 Solution

ID: past-exam-of-the-topics-in-fluid-mechanics-course-of-the-university-of-oxford/2023/1/b/i/solution

z

directed from the solid into the liquid. The downslope component of gravity is unchanged, but its normal component is now positive. Thus the hydrostatic pressure changes from

(h - z) cot α

- (h - z) cot α

. The resulting Benney equation is

h_{t} + 2 h^{2} h_{x} + ε \partial_{x} [h^{3} (+ \frac{2}{3} h_{x} cot α + S h_{xxx}) + \frac{8}{15} Re h^{6} h_{x}] = 0.

Only the normal-gravity contribution changes sign; the surface tension and inertial contributions retain their signs.

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