Problem 1:

A thin capillary of internal radius r_o = 0.5 mm and total length L = 5 cm is initially completely filled with water at 20 °C (ρ = 1000 kg/m, γ = 73 mN/m). Slowly, both ends of the capillary are open to the atmosphere, after which the liquids flows down to form a spherical drop of radius R that remains attached to the end of the capillary when the system achieves equilibrium, as shown in the figure (no water is spilled during the process). Assuming that h » R, h » r_o, and knowing that the contact angle of the liquid with the wall at the meniscus is Φ = 0, calculate h and R.

Problem 2:

The parallel plate viscometer consists of two parallel disks of radius R between which a liquid is placed (see figure). The upper disk rotates at a constant angular velocity ω, while the bottom disk is stationary

1. For low rotational speeds (creeping flow), we cm postulate a velocity field as follows:

v_r = v_z = 0, v_θ = rf(z). Find f(z).

2. Determine the torque necessary to keep the upper disk rotating.

Problem 3:

Consider the steady, incompressible flow of a liquid in a horizontal pipe (radius R) whose walls are porous, which allows some liquid to leak through the pipe wall into the surroundings. In this case, there will be a radi al component of the velocity inside the pipe, v_r(r,z). It is known that the mass transfer flux through the wall can be quantified by

v_r (R, z) K[P(z) - P_a]

where K is the permeability of the pipe wall (constant), P(z) is the pressure in the pipe (assumed to depend only on z), and Pa is the pressure of the surroundings (constant).

If the permeability is low enough, it is proposed that the axial component of the liquid velocity in the pipe will obey the solution corresponding to an impermeable wal, but using an average vel oci ty that depends on z, U (z) , i .e.

v_z(r, z) = 2U(z)[1 - (r/R)²]   -(2)

1. Use the continuity equation to express v_r(r,z) in terms of U.

2. From the solution of part 1 and equation (1), find Pin terms of U.

3. The solution proposed (equation 2) will be valid only if it is consistent with z- component of the Navier-Stokes equations. Substitute equation (2) and the solutions found in pats 1 and 2 into the z-component of the Navier-Stokes equai on, neglecting inertial terms, to find an ordinary differentia equation for U(z). Note that there is a mathematical contradiction because this equation contains r (which would mean that U cannot be an exclusive function of z, as assumed.) However, show that, if 4K μ/R << 1, the equation can be simplified to remove the contradiction.

4. Solve the simplified equation found in pat 3 to find U(z), using U(0)=Uo. Consider also that the pipe is so long that eventually al the flowing fluid will leak.

Problem 4:

A long cylindrical rod of radius R is positioned vertically in a lage container of liquid and rotated at an angular velocity ω (see figure). Far from the rod the liquid-air interface is at alevel z=0 and the liquid is stagnant.

1. Determine the velocity field in the liquid, v_θ(r).

2. Neglecting surface tendon effects, determine the height of the air-liquid interface as a function of r, z_s(r), r>R.