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# Fluid Mechanics

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on 5 September 2012#### Transcript of Fluid Mechanics

Well technically, the glass

is always full... of fluids! So what is Fluid? - a substance that continually deforms (flows) under an applied shear stress. And what are the kinds of fluids? Plasma Half Air Half Water Liquid Gas Fluid Mechanics is the branch of physics that studies fluids (liquids, gases, and plasmas) and the forces on them. So when did the study of fluid mechanics all started? - Going back to days of the ancient Greece, Archimedes investigated fluid statics and buoyancy and formulated his famous law known now as the Archimedes' principle, which was published in his work On Floating Bodies. The formula for Density: p m v where is the density and is the volume = p is the mass m v What is ? - The mass of a substance per unit volume. - The SI Unit for density is kg/m or g/cm. 3 3 DENSITY What is the density of a piece of wood that has a mass of 25.0 grams and a volume of 29.4 cm ? Sample Problem : 3 m p v = Solution : p = 25.0 grams 29.4 cm 3 p = 0.85 g/cm 3 What is ? PRESSURE - is the ratio of force to the area over which that force is distributed. The formula for Pressure : f P a = where P is the pressure f is the normal force and a is the area - the SI Unit for pressure is N/m or pascal Pa. 2 Calculate the pressure produced by a force of 800 N acting on an area of 2.0 m . Sample Problem : 2 f P a = Solution : 800 N P = 2.0 m 2 P = 400 N/m 2 What is Fluid Pressure? - The force per unit area of a fluid at any point at a given depth is called liquid pressure. The formula for Fluid Pressure : P = where P is the Fluid pressure p is the density g is the acceleration of gravity pgh and h is the height of the fluid above the object Find the pressure on a scuba diver when she is 12 meters below the surface of the ocean. Assume standard atmospheric conditions. Sample Problem : Solution : The density of sea water is 1.03 x 10 kg/m and the atmospheric pressure is 1.01 x 10 N/m. 3 3 5 2 P = pgh P = (1.03 x 10 kg/m ) (9.8 m/s ) (12 m) 3 3 2 P = f f f 1.21 x 10 N/m 5 2 P = t P P f atm + P = t (1.01 x 10 N/m ) + (1.21 x 10 N/m ) 2 5 P = 2.22 x 10 kPa t 10 2 2 Pressure in a Liquid -depends on the density of the liquid -pressure due to liquid is precisely equal to the product of weight density and depth. Formula for Liquid Pressure : Liquid Pressure = gravity x height x density Sample Problem : What is the pressure in the water at the base of a 10 m high dam wall? Solution: P = ghd P = (9.8 m/s )(10 m)(1000kg/m ) 2 3 P = 98000 N/m 2 P total = 2P total P = 196000 N/m 2 Archimedes' Principle An immersed object is buoyed up by a force equal to the weight of the fluid it displaces. Formula for Buoyant Force: F = pgV = mg B Where F is the buoyant force B

Fluid pressure is independent of the shape of its container. Important points to remember about fluid pressure: The forces exerted by a fluid on the walls of its container are always perpendicular to the walls. The fluid pressure is directly proportional to the depth of the fluid and to its density. At any particular depth, the fluid pressure is the same in all directions. A basketball floats in a bathtub of water. The ball has a mass of 0.5 kg and a diameter of 22 cm.

What is the buoyant force? Sample Problem : Solution : F = (0.5 kg)(9.8 m/s ) F = mg F = 4.9 N B B B 2 p is the density of the fluid g is the acceleration due to gravity and V is the Volume of the object. What is Fluid Flow? - is the motion of a fluid in which every particle in the fluid follows the same path as that followed by previous particles. Formula for Fluid Flow we can get the volume of a fluid in a given time interval : V = Avt where V is the volume A is the cross-section where the volume flows and vt is the average velocity v during a time interval t We can also get the rate of flow by using this formula : R = = vA Avt t In addition, if the fluid is incompressible, then the rate of flow will remain constant, thus we get this kind of formula : R = v A = v A 1 1 2 2 The water flows through a rubber hose 2 cm in diameter at a velocity of 4 m/s. What must be the diameter of the nozzle if water is to emerge at 20 m/s? Sample Problem : Solution : A v = A v 1 2 1 2 Since the area A is proportional to the square of the diameter, we can write : d v = d v 1 2 1 2 2 2 derivation for d 2 2 d v 1 2 1 2 2 2 v = d d = 2 d v 1 1 2 2 v = (2 cm) (4 m/s) 2 (20 m/s) d = 0.80 cm 2 2 d = 2 0.894 cm Bernoulli's Equation when a liquid is stationary, both v and v are zero. formula : 1 2 P - P = pg(h - h ) 2 1 1 2 Where P is pressure p is density g is acceleration of gravity h is the height of water or the displacement But if there is no change in pressure we can reduce the equation as : pgh + pv = pgh 1 1 2 2 2 or v = 2g(h - h ) = 2gh 2 1 2 1 This relationship is also known as Torricelli's theorem : v = 2gh Bernoulli's Principle - states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. - named after the Swiss scientist Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738. Sample problem : A crack in a water tank has a cross-sectional area of 1 cm. At what rate is water lost from the tank if the water level in the tank is 4 m above the opening? 2 Solution : R = A = (10 m ) 2gh (2)(9.8 m/s )(4 m) -4 2 2 R = (10 m )(8.85 m/s) -4 2 R = 8.85 x 10 m /s -4 3 FLUID MECHANICS

Presented by Group 3 Ronnel Gono James Mejia Florante Marasigan Redzl No Neil Francis Torio

