Consider steady flow in the two-dimensional channel in Fig. P5.2, in which the half-height varies as

Assume that q = Q⁄W is known, where Q is the volume flow rate and W is the channel width.

(a) Relate u(x), the mean velocity in the x direction, to q and h(x).

(b) If the height variations are gradual enough that 2πεh₀⁄L ≪ 1, it can be shown using the methods of Section 8.2 that

Derive an expression for v_y(x, y) in terms of q and h(x). Note that there is reflective symmetry about the plane y = 0.

Transcribed Image Text:

27.*). L h(x) = ho (1+ ε sin (₁. (P5.2-1)