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CE 576 Environmental Flows Spring 2012 Homework 3 13. 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 = ∂u/∂x + ∂v/∂y + ∂w/∂z) represents? c. What does the discussion in part b imply for ﬂuid elements in an incompressible ﬂow? u(x) time = t + Δ t time = t *14. The x-momentum equation in three dimensions is ρ ∂u ∂u ∂u ∂u +u +v +w ∂t ∂x ∂y ∂z = ∂τyx ∂τzx ∂τxx + + . ∂x ∂y ∂z Use the constitutive relations for the stresses and show that for an incompressible ﬂow: 2 ∂u ∂u ∂u ∂p ∂ u ∂ 2u ∂ 2u ∂u + 2 + 2 . +u +v +w =− +μ ρ ∂t ∂x ∂y ∂z ∂x ∂x2 ∂y ∂z *15. Reconsider the ﬂow (u, v) = (Γx, −Γy), which we considered in class. a. Compute the pressure if p = p0 at x = y = 0 and x and y are horizontal coordinates. b. Sketch isobars (lines of constant pressure) along with streamlines. c. Does your sketch make sense? Does the ﬂow go from high to low pressure? d. Compute the entire stress tensor (i.e., τxx , τxy , τyx , τyy ). 16. Suppose the viscous slot ﬂow is started from rest. Let T be the time since the ﬂow started. When can we consider the ﬂow to be steady? That is, use scaling to derive a condition for the ﬂow to be steady. 17. Watch the movie Low Reynolds Number Flows at the website below and answer the following: http://modular.mit.edu:8080/ramgen/iﬂuids/Low Reynolds Number Flow.rm (If that link does not work, try http://web.mit.edu/hml/ncfmf.html.) a. Explain the similarities between the ﬂow around bull spermatazoa 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? *18. 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 xdirection 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 = ρgh sin α. d. Explain why the result in part c is valid for laminar and turbulent ﬂow. α g h 1111111111111111111111111 0000000000000000000000000 τ0 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 z 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 x *19. We considered slow settling of a single spherical particle in a quiescent water body. However, that situation is likely to be rare in environmental ﬂows. a. Find the paper Cuthbertson and Ervine (2007, J. Hydraulic Engineering). b. Use ﬁgure 5 to explain how well Stokes’s law describes settling in open channel ﬂow. c. Critique the ﬁgure. Can you suggest ways to improve the presentation?