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Ean ( 4 ; 5) This equation can a l s o be r e w i t t e n , f o r f u t u r e convenience as -^-|-gradi=rgrad5. (4; 6 ) Let us examine the t o t a l d e r i v a t i v e of the c i r c u l a t i o n C with respect t o time 28 Unsteady Motion of Continiious Media Since then = rgrad S'dr- and grad i · rfr+y grad^^ · dr. therefore ^ = ^ [rgrad 5 + grad ( - f — ¿ ) ] · dr. (4; 7) Since the i n t e g r a l of the t o t a l d i f f e r e n t i a l , taken over a closed contour, vanishes i d e n t i c a l l y , we have I grad ^ - ^ _ / ) .
13) becomes the known B e r n o u l l i equation The B e r n o u l l i equation does not g e n e r a l l y s t a t e the conserva t i o n of energy l a w , and i s merely an i n t e g r a l of the d i f f e r e n t i a l equations of the motion of the medium. With t h i s , the quantity Q remains constant along each f l o w l i n e in the case of adiabatic motion. This i s evident, f o r the t r a j e c t o r y of any p a r t i c l e i s a f l o w l i n e and the entropy of each p a r t i c l e remains constant as the p a r t i c l e moves, i .
4 ; 2) Mathematical and Thermodynamic Methods of Gas Dynamics 2 7 We know from vector a n a l y s i s that (vV)O = j grad — [v rot v]. %Äiere q = ^ 2 + ^ + ^ i s the t o t a l v e l o c i t v of the p a r t i c l e in the medium, and consequently equation ( ^ . 2 ; can be r e w r i t t e n as ^ + i-grad^^-lf,rotül -^Igradp^O. ( 4 ; 3) Furthermore, since (Section 1) Λ =^+7-^5, (4; 4) where i i s the heat content of the medium, equaticm (3·3) be f i n a l l y w r i t t e n as % + grad = rot V] - h T-grad S.