Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów Problems 1 Lecture 1 1)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 Problems 2 Lecture 2 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. 2)Find the hydrostatic pressure distribution in a fluid whose density decreases with height: a) linearly b) exponentially

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów Problems 3 Lectures 3-4 1) 2)

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów Problems 4 Lectures 5-6 1)

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów Problems 5 Lectures 5-6 2)

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów Problems 6 Lectures 5-6 3)

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów Problems 7 Lectures 5-6 4)

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów Problems 8 Lecture 7 1) 2)

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów Problems 9 Lecture 10 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 Problems 10 Lecture 11 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 = 10 3 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 Stokes equation (with gravity)

Wstęp do Fizyki Środowiska - Podstawy mechaniki płynów Problems 11 Lecture 12 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 of Soil Physics: