Fluid mechanics formulas for gate || Civil Engineers

   Some of the basic formula for fluid mechanics which can help in competitive exam.

1. 𝜏=𝜇.𝑑𝑣𝑑𝑦
2. 𝜕𝜌𝜕𝑡+𝜕𝜕𝑥(𝜌𝑣𝑥)+𝜕𝜕𝑦(𝜌𝑣𝑦)+𝜕𝜕𝑧(𝜌𝑣𝑧)=0 (continuity equation)
3. 𝐷𝜌𝐷𝑡+𝜌∇.𝑉=0 (continuity equation)
4. 𝐹1−𝐹2=𝑀2−𝑀1𝑡=𝑚𝑣2−𝑚𝑣1𝑡=𝜌𝑄(𝑣2−𝑣1)
5. 𝐹=𝜌𝑎(𝑣+𝑢)[(𝑣+𝑢)−𝑢]=𝜌𝑎(𝑣+𝑢)𝑣(Jet propulsion moving with 𝑢 velocity)
6. 𝑝1𝜌𝑔+𝑣122𝑔+𝑧1=𝑝2𝜌𝑔+𝑣222𝑔+𝑧2
7. 𝜕𝜏𝜕𝑦=𝜕𝑝𝜕𝑥and 𝜇𝜕2𝑣𝜕𝑦2=𝜕𝑝𝜕𝑥
8. 𝜏=−𝜕𝑃𝜕𝑥. 𝑟2 and 𝜏=𝛾𝑤. ℎ𝐿𝐿.𝑟2
9. 𝑣=14𝜇. (−𝜕𝑝𝜕𝑥).(𝑅2−𝑟2)=𝑣𝑚𝑎𝑥.[1−(𝑟𝑅)2]
10. 𝑝1−𝑝2=32𝜇𝑉𝐿𝐷2=128𝜇𝑄𝐿𝜋𝐷4
11. 𝑓=64𝜇𝜌𝑉𝐷=64𝑅𝑒
12. 𝑣=𝑣𝑚𝑎𝑥. (1−𝑟𝑅)17⁄ (Turbulent)
13. ℎ𝐿=𝑓𝐿𝑣22𝑔𝐷
14. 𝐿𝐷5=𝐿1𝐷15+𝐿2𝐷25+𝐿3𝐷35 (Compound pipe)
15. 𝑝𝑎𝛾−ℎ=𝑝𝑠𝛾+𝑓𝑙𝑣22𝑔𝐷+0.5𝑣22𝑔+𝑣22𝑔 and 𝐻=𝑓𝐿𝑣22𝑔𝐷+1.5𝑣22𝑔 (Siphon)
16. 𝛿∗=∫(1−𝑣𝑉) ∞0𝑑𝑦
17. 𝜃=∫𝑣𝑉(1−𝑣𝑉) ∞0𝑑𝑦
18. 𝛿𝐸=∫𝑣𝑉(1−𝑣2𝑉2) ∞0𝑑𝑦
19. Blasius boundary layer thickness 𝛿𝑥=5√𝑅𝑒𝑥
20. Displacement thickness 𝛿∗𝑥=1.729√𝑅𝑒𝑥
21. Momentum thickness 𝜃𝑥=0.664√𝑅𝑒𝑥
22. 𝑐𝑑,𝑥=0.664√𝑅𝑒𝑥and 𝜏𝑥=𝑐𝑑,𝑥𝜌𝑉22
23. Drag force= 𝐶𝑑𝜌𝐴𝑉22 𝑎𝑛𝑑𝐹𝐴=𝜏0=𝐶𝑑𝜌𝑉22, 𝐶𝑑=1.328√𝑅𝑒𝐿
24. 𝛿′=11.6 𝜗√𝜏𝑜/𝜌=11.6𝜗𝑣∗(Laminar sub layer)
25. ℎ̅=x̅+IGAx̅ and ℎ̅=𝑥̅+𝐼𝐺𝐴𝑥̅. 𝑠𝑖𝑛2𝜃
26. 𝐺𝑀=𝐼𝑉−𝐵𝐺
27. Q= 815𝐶𝑑. √2𝑔.tan𝜃/2 𝐻52⁄
28. 𝑣=√𝑔𝑥²2𝑦
29. 𝑄=Cd𝑎1𝑎2√𝑎12−𝑎22. √2gh and ℎ=𝑥(𝑠𝑚𝑠𝑓−1)
30. 𝑄=𝜓1−𝜓2
31. 𝜕∅𝜕𝑥=−𝑣𝑥and 𝜕∅𝜕𝑦=−𝑣𝑦
32. 𝜕𝜓𝜕𝑥=𝑣𝑦and 𝜕𝜓𝜕𝑦=−𝑣𝑥
33. (𝑅𝑒)𝑟=𝜌𝑟𝑉𝑟𝐿𝑟𝜇𝑟=1
34. (𝐹𝑟)𝑟=𝑣𝑟√𝑔𝑟𝐿𝑟
35. 𝑎𝑥=𝑣𝑥.𝜕𝑣𝑥𝜕𝑥 +𝑣𝑦 .𝜕𝑣𝑥𝜕𝑦 +𝑣𝑧 .𝜕𝑣𝑥𝜕𝑧 +𝜕𝑣𝑥𝜕𝑡
36. 𝑎𝑦=𝑣𝑥.𝜕𝑣𝑦𝜕𝑥 +𝑣𝑦 .𝜕𝑣𝑦𝜕𝑦 +𝑣𝑧 .𝜕𝑣𝑦𝜕𝑧 +𝜕𝑣𝑦𝜕𝑡
37. 𝑎𝑧=𝑣𝑥.𝜕𝑣𝑧𝜕𝑥 +𝑣𝑦 .𝜕𝑣𝑧𝜕𝑦 +𝑣𝑧 .𝜕𝑣𝑧𝜕𝑧 +𝜕𝑣𝑧𝜕𝑡
38. 𝜔=12 𝑐𝑢𝑟𝑙𝑣=12(∇×𝑣)
39. Vorticity =2𝜔
40. Circulation =𝑣𝑜𝑟𝑡𝑖𝑐𝑖𝑡𝑦×𝑎𝑟𝑒𝑎
41. 𝐹=𝜌𝑎𝑣2
42. Q1=Q2[1+cosθ] 𝑎𝑛𝑑Q2=Q2[1−cosθ] (θ is with plate)
43. Ns=N√QH3/4, Ns=N√PH5/4
44. 𝑄2𝑔=𝐴3𝑇
45. 𝐹=𝑣√𝑔. 𝐴𝑇
46. 𝑦𝑐3=2𝑦12𝑦22𝑦1+𝑦2 and 𝐸= 𝑦12+𝑦1𝑦2+𝑦22𝑦1+𝑦2
47. 𝑑𝑦𝑑𝑥=𝑠𝑜−𝑠𝑓1−𝑄2𝑇𝑔𝐴3
48. 𝑃1+𝑀1 = 𝑃2+𝑀2 (specific force=pressure force+momentum per sec)
49. 𝑦2𝑦1[1+𝑦2𝑦1] = 2𝑞2𝑔𝑦13 (sequent depths for rectangular channel)
50. 𝐸=(𝑦2−𝑦1)34𝑦1𝑦2 (Energy loss in jump)