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Fluid Dynamics

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Floating An object floats on a fluid if its density is less than that of the fluid When floating F B = F W ρ f V disp g = ρ o V o g ρ f V disp = ρ o V o Fraction of an object submerged is given by the ratio of the density of the object to the fluid

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Example A rectangular block of wood (1.0m x 2.0m x 0.25m) floats so that 0.15 cm of its height is above the water level. a) find the density of the wood b) find the mass of the wood

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Example A continental block (ρ = 2800 kg/m 3 ) floats on the mantle (ρ = 3300 kg/m 3 ). Assuming the continent is 35km thick, estimate the height of the continent floating above the mantle

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Fluid Dynamics Describes motion of fluids Two types of fluid flow - streamline or laminar flow – smooth - turbulent flow – erratic small circles

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Streamline or Laminar Flow

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Turbulent Flow

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Flow Rate Mass of a fluid that passes a given point per unit of time Equation of continuity A 1 v 1 = A 2 v 2

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Example Water flows through a pipe of diameter of 3.0 cm at a flow speed of 6.0 m/s. The pipe expands to a diameter of 9.0 cm, what is the new flow speed?

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Example The radius of the aorta is about 1.0cm and the blood passing through it has a speed of about 30cm/s. A typical capillary has a radius of about 4 x10 -4 cm, and blood flows through it at a speed of about 5 x 10 -4 m/s. Estimate the number of capillaries in the body.

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Bernoulli’s Principle Where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high

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Bernoulli’s Equation P 1 + ½ρv 1 2 + ρgy 1 = P 2 + ½ρv 2 2 + ρgy 2

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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 0.50m/s through a 4.0cm diameter pipe in the basement under a pressure of 300 kPa, what will the flow speed and pressure be in a 2.6cm diameter pipe on the second floor 5.0m above?

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Applications A large tank, 25m in height and open at the tope is completely filled with water. An opening 5m from the bottom allows water to flow out. What velocity will the water leave the tank?

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