MEC 364: Homework #7

Due date: 11/15/12

Note: this homework counts double. To receive credit for a problem you must show your work in great detail

1. Consider the velocity field V = Axyi - (1/2)Ay2j in the xy-plane, where A = 0.25 m-1s-1, and the coordinates are measured in meters. Is this a possible incompressible flow field? Calculate the acceleration of a fluid particle at point (x, y) = (2, 1). ANS: a = (0.0625i+ 0.03125j) m/s2. 2. The variation in cross-sectional area with distance along a diffuser may be expressed as A = A1eax, where A1 is the cross-sectional area at the diffuser inlet. Assume that the flow is incompressible and uniform at any cross section. Sketch the variations of the area and velocity along the diffuser as functions of x. Develop an algebraic expression for the acceleration of a fluid particle in the diffuser in terms of V1 and x. ANS: ax = -a(V1)2e-2ax. 3. Incompressible liquid of negligible viscosity is pumped at total volume flow rate Q through two small holes into the narrow gap between closely spaced circular parallel plates. The spacing between the plates is h. The liquid flowing away from the holes has only radial motion. Assume uniform flow across any vertical section and discharge to atmospheric pressure at r = R. (a) show that the radial velocity in the narrow gap is Vr = Q/(2πrh). (b) Derive an expression for the acceleration of a fluid particle in the gap. ANS: a = -[Q/(2πh)]2(1/r3)er. 4. A steady, two-dimensional velocity field is given by V = Axi – Ayj, where A = 1 s-1. Show that the streamlines for this flow are rectangular hyperbolas, xy = C. Plot streamlines that correspond to C = 0, 1, and 2 m2. Obtain a general expression for the acceleration of a fluid particle in this velocity field. Calculate the acceleration of fluid particles at the points (x, y) = (0.5, 2), (1, 1), and (2, 0.5), where x and y are measured in meters. Show the accel-eration vectors on the streamline plot. ANS: a = (0.5i + 2j) m/s2; a = (i + j) m/s2, a = (2i + 0.5j) m/s2. 5. A velocity field is represented by the expression V = (Ax –B)i + Cyj + Dtk, where A = 2 s-1, B = 4 m s-1, and D = 5 m s-2. Determine the proper value for C if the flow field is to be incompressible. Calculate the acceleration of a fluid particle located at the point (x, y) = (3, 2). Sketch the flow streamlines in the xy-plane. ANS: C = -2 s-1, a = (4i + 8j + 5k) m/s2. 6. A flow is represented by the velocity field V = 10xi – 10yj +30k. Determine if the flow field is (a) a possible incompressible flow, and (b) irrotational. 7. Consider the two-dimensional incompressible flow field in which u = Axy and v = By2, where A = 1 m-1s-1 and B = -0.5 m-1s-1. Determine the rotation at point (x, y) = (1,1). Evaluate the circulation about the “curve” bounded by y = 0, x = 1, y =1, and x = 0. ANS: w = -0.5k rad s-1, Γ = -0.5 m2 s-1. 8. Consider the pressure-driven flow between stationary parallel plates separated by distance b. Coordinate y is measured from the bottom plate. The velocity field is given by u = U(y/b)[1-(y/b)]. Obtain an expression for the circulation about a closed curve of height h and length L. Evaluate when h = b/2 and when h = b. Show that the same result is obtained from the area integral of the Stokes theorem (Eq. 5.18 in reference book by Fox et al.). ANS: Γh = b/2 = -UL/4, Γh = b = 0. 9. Consider the pressure-driven flow between stationary parallel plates separated by distance 2b. Coordinate y is measured from the channel centerline. The velocity field is given u = umax[1-(y/b)2]. Evaluate the rate of linear and angular deformation. Obtain an expression for the vorticity vector ζ . Find the location where the vorticity is maximal. ANS: (dγ/dt) = -2umaxy/b2, ζ = 2umaxy/b2k.

[Notation: Bold represents a vector quantity]