Fluid Mechanics Formulas Shortcuts
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MECHANICAL ENGINEERING – – F FLUID MECHANICS
MECHANICAL ENGINEERING – FLUID MECHANICS Pressure (P ):
If F be the normal force acting on a surface of area A in contact with liquid, then pressure exerted by liquid on this surface is: P F / A
Units : N / m 2 or Pascal (S.I.) and Dyne/cm (C.G.S.)
2
[ F ]
2
]
[ ML1 T 2 ]
Dimension : [ P ]
Atmospheric pressure: Its value on the surface of the earth at sea level is nearly 1 . 013 10 5 N / m 2 or Pascal in S.I. other practical units of pressure are atmosphere, bar and torr (mm of Hg )
[ A]
[ MLT 2
[ L ]
5 1atm 1.01 10 Pa 1 .01 bar 760 torr
Fluid Pressure at a Point:
dF dA
Density ( ρ ): m dm V dV
In a fluid, at a point, density ρ is defined as: lim
In case of homogenous isotropic substance, it has no directional properties, so is a scalar.
It has dimensions [ ML3 ] and S.I. unit kg /m while C.G.S. unit g /cc with
V 0
3
1 g / cc 10 kg / m 3
3
Density of body = Density of substance
Relative density or specific gravity which is defined as : RD
If m 1 mass of liquid of density 1 and m 2 mass of density 2 are mixed, then as m m1 m 2 and V (m 1 / 1 ) (m 2 / 2 )
m V
m1 m 2 (m1 / 1 ) (m 2 / 2 )
If m1 m 2 ,
2 1 2
1 2
Density of body Density of water
[As V m / ]
mi (m i / pi )
Harmonic mean
If V 1 volume of liquid of density 1 and V 2 volume of liquid of density 2 are mixed, then as: m 1 V 1 2 V 2 and V V 1 V 2 [As m / V ] If V 1 V 2 V ( 1 2 ) / 2 = Arithmetic Mean
MECHANICAL ENGINEERING – FLUID MECHANICS
With rise in temperature due to thermal expansion of a given body, volume will increase while mass will remain unchanged, so density will decrease, i.e., 0
(m / V ) (m / V 0 )
V 0 V
V 0
V 0 (1 )
[As V V 0 (1 ) ]
or 0
~ 0 (1 ) –
With increase in pressure due to decrease in volu me, density will increase, i.e., 0
(1 )
(m / V ) (m / V 0 )
V 0
[As
V
By definition of bulk-modulus : B V 0 p 0 1 B
Specific Weight (
w
1
m V
]
p p i.e., V V 0 1 V B ~ 0 1
p B
):
It is defined as the weight per unit volume.
Specific weight
Weight Volume
m.g Volume
. g
Specific Gravity or Relative Density (s):
It is the ratio of specific weight of fluid to the specific weight of a standard fluid. Standard fluid is water in case of liquid and H 2 or air in case of gas. s
w
. g w. g
w
Where, w Specific weight of water, and w Density of water specific.
Specific Volume (
v
):
Specific volume of liquid is defined as volume per unit mass. It is also defined as the reciprocal of specific density.
Specific volume
V m
Inertial force per unit area =
1
dp / dt A
v(dm / dt ) A
=
v Av A
= v 2
MECHANICAL ENGINEERING – FLUID MECHANICS Viscous force per unit area: F / A Reynold’s number:
N R
v r
Inertial force per unit area Viscous force per unit area
v 2 v / r
v r
Pascal’s Law: p x p y p z ; where, p x , p y and p z are the pressure at point x,y,z respectively. Hydrostatic Law: p pg or dp pg dz z p
p pgh and h
o
dp pg
h
o
dz
p pg
; where, h is known as pressure head.
Pressure Energy
Potential energy
Kinetic energy
It is the energy possessed by a liquid by virtue of its pressure. It is the measure of work done in pushing the liquid against pressure without imparting any velocity to it.
It is the energy possessed by It is the energy possessed by a liquid by virtue of its height or liquid by virtue of its motion or position above the surface of velocity. earth or any reference level taken as zero level.
Pressure energy of the liquid PV
Potential energy of the liquid Kinetic energy of the liquid 2 mgh mv /2
Pressure energy per unit mass of the liquid P / ρ
Potential energy per unit mass of Kinetic energy per unit mass of 2 the liquid gh the liquid v /2
Pressure energy per unit volume Potential energy per unit volume Kinetic energy per unit volume 2 of the liquid P of the liquid ρ gh of the liquid ρ v /2
Quantities that Satisfy a Balance Equation Quantit mass x momentum y momentum z Energy Species y momentum m mu mv mw E + mV /2 m 1 u v w e + V /2 W In this table, u, v, and w are the x, y and z velocity components, E is the total thermodynamic internal energy, e is the thermodynamic internal energy per unit mass, (K) (K) and m is the mass of a chemical species, K, W is the mass fraction of species K.
