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# Lecture 1 - PowerPoint PPT Presentation

Lecture 1.

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• In a vertical capillary filled with water air bubbles are rising Sketch the diagram of the total mass contained in the control volume as a function of time Sketch the diagram of the flux of mass and volume through the top surface of the control volume as a function of time

water

air

Click on the picture to see the bubbles move

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

1) Ellipsoid of revolution is given by the equation where is vertical coordinate, and are coordinates in the horizontal plane Calculate the coordinates of the lowest point. The ellipsoind is filled with fluid ehose density at the lowest point is equal and decreases linearly with height Calculate the total mass of fluid contained in the ellipsoid For any level calculate the mass of the fluid below this level.

• Find the hydrostatic pressure distribution in a fluid whose density decreases with height:

• a) linearly

• b) exponentially

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

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2)

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

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Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

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Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

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Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

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Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

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Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

1)

Consider a cylindrical capillary with circular cross-section

with radius R and length L. Let it be closed at one end. How high does water

rise if the material is perfectly water-wet and if air can be described as an

ideal gas? Does this height depend on temperature?

2) Two Grains with Water

Consider the figure and assume that the two

grains are fixed in space. Discuss the shape of the air-water interface and

explain if it is consistent with the Young-Laplace equation? What

force acts on one of the grains and what movement will it perform when it is

free to move?

3) Cohesion of Sand

Walking along a sandy beach, you notice that at a

certain distance from the water, there extends a strip where you can walk

easily without sinking in. Nearer to water as well as farther away from it,

this is not the case anymore and walking is dicult. Explain!

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

1)

Water Infiltrating in Soil

Consider a heavy rainfall event that has

10 mm of water infiltrate a sandy soil within 1 hour. As characteristic time for

the external forcing choose τ = 103 s and as characteristic length choose the

radius of a large pore, . = 100 μm. To estimate a typical velocity, assume that

the infiltrating water passes through 1/3 of the volume

Calculate the Strouhal number, the Froude number and the Reynolds number.Show that the momentum equation can be well approximatet by the Stokesequation (with gravity)

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów

1) Units of Conductivity

Darcy's law may be written in terms of the hydraulic potential, or in terms of the hydraulic head,

For simplicity, the same symbol is often used for the hydraulic

conductivity K. What are its units in the two formulations?

2)

Hint: read chapter 3.2.5 of Soil Physics:

http://www.iup.uni-heidelberg.de/institut/forschung/groups/ts/students/lecture_notes05/lecture_notes05/sp-v03-ch3.pdf

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów