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𝜕𝑥 and 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. Blassius 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√𝑅𝑒𝑥 and 𝜏𝑥=𝑐𝑑,𝑥𝜌𝑉22

23. Drag force= 𝐶𝑑𝜌𝐴𝑉22 𝑎𝑛𝑑 𝐹𝐴=𝜏0=𝐶𝑑𝜌𝑉22, 𝐶𝑑=1.328√𝑅𝑒 𝐿

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=−𝑣𝑥 and 𝜕∅𝜕𝑦=−𝑣𝑦

32. 𝜕𝜓𝜕𝑥=𝑣𝑦 and 𝜕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 𝑐𝑢𝑟𝑙 𝑣=12(∇×𝑣)

39. Vorticity =2𝜔

40. Circulation =𝑣𝑜𝑟𝑡𝑖𝑐𝑖𝑡𝑦×𝑎𝑟𝑒𝑎

Fluid mechanics formulas for gate
 Fluid mechanics formulas for gate

41. 𝐹=𝜌𝑎𝑣2

42. Q1=Q2[1+cosθ] 𝑎𝑛𝑑 Q2=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)