, Continuity equation (Sections ) The mass flow rate is defined as The volume flow rate is m v t V v t The continuity equation for an incompressible fluid equates the volume flow rates past two different points, V t v V t v The continuity equation is a consequence of conservation of mass. Lesson, page 7

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