3 Hydrostatic pressure P = ρgh ρ=fluid/gas density g=acceleration due to gravityh=heightP = ρghPressure in liquid/gas is isotropic. It acts equally in all directionsPressure is force per unit areaDue 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
8 Bernoulli’s principle For a non-turbulent flow of fluid or gasAs speed increases, the pressure in the fluid or gas decreases.
9 Bernoulli’s equation P + ½ ρv2+ ρgh = const P=pressure of the fluid/gas along the streamlinev=velocity of the fluid/gas along the streamlineg=acceleration due to gravityh=heightρ=fluid/gas densityThe Bernoulli’s equation expresses conservation of enegy. It assumes that:The fluid/gas has a constant densityThe fluid/gas is traveling in a steady flowThere is no frictionThe fluid/gas is non viscous and incompressable
12 Derivation of Bernoulli Distance lAccelerationa-aAcceleration in the non-inertial frame moving with the flowBecause 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.
13 Derivation of Bernoulli Distance lAccelerationa-aAcceleration in the non-inertial frame moving with the flowThe equivalence principle:In an accelerated reference frame moving with the flow we can calculate the pressure difference as if it were a pressuredifference in a gravitational field, 𝚫P = P2 - P1 = ρ a l
16 Inertial force and the equivalence principle 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.
17 Derivation of Bernoulli Distance lAccelerationa-aAcceleration in the non-inertial frame moving with the flowKinematics 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.
18 Derivation of Bernoulli Distance lAccelerationa-aAcceleration in the non-inertial frame moving with the flowCombining the two results gives the Bernoulli equation,𝚫P = P2 - P1 = ρ a l = ρ (v22 - v12 )/2
28 Ships passing on parallel course 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
31 Airfoil lift schematics 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.
33 The Magnus effectWhere the cylinder is turning into the airflow, the air is moving faster and the pressure is lowerWhere the cylinder is turning away from the airflow, the air is moving slower and the pressure is greaterThe cylinder moves towards the low pressure zone
34 Curveballs The Magnus effect! stitches help the ball to catch the air the baseball curves towards the lower air pressure
35 Physics of golf: dimples on the ball and the Magnus effect Typical ball spin-rates are:3,600 rpm when hit with a 10Â° driver (8Â° launch angle) at a velocity of 134 mph7,200 rpm when hit with a 5 iron (23Â° launch angle) at a velocity of 105 mph10,800 rpm when hit with a 9 iron (45Â° launch angle) at a velocity of 90 mphDimples 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.