LECTURE 4FLUID FLOW & SURFACE TENSION

Lecture Instructor: Kazumi Tolich

Lecture 4

¨ Reading chapter 15.6 to 15.9¤ Continuity equation¤ Bernoulli’s equation

n Torricelli’s law

¤ Viscosity¤ Surface tension

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Equation of continuity

¨ For incompressible fluids, density is constant throughout the fluid, and any amount of fluid passing point 1 in a given time, must also flow past point 2 in the same time.

¨ Equation of continuity for an incompressible fluid is given by

𝐴"𝑣" = 𝐴%𝑣%

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Bernoulli’s equation¨ Bernoulli’s equation for an incompressible and non-viscous fluid in a steady

streamline flow:

𝑃" + 𝜌𝑔𝑦" +12𝜌𝑣"

% = 𝑃% + 𝜌𝑔𝑦% +12𝜌𝑣%

%      or 𝑃 + 𝜌𝑔𝑦 +12𝜌𝑣

% = constant

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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Example: 1

¨ Water flows through a horizontal tube of diameter 𝐷" = 2.8 cm that is joined to a second horizontal tube of diameter 𝐷% = 1.6 cm. The pressure difference between the tubes is ∆𝑃 = 7.5 kPa. Find the speed of flow in the first tube, 𝑣".

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Quiz: 1

¨ Water is discharged from the tank in the manner shown. At which point or points is the pressure the least? A. 1

B. 2

C. 3

D. 4

E. 1 & 2

F. 3 & 4

G. 1 & 4

H. 2 & 3

I. The pressure is the same at all points.

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Quiz: 4-1 answer

¨ 2¨ The higher the position, the smaller the pressure.

¨ The faster the flow (the narrower the pipe), the smaller the pressure.

¨ 𝑃 + 𝜌𝑔𝑦 + "%𝜌𝑣% = constant

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Common pitfalls & misapplications of Bernoulli’s equation

¨ 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.

¨ There are many misapplications of Bernoulli’s equation.

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Torricelli’s law

¨ Torricelli's law can be derived from Bernoulli’s equation, and it 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.

𝑣%% = 2𝑔ℎ

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Viscosity & viscous flow

¨ Viscosity, fluid friction or thickness, slows the flow along surfaces, and causes the pressure to drop in the length of the tube.

𝑃" − 𝑃% = 8𝜋𝜂𝑣𝐿𝐴

where 𝜂 is the coefficient of viscosity of the fluid.

¨ The volume flow rate (Poiseuille’s equation) is given by

∆𝑉∆𝑡 = 𝐴𝑣 =

𝑃" − 𝑃% 𝜋𝑟?

8𝜂𝐿

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Quiz: 2

¨ When paddling a canoe upstream, where should you travel to move efficiently?A. Close to the shoreB. Close to the middle of the streamC. It makes no difference where you are.

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Quiz: 4-2 answer

¨ Close to the shore

¨ Due to the viscosity of water, the water’s velocity is the greatest near the middle of the stream, and smallest near the shore.

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Quiz: 3

¨ A liquid, such as water, with a low but not negligible viscosityis flowing from A to C. The cross-sectional area at A is equal to that at C, and the cross-sectional area at B is significantly smaller. Rank the pressures at different positions at A, B and C, denoted by 𝑃@, 𝑃A, and 𝑃B, respectively, smallest first.

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Quiz: 4-3 answer

¨ 𝑃A < 𝑃B < 𝑃@¨ Due to the viscosity of water, the pressure drops as it flows in a pipe. So, 𝑃B < 𝑃@.

¨ The pressure in the pipe that is significantly narrower (region B), the water has to flow much faster, so the pressure in region B is much less than in region A or C.

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Demo: 1

¨ Venturi tubes

¤ Demonstration of n Bernoulli’s equation: 𝑃 + 𝜌𝑔𝑦 + "

%𝜌𝑣% = constant

n Viscous flow

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Example: 2

¨ Water at 20 °C flows through a horizontal hose at the rate of ∆𝑉 ∆𝑡⁄ = 5.0  ×  10I? m3/s. The diameter of the garden hose is 𝐷 = 2.5 cm.

a) What is the average water speed in the hose?

b) What is the pressure drop across a length of 𝐿 = 15 m in the hose?

c) Suppose that the cross-sectional area of the hose is halved, but the length and the pressure drop remain the same. By what factor does the volume flow rate change? By what factor does the average water speed change?

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Surface tension17

¨ Surface tension is caused by molecular attractions.

¨ This tendency to pull surface molecules into the liquid causes the surface to become as small as possible.

¨ Rain drops are nearly spherical, not tear-drop shaped!

¨ Water resists the distortion of on the surface when an insect is on it with a force sufficient to support the weight of the insect.

Demo: 2

¨ Soap film pull up¤ Demonstration of surface tension of water

¨ Surface tension disk¤ Measurement of force needed to overcome surface tension

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Quiz: 4

¨ Ducks have greater density than water, but they can sit on water. Can they sit on soapy water? The densities of pure and soapy water are basically the same.A. Yes.B. No.

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Quiz: 4-4 answer

¨ No.

¨ A duck’s feather has a crisscross grid of barbules that form a mesh.

¨ The water surface has to stretch to push through this mesh, but surface tension in pure water resists such stretching.

¨ So pure water does not penetrate the mesh, and duck’s skin remains dry.

¨ Soapy water does not have nearly as much surface tension as pure water.

¨ With much reduced water tension, ducks will sink because their density is greater than the density of water.

¨ Soapy water is used for laundry so that water can penetrate the closely spaced fibers of clothing.

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