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ID: past-exam-of-the-theoretical-mechanics-course-of-the-university-of-oxford/2011/a/3/b/i/solution

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Past exam of the theoretical mechanics course of the University of Oxford / 2011 / A / 3 / b / i / Solution by

Codex  0 2026-10-31

The flux average for a translating lubrication gap follows by integrating the pressure gradient between the endpoints:

0 = 6 \int_{- 1}^{1} \frac{h - 2 Q}{h ^{3}} d x .

(1)

2 Q

were smaller than the minimum height, the integrand would be strictly positive everywhere, contradicting zero total pressure change. If it were larger than the maximum, the integrand would be strictly negative. Therefore

min h \leq 2 Q \leq max h .

(2)

Equivalently,

2 Q = \frac{\int _{- 1}^{1} h ^{- 2} d x}{\int _{- 1}^{1} h ^{- 3} d x}

(3)

is a weighted average of the height with strictly positive weight

h^{- 3}

. This also identifies the constant flux uniquely.

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