Question 1: In class we showed that the rate of angular strain is related to velocity gradients like ∂u/∂y and ∂v/∂x-that is, derivatives in a direction normal to the velocity component. In this problem you will examine the role of derivatives in the same direction as the velocity component.
a. Consider a fluid element in a flow u(x). Show that the rate of linear strain is ∂u/∂x.
b. What do you think the divergence of the velocity (i.e., ∇ • u) represents?
c. What does the discussion in part b imply for fluid elements in an incompressible flow?
Question 2: Explain why the pressure gradient-not the pressure-appears in the Navier Stokes equations.
Question 3: Consider the flow (u, v) = (Γx,-Γy). Assume that the fluid has a dynamic viscosity μ. If you have not done problem 6, do parts a and b; otherwise, just refer to problem 6 for those parts.
a. Sketch streamlines, which are lines tangent to the velocity vector. (You can compute and plot the streamlines, but I want you simply to sketch them.)
b. What might this flow represent? In other words, if your sketch in part a revealed sinusoidal streamlines, you might say that the flow represents flow over sand ripples.
c. Compute the pressure if p = p0 at x = y = 0. (See "Example of computing stressand pressure" under "Handouts" in Blackboard.)
d. Sketch isobars (lines of constant pressure) along with streamlines.
Question 4: Watch the movie Low Reynolds Number Flows at the website and answer the following:
a. Explain the similarities between the flow around bull spermatozoa and the flow of glaciers.
b. Explain hydrodynamic lubrication.
c. Suppose a sphere of diameter 3 mm falls slowly in a viscous fluid. How much faster will it fall than a sphere (of the same material) of diameter 2 mm
d. Explain why waving a rigid rudder does not propel a body in highly viscous fluid.
e. What was your favorite part of the movie? Why?
Question 5: Consider steady, uniform, two-dimensional, incompressible, laminar flow down a solid, impermeable plane inclined at an angle α.
a. Solve for the velocity u parallel to the plane. (Hint: Remember that the x direction will have a component of gravity, and assume that the shear stress (e.g., from wind) is zero at the air-fluid interface at z = h.)
b. Give at least two reasons why your expression for u is plausible.
c. Show in two ways that the shear stress on the bottom is τ0 = ρghsinα andcompute the friction factor.
d. Explain why the result in part c is valid for laminar and turbulent flow.
Question 6: In class we discussed the connection between the viscous slot flow and groundwater. Evaluate whether the two main assumptions of the viscous slot flow solution hold for the flow of groundwater through sand of average grain diameter 1mm with a gradient of piezometric head of 0.01. In particular, estimate the length of the pore needed for the assumptions to hold.
Question 7: (From 2013 midterm) A viscometer, or a device to measure the viscosity of a fluid, is designed so that a long, wide plate is drawn steadily through a viscous fluid at speed U0. The device is constructed so that the plate can held at a constant distance h1 from the upper wall and a constant distance h2 from the lower wall. The force F required to draw the plate through the fluid is measured, and fluid is pumped from one end of the device to the other.
a. Assume that the conditions for a two-dimensional viscous slot flow apply to this problem and derive an expression for the viscosity in terms of the force F, the plate's width B, the length L, the speed U0, and the distances h1 and h2. Assume the pressure gradient to be zero and consider only the force of the fluid on the plate.
b. If h1 = 2 mm, h2 = 4 mm, L = 300 mm, B = 300mm and U0 = 3mm/s, and the fluid's density and dynamic viscosity are 800 kg/m3 and 2 Pa.s, respectively, do the assumptions behind the viscous slot flow apply in this problem?
c. Estimate how long one must wait after the plate starts moving for the flow to be steady.
Question 8: A cylinder of radius R1 is pulled along the centreline of a cylinder of radius R2. A pressure gradient dp/dx is also applied.
a. State the assumptions for this problem to be treated as a viscous slot flow.
b. Compute the velocity profile.
c. Derive an expression for the drag (per unit length) on the inner cylinder.
Question 9: The settling velocity of E. coli was specified. In general, one would need to measure or estimate the settling velocity. Of course, settling of sediment in streams is more complicated than settling of uniform particles in a quiescent water body.
a. Compute the diameter of a sand particle corresponding to a settling velocity of 5× 10-3 m/s [75 microns]. Compare this value to the diameter of 10 microns for a large clay or small silt particle.
b. In class we used scaling to estimate the time for a particle released from rest to reach its terminal settling velocity. Compute that time more precisely by solving a more complete version of the force balance that includes the added mass effect:
(s + Cm)dws/dt= (s - 1)g - FD/ρVs
Where Vs is the volume of the sediment particle and Cm is the added mass coefficient, which equals 1 for a sphere. When does a 10-micron particle reach 99% of its terminal settling velocity? [9 × 10-5 s] Is that time large or small?
c. Streams are turbulent, and the velocity varies from zero at the bed to a maximum at the water surface. Find a journal article (using Web of Science, for example) by Cuthbert son and Ervine on settling in open channel flow, and explain whether particles in a stream would settle faster or slower than predicted by Stokes's law.
d. Bacteria can flocculate, or form assemblages (flocs) that consist of sediment grains, water, bacteria, and extracellular polymeric substances. Use the work of Droppo or others to explain whether Stokes's law accurately predicts the settling velocity of flocs.
e. Sediment in streams is characterized by a grain size distribution. That is, many diameters are present. Explain briefly how you might compute the settling velocity in that case.
f. Propose two more reasons why settling velocities for particles in streams might deviate from Stokes's law.