LECTURE 4 FLUID FLOW & BERNOULLI Lecture Instructor: Kazumi Tolich

Lecture 4

¨  Reading chapter 12-1 to 12-3 ¤ Continuity equation ¤ Bernoulli’s equation

n Torricelli’s law

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Equation of continuity ¨  Any amount of fluid passing point 1 in a given time, Δt, must also flow past

point 2 in the same time.

¨  For incompressible fluids, density, ρ, is constant throughout the fluid.

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Δm1 = Δm2 → ρA1v1Δt = ρA2v2Δt

A1v1 = A2v2 Equation of continuity for an incompressible fluid

ρ ρ

Example 1 ¨  A garden hose having an internal

diameter of d1 = 0.75 inches is connected to a stationary lawn sprinkler that consists of an enclosure with 24 holes, each d2 = 0.050 inches in diameter. If the water in the hose has a speed of v1 = 3.0 ft/s, at what speed does it leave the sprinkler holes?

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Bernoulli’s equation ¨  Bernoulli’s equation for a fluid whose speed and height changes.

P is pressure, ρ is the fluid density, g is acceleration due to gravity, y is the height, and v is velocity along the streamline.

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P1 + ρgy1 + 12 ρv1

2 = P2 + ρgy2 + 12 ρv2

2 P + ρgy+ 12 ρv

2 = constant

Bernoulli’s equation conditions ¨  Incompressible

¨  Non-viscous ¤  Fluid does not resist against flowing.

¨  Steady streamlined flow ¤  The particles of the fluid move along straight or smoothly curved paths

that do not intersect.

¨  These conditions generally do not apply. However, Bernoulli’s equation may be a good starting point, and useful for describing the general behavior.

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Clicker question: 1 7

Arterial flutter ¨  If an artery is narrowed due to buildup of plaque on its inner

wall, the blood through the constriction is faster, and the pressure lower. The lower pressure causes the soft arterial walls to collapse until enough pressure is built up to reopen it. This process repeats itself.

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Example: 2 ¨  Water flows through a horizontal tube

of diameter D1 = 2.8 cm that is joined to a second horizontal tube of diameter D2 = 1.6 cm. The pressure difference between the tubes is ΔP = 7.5 kPa. a)  Which tube has the higher pressure?

b)  Which tube has the higher speed of flow?

c)  Find the speed of flow in the first tube, v1.

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Demo: 1

¨  Venturi tubes ¤ Demonstration of Bernoulli’s equation.

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P + 12 ρv

2 = constant

Clicker question: 2 11

Common pitfalls

¨  The speed of fluid flow changes due to a difference in pressure from one point to another. A constant fluid flow does not cause reduced pressure.

¨  Bernoulli’s equation applies only to one system, not a comparison of two systems. It compares pressure within the same streamline.

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Misapplications of Bernoulli’s equation

¨  There are many misapplications of Bernoulli’s equation.

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Torricelli’s law ¨  Torricelli's law states that

For an open tank, the speed of the liquid leaving a hole a distance h below the surface is equal to that acquired by an object falling freely through a distance h.

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v22 = v1

2 + 2gh

P1 + ρgy1 + 12 ρv1

2 = P2 + ρgy2 + 12 ρv2

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Pat + ρgh+ 12 ρv1

2 = Pat + ρg0+ 12 ρv2

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Example: 3 ¨  A cylinder that is H = 20 cm tall is

filled with water. If a hole is made in the side of the cylinder, h = 5 cm below the top level, how far from the base of the cylinder will the stream land? Assume that the cylinder is large enough so that the level of the water in the cylinder does not drop significantly.

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