Download Elementary Classical Hydrodynamics by B. H. Chirgwin PDF

By B. H. Chirgwin

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A3/ The velocities of points situated at distances ll9 /2, / 3 from A on these principal directions are ai/i, a2/2, zh respectively. Consider now a cuboidal element of fluid having ll9 /2, / 3 as its edges and a volume x — /i /2 /3. The edges of this element increase at the given rates and so the volume of the element increases at the rate dx , = T(aiH-a2 + a 3 ). at The sum ai + a 2 + a 3 is the sum of the diagonal terms (the trace of etJ regarded as a matrix) which is an invariant for rotation of the coordinate axes so that in general 1 dr „ 7~dT = ?

1 dp dV dz dz Q (dV dV dV \ a2(x dx+y dy)-\^d * + - ^ - d ^ - ^ r d*) . 2 e where C is an arbitrary constant. 2r2. *. - + F + | « 2 = (a2 + y £ 2 ) r 2 + C . 3 SOME GENERAL THEOREMS 47 Since the streamlines all lie on surfaces r = constant we see that Bernoulli's equation for steady motion is satisfied, for Q 2 2 2 2 where Q = (a + f /? )r + C and varies from one streamline to another as r varies. (ii) A pipe of variable circular cross-section is given by r = a(cosh az) 1 ' 4 , where r, 0, z are cylindrical polar coordinates with the z-axis vertical.

Similarly, %zP = gQ [~Y~ sin2 cj) +a-s cos (j) + h\(-a + s cos

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