1. Consider an irrotational and inviscid flow of water created using a superposition of a rankine oval and uniform flow. The sink-source pair of the oval is on the x–axis, with the source on the left (c = ?1.5 m) and the sink on the right (c = +1.5 m). The strength of the sink and source is E = 40 m2/s.

(a) Pick an arbitrary value for the magnitude of the uniform flow’s velocity. Then, derive and plot the contours for the stream and potential functions of this uniform flow as it flows over the Rankine oval.

(b) Derive the velocity vectors (u and v). Then, find the velocity of the uniform flow to yield a rankine oval with a total length 2l = 10 m.

(c) Assume that the upstream pressure is zero. Calculate and plot the pressure distribution on the surface of the rankine oval, for the uniform flow velocity found in part

(b).

2. A Newtonian fluid flows steadily through a channel (see figure below). The channel extends infinitely in the x1–direction and x3–direction (normal to the x1 and x2 plane). The distance between the channel walls is h. The upper plate is held at a higher temperature (TU) than the lower plate, which has a temperature of TL. The linear temperature distribution in the gap is described by the following equation:

,

where: ?T = TU ? TL.

The viscosity µ is a function of temperature T and is described as:

µ(T) = µLe??(T?TL),

where: µL is the viscosity at TL, and ? is a constant.

The pressure gradient in the x1–direction is constant and is described as , while

= 0. Also, the velocity and the fluid density ? depend only on x2. Finally, assume that body forces and the dissipation of heat from the plate are negligible.

Channel with plane flow.

(a) Provide the boundary conditions for the velocity components.

(b) Using the continuity equation and the boundary conditions for this viscous flow, calculate the distribution of the velocity component in the x2–direction: u2.

(c) Using the Cauchy-Poisson law (Lecture 06, Slide 13), determine the components of the stress tensor and write your answer in matrix form.

(d) Compute the distribution T12(y), from Cauchy-Poisson’s law, to obtain the shear stress in the gap up to a constant.

(e) Compute the velocity field u1(x2) by using the results from part (b) and (c), along with the no-slip boundary condition. Start with the Navier-Stokes equation described in Lecture 06 (Slide 19).

(f) Using the results from part (e), plot the velocity distribution for the following cases and state which K will yield no flow:

1. K = 0

2. K >0

3. K <0 3. Wind strikes the side of a simple residential structure and is deflected up over the top of the structure, as depicted by the figure below. Use the control volume shown in the figure to answer this problem. Assume the following: (i) two-dimensional steady inviscid constant-density flow, (ii) uniform upstream velocity profile, (iii) linear velocity gradient profile at the downstream flow location (velocity is U at the upper boundary and the velocity is zero at the lower boundary (see figure below), (iv) there is no flow through the upper boundary of the control volume, and (v) there is constant pressure on the upper boundary of the control volume. Residential structure. (a) Determine h2 in terms of U and h1. (b) Determine the direction and magnitude of the horizontal force on the house per unit depth into the page, in terms of the fluid density ?, the upstream velocity U, and the height of the house h1. (c) Evaluate the magnitude of the force (from part b) for a house that is 10 m tall and 20 m long in wind of 27 m/s (approximately 60 miles per hour). (d) Assume that the length of the house could be approximated as a flat plate, and that the air flow is viscous with ?air= 1.5 × 10?5. Could you use Blasius’ solution to approximate the boundary layer thickness, using the details of part (c)? Justify your answer.