The mechanics of nonviscous fluids Chapter 13 The mechanics of nonviscous fluids 13. 2 equation of continuity 13.3 Bernoulli’s Equation 13.4 static consequences of Bernoulli's equation 13-6 Blood Pressure measurements using the sphygmomanometer T . Norah Ali Al moneef

Steady/Unsteady flow Turbulent Flow In steady flow the velocity of particles is constant with time Unsteady flow occurs when the velocity at a point changes with time. Turbulence is extreme unsteady flow where the velocity vector at a point changes quickly with time e.g. water rapids or waterfall. When the flow is steady, streamlines are used to represent the direction of flow. Steady flow is sometimes called streamline flow Streamlines never cross. A set of streamlines can define a tube of flow, the borders of which the fluid does not cross Turbulent Flow Turbulent flow is an extreme kind of unsteady flow and occurs when there are sharp obstacles or bends in the path of a fast-moving fluid. In turbulent flow, the velocity at a point changes erratically from moment to moment, both in magnitude and direction. T . Norah Ali Al moneef

Viscous/Non viscous flow A viscous fluid such as honey does not flow readily, it has a large viscosity. Water has a low viscosity and flows easily. A viscous flow requires energy dissipation. Zero viscosity – requires no energy. with no dissipation of energy. Some liquids can be taken to have zero viscosity e.g. water. An incompressible, non viscous fluid is said to be an ideal fluid T . Norah Ali Al moneef

13. 2 Stream flow T . Norah Ali Al moneef T . Norah Ali Al moneef

The volume flow rate, Q is defined as the volume of fluid that flows past an imaginary (or real) interface. A v represents the volume of fluid per second that passes through the tube and is called the volume flow rate Q T . Norah Ali Al moneef

If the area is reduced the fluid must speed up! If the fluid is incompressible, the density remains constant throughout Av represents the volume of fluid per second that passes through the tube and is called the volume flow rate Q This just means that the amount of fluid moving in any “section of pipe” must remain constant. If the area is reduced the fluid must speed up! Speed is high where the pipe is narrow and speed is low where the pipe has a large diameter T . Norah Ali Al moneef

The product of the area and the fluid speed at all points along a pipe is constant for an incompressible fluid. T . Norah Ali Al moneef

Q: Have you ever used your thumb to control the water flowing from the end of a hose? A: When the end of a hose is partially closed off, thus reducing its cross-sectional area, the fluid velocity increases. This kind of fluid behavior is described by the equation of continuity. Example: Oil is flowing at a speed of 1.22 m/s through a pipeline with a radius of 0.305 m. How much oil flows in 1 day? T . Norah Ali Al moneef

example Each second 5525m3 of water flows over the 670 m wide cliff of the Horseshoe falls portion of Niagara falls .the water is a approximately 2m deep as it reach the cliff . What is its speed at that instant example A river flows in a channel that is 40. m wide and 2.2 m deep with a speed of 4.5 m/s. The river enters a gorge that is 3.7 m wide with a speed of 6.0 m/s. How deep is the water in the gorge? The area is width times depth. A1 = w1d1 Use the continuity equation. v1A1 = v2A2 v1w1d1 = v2w2d2 Solve for the unknown d2. d2 = v1w1d1 / v2w2 =(4.5 m/s)(40. m)(2.2 m) / (3.7m)(6.0 m/s) = 18 m T . Norah Ali Al moneef

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example If pipe 1 diameter = 50mm, mean velocity 2m/s, pipe 2 diameter 40mm takes 30% of total discharge and pipe 3 diameter 60mm. What are the values of discharge and mean velocity in each pipe? T . Norah Ali Al moneef

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Example: The volume rate of flow in an artery supplying the brain is 3.6x10-6 m3/s. If the radius of the artery is 5.2 mm, A) determine the average blood speed. B) Find the average blood speed if a constriction reduces the radius of the artery by a factor of 3 (without reducing the flow rate). T . Norah Ali Al moneef

Example: Example,decrease in area of stream of water coming from tap. So v2 > v1 T . Norah Ali Al moneef

Example if the cross-section area, A, is 1.2 x 10-3m2 and the discharge, Q is 2.4 l / s , then the mean velocity, Example if the area A 1 = 10 x10-3 m2 and A 2 = 3 x10-3 m 2 and the upstream mean velocity, 1 = 2.1 m / s , then the downstream mean velocity T . Norah Ali Al moneef

