Exercises of the course

Pagina 1

Compressible fluid dynamics exercises b)Eix.y)= 2×1-1×+41I • S'MPLT CONNECTED DOMAIN + IRROTATIONAL = CONSERVATIVE FIELD i IK • I= n F = det 7×24Iz = IFY - IF× '≤= - 11=0 NOT CONSERVATIVE- - Ix 34Fx FyFz a) -0= 20 I+ 70 I+ 70 E= zxz >e #+""I+ gzz@ ✗2+44I+ zz@ ✗2+44± JX 04 OZ b)I= -0 + 2e ✗2+44e - ° Il = /22-2@✗' +44 + 4×22-2e #+44 )+ 1gZZ@✗' +44 ≠0 not SOLENOIDAL i iK • _ ^ I = det Ix24Iz = Tmz - Jun è+ Dux - Suz I+ Tiny - JMX I= 74 IZ JZ 2x IX 74Mx mylez = /gz@ ×"""_ gz@ ✗'+"" )è+ /c.✗ze ✗⊥44_ a✗zè""")I+ (g✗ZZe ✗'+""- 8×2-2@✗'+""I±= IRROTATIONAL al ◦ FIRST OF all , LET'S CHECK THAT THE F IS 1220TATIONAL ,SO THAT : 'RTDOTATIONAL +S'MPLYCONNECTED= CONSERVATIVE WEcan- DEFINE A POTENTIAL FUNCTION i IK • ^ I = det Ix24Iz = 7ft - JFY è+ 3Fx - JFZ I+ 71=4- JFx ±= - 34 IZ 77 Ix IX 74Fx FyFZ = Gxy - ^ - 4×4+ 1 = 0 2 ✗ 2 ✗ ∅= | 6×2-2×4>+ 4 d.✗= 2×3- ✗242+ y✗ + C 2 ✗• E= _ 0 0= | - 2×24+4+ ✗ dy= - ✗242-1 44+ c • ∅= 2×3- ✗242+ y✗ +44+ C • Il/×,4,t)= /0,3✗+244)è+ |1,64+1,2)I • g= f/ti • Flo)= 1,2kgn } • MASS CONSERVATION : 79 + I° - S+ f- ° Il = ☐ set • - ° Il= 0,3t 1,6 = 1,9 §+ 1,99= o i9101= 1,2 • g= gol" j= foie " go/✗+ 1,9)e " = o ✗= - 1,9 9= goe " ◦ | g= 1,2e-""ᵗ 1,2= Io gIo)= 1,2