Full transcriptis always full... of fluids! So what is Fluid? - a substance that continually deforms (flows) under an applied shear stress. And what are the kinds of fluids? Plasma Half Air Half Water Liquid Gas Fluid Mechanics is the branch of physics that studies fluids (liquids, gases, and plasmas) and the forces on them. So when did the study of fluid mechanics all started? - Going back to days of the ancient Greece, Archimedes investigated fluid statics and buoyancy and formulated his famous law known now as the Archimedes' principle, which was published in his work On Floating Bodies. The formula for Density: p m v where is the density and is the volume = p is the mass m v What is ? - The mass of a substance per unit volume. - The SI Unit for density is kg/m or g/cm. 3 3 DENSITY What is the density of a piece of wood that has a mass of 25.0 grams and a volume of 29.4 cm ? Sample Problem : 3 m p v = Solution : p = 25.0 grams 29.4 cm 3 p = 0.85 g/cm 3 What is ? PRESSURE - is the ratio of force to the area over which that force is distributed. The formula for Pressure : f P a = where P is the pressure f is the normal force and a is the area - the SI Unit for pressure is N/m or pascal Pa. 2 Calculate the pressure produced by a force of 800 N acting on an area of 2.0 m . Sample Problem : 2 f P a = Solution : 800 N P = 2.0 m 2 P = 400 N/m 2 What is Fluid Pressure? - The force per unit area of a fluid at any point at a given depth is called liquid pressure. The formula for Fluid Pressure : P = where P is the Fluid pressure p is the density g is the acceleration of gravity pgh and h is the height of the fluid above the object Find the pressure on a scuba diver when she is 12 meters below the surface of the ocean. Assume standard atmospheric conditions. Sample Problem : Solution : The density of sea water is 1.03 x 10 kg/m and the atmospheric pressure is 1.01 x 10 N/m. 3 3 5 2 P = pgh P = (1.03 x 10 kg/m ) (9.8 m/s ) (12 m) 3 3 2 P = f f f 1.21 x 10 N/m 5 2 P = t P P f atm + P = t (1.01 x 10 N/m ) + (1.21 x 10 N/m ) 2 5 P = 2.22 x 10 kPa t 10 2 2 Pressure in a Liquid -depends on the density of the liquid -pressure due to liquid is precisely equal to the product of weight density and depth. Formula for Liquid Pressure : Liquid Pressure = gravity x height x density Sample Problem : What is the pressure in the water at the base of a 10 m high dam wall? Solution: P = ghd P = (9.8 m/s )(10 m)(1000kg/m ) 2 3 P = 98000 N/m 2 P total = 2P total P = 196000 N/m 2 Archimedes' Principle An immersed object is buoyed up by a force equal to the weight of the fluid it displaces. Formula for Buoyant Force: F = pgV = mg B Where F is the buoyant force B

Fluid pressure is independent of the shape of its container. Important points to remember about fluid pressure: The forces exerted by a fluid on the walls of its container are always perpendicular to the walls. The fluid pressure is directly proportional to the depth of the fluid and to its density. At any particular depth, the fluid pressure is the same in all directions. A basketball floats in a bathtub of water. The ball has a mass of 0.5 kg and a diameter of 22 cm.

What is the buoyant force? Sample Problem : Solution : F = (0.5 kg)(9.8 m/s ) F = mg F = 4.9 N B B B 2 p is the density of the fluid g is the acceleration due to gravity and V is the Volume of the object. What is Fluid Flow? - is the motion of a fluid in which every particle in the fluid follows the same path as that followed by previous particles. Formula for Fluid Flow we can get the volume of a fluid in a given time interval : V = Avt where V is the volume A is the cross-section where the volume flows and vt is the average velocity v during a time interval t We can also get the rate of flow by using this formula : R = = vA Avt t In addition, if the fluid is incompressible, then the rate of flow will remain constant, thus we get this kind of formula : R = v A = v A 1 1 2 2 The water flows through a rubber hose 2 cm in diameter at a velocity of 4 m/s. What must be the diameter of the nozzle if water is to emerge at 20 m/s? Sample Problem : Solution : A v = A v 1 2 1 2 Since the area A is proportional to the square of the diameter, we can write : d v = d v 1 2 1 2 2 2 derivation for d 2 2 d v 1 2 1 2 2 2 v = d d = 2 d v 1 1 2 2 v = (2 cm) (4 m/s) 2 (20 m/s) d = 0.80 cm 2 2 d = 2 0.894 cm Bernoulli's Equation when a liquid is stationary, both v and v are zero. formula : 1 2 P - P = pg(h - h ) 2 1 1 2 Where P is pressure p is density g is acceleration of gravity h is the height of water or the displacement But if there is no change in pressure we can reduce the equation as : pgh + pv = pgh 1 1 2 2 2 or v = 2g(h - h ) = 2gh 2 1 2 1 This relationship is also known as Torricelli's theorem : v = 2gh Bernoulli's Principle - states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. - named after the Swiss scientist Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738. Sample problem : A crack in a water tank has a cross-sectional area of 1 cm. At what rate is water lost from the tank if the water level in the tank is 4 m above the opening? 2 Solution : R = A = (10 m ) 2gh (2)(9.8 m/s )(4 m) -4 2 2 R = (10 m )(8.85 m/s) -4 2 R = 8.85 x 10 m /s -4 3 FLUID MECHANICS

Presented by Group 3 Ronnel Gono James Mejia Florante Marasigan Redzl No Neil Francis Torio