Example A sink has an area of about 0.25 m2. The drain has a diameter of 5 cm. If the sink drains at 0.03 m/s, how fast is water flowing down the drain? Ad = pr2 = (p)(0.025 m)2 = 1.96 X 10-3 m3 Advd = Asvs vd = Asvs/Ad=[(0.25 m2)(0.03 m/s)]/(1.96 X 10-3 m3) vd = 3.82 m/s T . Norah Ali Al moneef

Example Aava = NAcvc (N is the number of capillaries) The radius of the aorta is about 1.0 cm and blood passes through it at a speed of 30 cm/s. A typical capillary has a radius of about 4 X 10-4 cm and blood flows through it at a rate of 5 X 10-4 m/s. Estimate how many capillaries there are in the human body. Aava = NAcvc (N is the number of capillaries) Aa = pr2 = (3.14)(0.01 m)2 = 3.14 X 10-4 m2 Ac = pr2 = (3.14)(4 X 10-6 cm)2 = 5.0 X 10-11 x10-4 m2 N = Aava/ Acvc (N is the number of capillaries) N = (3.14 X 10-4 m2)(0.30 m/s) = ~ 4 billion (5.0 X 10-11 x10-4 m2)(5 X 10-4 m/s) T . Norah Ali Al moneef

Example: same Example: The figure shows 3 straight pipes through which water flows. The figure gives the speed of the water in each pipe. Rank them according to the volume of water that passes through the cross-sectional area per minute, greatest first. 6 same Example: Water flows smoothly through the pipe shown in the figure, descending in the process. Rank the four numbered sections of pipe according to (a) the volume flow rate Q through them (b) the flow speed v through them (c) the water pressure p within them, greatest first. a)Same b) 1,2,3,4 c ) 4,3,2,1 T . Norah Ali Al moneef

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(II) A (inside) diameter garden hose is used to fill a round swimming pool 6.1 m in diameter. How long will it take to fill the pool to a depth of 1.2 m if water issues from the hose at a speed of The volume flow rate of water from the hose, multiplied times the time of filling, must equal the volume of the pool. T . Norah Ali Al moneef

Pressure Pressure is the ratio of a force F to the area A over which it is applied: A = 2 cm2 1.5 kg P = 73,500 N/m2 The greater the force, the greater the pressure is. The greater the area, the smaller the force is. example a) Calculate the total force of the atmosphere acting on the top of a table that measures (b) What is the total force acting upward on the underside of the table? (a) The total force of the atmosphere on the table will be the air pressure times the area of the table. (b) Since the atmospheric pressure is the same on the underside of the table (the height difference is minimal), the upward force of air pressure is the same as the downward force of air on the top of the table, T . Norah Ali Al moneef

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Example The mattress of a water bed is 2.00m long by 2.00m wide and 30.0cm deep. a) Find the weight of the water in the mattress. The volume density of water at the normal condition (0oC and 1 atm) is 1000kg/m3. So the total mass of the water in the mattress is Therefore the weight of the water in the mattress is b) Find the pressure exerted by the water on the floor when the bed rests in its normal position, assuming the entire lower surface of the mattress makes contact with the floor. Since the surface area of the mattress is 4.00 m2, the pressure exerted on the floor is T . Norah Ali Al moneef T . Norah Ali Al moneef

13.3 Bernoulli’s Equation Relates pressure to fluid speed and elevation Bernoulli’s equation is a consequence of Work Energy Relation applied to an ideal fluid Assumes the fluid is incompressible and nonviscous, and flows in a nonturbulent, steady-state manner T . Norah Ali Al moneef T . Norah Ali Al moneef

This work is negative because the force on the segment of fluid is to the left and the displacement is to the right. Thus, the net work done on the segment by these forces in the time interval T . Norah Ali Al moneef

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States that the sum of the pressure, kinetic energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline T . Norah Ali Al moneef

13.4 static consequences of Bernoulli's equation Fluid At Rest In a Container : Pressure in a continuously distributed uniform static fluid varies only with vertical distance and is independent of the shape of the container. • The pressure is the same at all points on a given horizontal plane in a fluid. • For liquids, which are incompressible, we have: pb + ρgh1 = patm + ρgh2 pb = patm + ρg (h2 - h1) = patm + ρgd Y a = y b = y c = yd y A = y B = y C = y D h1 h2 d T . Norah Ali Al moneef T . Norah Ali Al moneef

