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Bernoulli’s law and Magnus force Blaise Pascal P = ρgh ρ=fluid/gas density g=acceleration due to gravity h=height P = ρgh • Pressure in liquid/gas is isotropic. It acts equally in all directions • Pressure is force per unit area • Due to the gravity, pressure at a given level equals to the weight of the column of liquid/gas above this level over a unit area P = ρgh • For a non-turbulent flow of fluid or gas • As speed increases, the pressure in the fluid or gas decreases. P + ½ ρv2+ ρgh = const P=pressure of the fluid/gas along the streamline v=velocity of the fluid/gas along the streamline g=acceleration due to gravity h=height ρ=fluid/gas density The Bernoulli’s equation expresses conservation of enegy. It assumes that: The fluid/gas has a constant density The fluid/gas is traveling in a steady flow There is no friction The fluid/gas is non viscous and incompressable Distance l Acceleration a -a Acceleration in the non-inertial frame moving with the flow Because velocity of the fluid/gas flow has changed (increased) from v1 to v2 , there must be a force which causes it to accelerate while passing the distance l. For simplicity, let us assume constant acceleration a. Distance l Acceleration a -a Acceleration in the non-inertial frame moving with the flow The equivalence principle: In an accelerated reference frame moving with the flow we can calculate the pressure difference as if it were a pressure difference in a gravitational field, P = P2 - P1 = ρ a l The inertial mass relates force and acceleration in the Newton’s first law of motion: F = ma. The gravitational mass determines force of gravitational attraction in the Newton’s law of gravity: (= mg). The inertial mass and the gravitational mass are equal. Distance l Acceleration a -a Acceleration in the non-inertial frame moving with the flow Kinematics of motion with constant acceleration, a, gives, v2 = v1 + at, l = v1t + ½ at2 = (v22 - v12 ) /(2a) where t is the time it took the flow to pass the distance l. Distance l Acceleration a -a Acceleration in the non-inertial frame moving with the flow Combining the two results gives the Bernoulli equation, P = P2 - P1 = ρ a l = ρ (v22 - v12 )/2 Patm Patm ρ v2/2+Patm= ρgh+Patm => v2 = 2gh Ventouri detergent intake system in a powerwasher Ventouri wine aerator Dental Saliva Ejector Hose With Water Venturi Suction System Becomes important for wind velocity v > √2gh (≈ 10 m/s for h ≈ 5 m). Ships sailing side by side can get too close together (as in picture above, at a certain point during the refueling). When this happens, the Venturi effect takes over, and the ships will head toward an unavoidable collision An airfoil creates a region of high pressure air below the wing, and a low pressure region above it. The air leaving the wing has a downward flow creating the Newtonian force. Bernoulli pressure field creates the downwash. Where the cylinder is turning into the airflow, the air is moving faster and the pressure is lower Where the cylinder is turning away from the airflow, the air is moving slower and the pressure is greater The cylinder moves towards the low pressure zone The Magnus effect! • stitches help the ball to catch the air • the baseball curves towards the lower air pressure Typical ball spin-rates are: 3,600 rpm when hit with a 10Â° driver (8Â° launch angle) at a velocity of 134 mph 7,200 rpm when hit with a 5 iron (23Â° launch angle) at a velocity of 105 mph 10,800 rpm when hit with a 9 iron (45Â° launch angle) at a velocity of 90 mph Dimples cause the air-flow above the ball to travel faster and thus the pressure on the ball from the top to be lower than the air pressure below the ball. This pressure difference (i.e. more relative pressure from below than on top) causes the ball to lift (Magnus effect) and stay in the air for a longer time. Topping the ball (i.e. when the bottom of the club-face hits the ball above its center) will cause the ball to spin in the other direction - i.e. downward - which will cause the ball to dive into the ground. 1927