Fluid Mechanics Formulae Sheet



m V

SG l 

W  g V Weight density of liquid  Weight density of water w



v

 

p Liquid Jet 

  2L

w

Ld

bd 3  12

8 d

 z

   u v x y dp  w  g p  gz dz

 AV   A V 1 11

2 2 2

bh3 36

d 4

     ( u )  ( v)  ( w)  0 t x y z

u v w   0 x y z

 a 2  4ab  b 2  3    h  Trape  64  36(a  b)     VD VD R   V  ui  vj  wk  u 2  v 2  w2 e   I G circle 

pHollowbubble 

V 1  m 

Ah

I G Trian 

 g   0  t  t 2 4 d

h



v

Q  A V 

IG

h* 

I G Rec

 du     dy    1  l   0  2   1  t  t 

pliquiddroplet 



F  gA h

IG

1  v u  W     z 2  x y 

du u u u u a  u v w  x dt x y z  t   

1  w v  W     x 2  y z 

dv v v v v a  u v w  y dt x y z t dw w w w w a  u v w  z dt x y z t

1  u w  W    y 2  z x 

p v2   z  const g 2 g

Vorticity  2 W

   Aa i a j a k  a 2 a 2 a 2 x y z x y z

S   S  h  x  h  1 h  x 1  l   S0   S0 

u

 x  v y

 i 1 1  W    V   2 2 x u

vtheoritical  2 gh   (For pito t-tube) vactual  Cv 2 gh 

 2  2  0 x 2 y 2

Pab  Patm  Pgauge

4 f  L v hf  (Darcy-Weisbach) 2 g  d P 2

ρ

     0 x 2 y 2 z 2 2

2

2

 j  y v

 k  z w

    y x       x y 

Qventurimeter  a 2 v 2  C d 

 gdz  vdv  0

Cd  Cv  Cc

a1 a 2 a1  a 2 2

2

 2 gh

f 

0.079 for Re  4000-106 4 Re

H series 

4 f  L1V12 L2V22 L3V32      2 g  d1 d2 d3 

S  h  x  h  1  S0   S  h  x 1  l   S0  h  hf

2

 4Q  4 fL 2  2 Cd   πd   4 16Q f  L  H equivalent  d  2g π 2  2 g  d 5 

Fe 

Fi V  Fg Lg

FL    0 cos  dA   P sin dAC L  A 

v 2 2

v 2 A 2

S  ρAV(v2  v1 ) C  RT dp c  dρ

he

k

 k  1 2  k 1 P0  P1 1  M1  2   dA dv  v 2  dv 2  M 1   1  A v  c2  v v2 



k 1  2k p1  p ρ  2k p1   1  2  1    1  n k  k  1 ρ1  ρ2 p1  k  1 ρ1  

dA dv  v 2  dv 2   2  1  M 1 A v c  v



n

k 1 k

p    2   p1 

k 1 k

2  k 1

P



Re 

m

d 4

ρg h f f' L

V 

1

 V2 2g



2

Eu 

Fi V  Fp P/ρ

We 

V σ/ρL Fi Fe



ρVL VL V  d   μ v v

dp v 2   gz  const p 2

p v2 log e p   z  const ρg 2g 2  k  p v   z  const    k  1  ρg 2 g

1

 k  1 2  k 1 ρ0  ρ1 1  M1  2   V C

FR  FD2  FL2



k 1  k2  2k p1 ρ1 n  n k  k 1  



k 1  2  2k p1 ρ1 n k  n k  k 1  

m  A2 m  A2

k 1

mmax  A2

2 2k  2  k 1    k  1 k(k  1 )  k  1 

 k 1 2  T0  T1 1  M1  2  

v2 

L1V12 L2V22  2 gd1 2 gd 2

dv dA dρ   0 v A ρ

K C ρ

v M C

f' P   L  v 2 (Chezy's formula) ρg A

L L L L  15  25  35 5 d d1 d 2 d 3

2τ 0 ρv 2

M

hf 

Parallel p ipes:

V 2 gh

V

K C  KRT ρ



f 

16 for Re  2000 i.e. Visc ous Flow Re

h

Cv 

FD   P cos dA    0 sin dA  CD  A 

f 

2k p1  k  1 ρ1



m venturimeter 

k 1   2k p1   p2  k  1    k  1 1   p1     2

 A2  p  k 1   22  2   A1  p1 

 xx 



HGL =

1 DV u v w   xx   yy   zz    V Dt x y z

u v w ,  yy  ,  zz  x y z



h

h

1  u

v 

1  w

u 

dBsys

EGL =

dt



CV

  b  dV  CS bV ndA t

Skin friction

(for turbulent)

(for laminar) Laminar flow in pipe

Laminar flow in pipe

Laminar flow in pipe

u(r) =

Poiseuille’s law

laminar

laminar

(turbulent flow) von Karman eq. Prandtl eq.

1  v

w 

 xy     ,  zx     ,  yz     2  y x  2  x z  2  z y 

Colebrook equation

(parallel pipe)

;

;

; ; flow through a converging nozzle Fanno line eq.

Rayleigh line eq.