Absolute Pressure and Gauge Pressure The pressure P is called the absolute pressure Remember, P = Patm + ρ g h P – Patm = ρ g h ( so ρ g h is the gauge pressure) , because it is the actual value of the system’s pressure. Gauge pressure: Pressure expressed as the difference between the pressure of the fluid and that of the surrounding atmosphere Usual pressure gauges record gauge pressure. To calculate absolute pressure: P = P atm + P gauge As a fluid moves along if it changes speed or elevation then the pressure changes and vice versa. Bernoulli’s equation is really just conservation of energy for the fluid Bernoulli’s equation. P + ½ ρv2 + ρg h = constant Bernoulli’s equation shows that as the velocity of a fluid speeds up it’s pressure goes down…..this idea used in airplane wings and frisbees (difference in pressure leads to upward force we call Lift) T . Norah Ali Al moneef

Atmospheric Pressure and Gauge Pressure The pressure p1 on the surface of the water is 1 atm, or 1.013 x 105 Pa. If we go down to a depth h below the surface, the pressure becomes greater by the product of the density of the water , the acceleration due to gravity g, and the depth h. Thus the pressure p2 at this depth is In this case, p2 is called the absolute pressure -- the total static pressure at a certain depth in a fluid, including the pressure at the surface of the fluid The difference in pressure between the surface and the depth h is gauge pressure Note that the pressure at any depth does not depend of the shape of the container, only the pressure at some reference level (like the surface) and the vertical distance below that level. T . Norah Ali Al moneef

example (II) What gauge pressure in the water mains is necessary if a fire hose is to spray water to a height of 15 m? Apply Bernoulli’s equation with point 1 being the water main, and point 2 being the top of the spray. The velocity of the water will be zero at both points. The pressure at point 2 will be atmospheric pressure. Measure heights from the level of point 1. example A tire gauge reads 220 kPa, what is the absolute pressure? P - Patm = PG P = Patm + PG P = 101.3 kPa + 220 kPa = 321 ka T . Norah Ali Al moneef

PA>PB PA=PC PA – Patm = ρ g h PA = PB = PC = PD P atm is atmospheric pressure =1.013 x 105 Pa The pressure does not depend upon the shape of the container T . Norah Ali Al moneef

Variation of Pressure with Depth • If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium • All points at the same depth must be at the same pressure – Otherwise, the fluid would not be in equilibrium Pressure changes with elevation The pressure gradient in the vertical direction is negative The pressure decreases as we move upward in a fluid at rest Pressure in a liquid does not change due to the shape of the container The fluid would flow from the higher pressure region to the lower pressure region the pressure at a point in a fluid depends only on density, gravity and depth T . Norah Ali Al moneef

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example Water circulates throughout a house in a hot-water heating system. If the water is pumped at a speed of 0.5m/s through a 4.0cm diameter pipe in the basement under a pressure of 3.0atm, what will be the flow speed and pressure in a 2.6cm diameter pipe on the second 5.0m above? Assume the pipes do not divide into branches. Using the equation of continuity, flow speed on the second floor is Using Bernoulli’s equation, the pressure in the pipe on the second floor is T . Norah Ali Al moneef

Pressure Measurements: The manometer Manometers are devices in which one or more columns of a liquid are used to determine the pressure difference between two points. U-tube manometer One end of the U-shaped tube is open to the atmosphere The other end is connected to the pressure to be measured T . Norah Ali Al moneef

Pressure at B is P=Patm+ρgh Pressure at A > Patm Pressure in a continuous static fluid is the same at any horizontal level so, Pressure at A = Pressure at B P A = P B = Patm+ ρgh T . Norah Ali Al moneef T . Norah Ali Al moneef T . Norah Ali Al moneef

Pressure at c =pressure at D 13-6 Blood Pressure measurements using the sphygmomanometer Pressure at c =pressure at D PC = PD PC = PA + ρsgh PB = Patm + ρ gh PA + ρsgh = Patm + ρ gh PA = Patm + ρ g h – ρsgh Pblood =PA = Patm + ρ g h – ρsgh T . Norah Ali Al moneef

Blood Pressure Sphygmometer Blood pressure is measured with a special type of manometer called a sphygmomano-meter Pressure is measured in mm of mercury 26.8 K P a T . Norah Ali Al moneef

Pressure with depth Estimate the amount by which blood pressure changes in an actuary in the foot P2 and in the aorta P1 when the person is lying down and standing up Take density of blood = 1060kg/m3 T . Norah Ali Al moneef

example A fluid of constant density ρ = 960 kg / m3 is flowing steadily through the above tube. The diameters at the sections are d 1 =100 mm and d 2 = 80 mm . The gauge pressure at 1 is p1 = 200 k N/ m2 ,and the velocity here is u 1 = 5 m /s . We want to know the gauge pressure at section 2. The tube is horizontal, with y1 = y2 so Bernoulli gives us the following equation for pressure at section 2: P2 = 2 x10 5 + 960 { 52 – ( 7.8 )2} /2 = 182796.8 pa T . Norah Ali Al moneef

