Elementary Classical Hydrodynamics by B. H. Chirgwin

By B. H. Chirgwin

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There are three directions, those of the principal axes of the quadric, in which the divergence of the fluid is in a direct line through A, and if we choose the coordinate axes in these directions (or rotate them to coincide with these directions) then etj = / a i 0 \0 0 0\ a2 0 01. 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.

6) has the same value at all points of the region of irrotational motion, and not merely along a streamline. 1 SOME GENERAL THEOREMS 37 3. e. w = f - ^ + F - f - ^ v 2 = constant. 7) For an incompressible fluid Q is constant and 1 iv = — + V+— v2 = constant. 7a) hold throughout the region of irro­ tational motion for all times. 7) which apply to steady motion are usually known as Bernoulli's equations. All these results have been obtained by integrating the equation of motion with respect to the space variables and so constitute first integrals of the equations of motion.

2 9) - This shows that div v is a measure of the rate at which fluid diverges from a point. e. div v = 0. 2 41 SOME GENERAL THEOREMS In this case co^ is anti-symmetric, in fact (0 \-C2 so that 2(w2)l = C2J3-C3J2, -Cs Ca\ Ci 0/ 2(«2>2 = C3^1~sl>;3? 2(W2)3 = Ciy2~t2yi or 112= K x y . 10) Hence this contribution to the velocity is the same as that of a rigid body rotating about A with angular velocity \ £. ) If we write down the expressions for £1, £2, £3, we obtain c. e. 3^2 ^ , OX3 &>21 = «. S2 = _ 3vi &>13 — "^ OX3 3^2 dvi dxi dx2 ' C = curl v.

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