What is the gauge pressure if: h1 = 0.4m and h2 = 0.9m? An example of the U-Tube manometer Using a u-tube manometer to measure gauge pressure of fluid density ρ = 700 kg/m3, and the manometric fluid is mercury, with a relative density of 13.6. What is the gauge pressure if: h1 = 0.4m and h2 = 0.9m? b) h1 = 0.4 and h2 = -0.1m? pB = pC pB = pA + ρgh1 pB = pAtmospheric + ρman gh2 Subtract patmospheric to give gauge pressure pA = ρman gh2 - ρgh1 a) pA = 13.6 x 103 x 9.8 x 0.9 - 700 x 9.8 x 0.4 = 117 327 N, b) pA = 13.6 x 103 x 9.8 x (-0.1) - 700 x 9.8 x 0.4 = -16 088.4 N, The negative sign indicates that the pressure is below atmospheric T . Norah Ali Al moneef

Example: Fluid is flowing from left to right through the pipe. Points A and B are at the same height, but the cross-sectional areas of the pipe differ. Points B and C are at different heights, but the cross-sectional areas are the same. Rank the pressures at the three points, from highest to lowest. A) A and B (a tie), C B) C, A and B (a tie) C) B, C, A D) C, B, A E) A, B, C E) PA > PB > PC T . Norah Ali Al moneef

Pressure Measurements: A long closed tube is filled with mercury and inverted in a dish of mercury Measures atmospheric pressure as ρ g h Since the closed end is at vacuum, it does not exert any force. p=0 Vacuum! Vacuum pressure = 0 patm h For mercury, h = 760 mm. How high will water rise? No more than h = patm/g = 10.3 m T . Norah Ali Al moneef T . Norah Ali Al moneef

example We can set (assume the hole is on the ground or is where we measure height from). We also have 1 atm. So we have v2 h T . Norah Ali Al moneef

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example P2 =p1 + ρ (v12 –v22 ) / 2 = 180X103 + 103X(22 – 182) = 20X 103 pa T . Norah Ali Al moneef

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If a fluid is at rest in a container, all portions of the fluid must be in static equilibrium If the height doesn’t change much, Bernoulli becomes: y1 = y2 Where speed is higher, pressure is lower. Speed is higher on the long surface of the wing – creating a net force of lift. FL T . Norah Ali Al moneef

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example 1.5m Water flows through the pipe as shown at a rate of .015 m3/s. If water enters the lower end of the pipe at 3.0 m/s, what is the pressure difference between the two ends? A2 = 20 cm2 T . Norah Ali Al moneef

Example Estimate the force exerted on your eardrum due to the water above when you are swimming at the bottom of the pool with a depth 5.0 m. We first need to find out the pressure difference that is being exerted on the eardrum. Then estimate the area of the eardrum to find out the force exerted on the eardrum. Since the outward pressure in the middle of the eardrum is the same as normal air pressure Estimating the surface area of the eardrum at 1.0cm2=1.0x10-4 m2, we obtain T . Norah Ali Al moneef

Example. A diver is located 20 m below the surface of a lake (r = 1000 kg/m3). What is the pressure due to the water? The difference in pressure from the top of the lake to the diver is: h r = 1000 kg/m3 DP = rgh h = 20 m; g = 9.8 m/s2 DP = 196 kPa T . Norah Ali Al moneef

Example (a) What are the total force and the absolute pressure on the bottom of a swimming pool 22.0 m by 8.5 m whose uniform depth is 2.0 m? (a)The absolute pressure is given by Eq. 10-3c, and the total force is the absolute pressure times the area of the bottom of the pool. T . Norah Ali Al moneef

Example: a) At what water depth is the pressure twice the atmospheric pressure? b) What’s the pressure at the bottom of the 11-km-deep Marianas Trench, the deepest point in the ocean? Take 1 atm = 100 kPa & water = 1000 kg/m3 . (a)  (b) Pressure increases by 1 atm per 10 m depth increment.  T . Norah Ali Al moneef

Applications of Bernoulli's Equation Air speeds up in the constricted space between the car & truck creating a low-pressure area. Higher pressure on the other outside pushes them together. Hold two sheets of paper together, as shown here, and blow between them. No matter how hard you blow, you cannot push them more than a little bit apart! T . Norah Ali Al moneef

Summary Fluid at Rest: PA - PB = rgh Bernoulli’s Theorem: Streamline Fluid Flow in Pipe: PA - PB = rgh Horizontal Pipe (h1 = h2) Fluid at Rest: Bernoulli’s Theorem: T . Norah Ali Al moneef