MECHANICAL ENGINEERING – FLUID MECHANICS The other energy term, mV /2, is the kinetic energy.
(m ) ( x y z ) ( ) x y z t t t t Inflow u x y z v y x z w z y x
Outflow u x x y z v y y x z w z z y x
Source S x y z
u x x u x v y y v y t x y
u v w S * t x y z
* S
Storage
Lim
x y z 0
w
z z
z
S
The Mass Balance Equations:
ui 0 t xi
u v w 0 t x y z
u ui i 0 t xi xi
u v w u v w 0 x y z x y z t
D Dt
u v w t x y z
u v w 0 Dt x y z
D
u v w 0 x y z
or
or
D Dt
D Dt
ui 0 xi
ui 0 xi
ui t xi D Dt
w
0
z
S
MECHANICAL ENGINEERING – FLUID MECHANICS
ui S u i t t x x i i
ui S t xi
Momentum Balance Equation:
Net j direction source term
u j t
ui u j xi
ij xi
B j
1 j x1
x2
3 j x3
B j
ij
x j
u j t u j t
ui u j xi ui u j xi
xi
u j 2 ( ) ij 3 xi
u u j i x x x j i j
P x j xi
u uu vu wu B x t x y z
2 ( ) B j 3
j 1,3
j 1,3
P u v u w u 2 2 ( ) x x x y x y z x z x 3
Energy Balance Equation:
B j
ui u j 2 ( ) ij B j P ij x x 3 j i
xi
For a Newtonian fluid, the stress, σij, is given by the following equation:
ui
2 j
j 1,3
ij P ij
This directional heat flux is given the symbol qi: q i k
Net xDirectionheat Unit Volume
q x x x q x x
x y z
Limit Net xDirection heat source
x 0
Unit Volume
y z
q x x
T xi
q x x x q x x
x
MECHANICAL ENGINEERING – FLUID MECHANICS
Heat Rate
q x q y q x q i x y x xi
Body- force work rate = ρ(uB x + vB y + wB z ) =ρu B i i
The work term on each face is given by the following equation: y-face surface force work = (uσ yx + vσ yy + wσ yz )Δx Δz =uiσ iy Δx Δz
Net yFace Surface Force Work
Net Surface Force Work
Energy balance equation:
(u yx v yy w yz ) y
ui yi y
ui xi ui yi ui zi ui ji x y z x j
q ui ji (e V 2 / 2) ui (e V 2 / 2) i ui Bi t xi xi x j
Substitutions for Stresses and Heat Flux: Using only the Fourier Law heat transfer, the source term involving the heat flux in the energy balance equation:
qi T T T T T k k k k k xi xi xi xi xi x x y y z z
u u j e ui e T ( 2 ) ij ui k P ij i x t xi xi xi 3 x j j xi
ij
u u j ui 2 e ui e T k P i ( )2 x j xi x j 3 t xi xi xi
ui u j x j x j
Dissipation to avoid confusion with the general quantity in a balance equation:
Φ D
u u j ui 2 i ( )2 x j xi x j 3
MECHANICAL ENGINEERING – FLUID MECHANICS
e ui e T k P Φ t xi xi xi
D
The temperature gradient in the Fourier law conduction term may also be written as a gradient of enthalpy or internal energy:
1 T 1 e 1 T P P 2 xi cv xi cv T xi
T 1 h 1 T P P c p xi xi c p xi
1 e ui e k e 1 T P P Φ P 2 t xi xi cv xi xi cv T xi
h ui h k h 1 T P P DP Φ t xi xi c p xi xi c p xi Dt
c p
D
D
T ui T T DP k T Φ P Dt xi xi xi t D
MECHANICAL ENGINEERING – FLUID MECHANICS General Balance Equations φ
φ
φ 1 u = ux = u1
c 1 1
Γ 0 μ
S 0
v = uy = u2
1
μ
w = uz = u3
1
μ
e
1
k/cv
P u v w 2 ( ) B x 3 x x x y x z x x P u v w 2 ( ) B y y x y y y z y y 3 P u v w 2 ( ) B z 3 z x z y z z z z
P Φ D
h
1
k/cP Φ D
T
cP
T
k
cv
W
xi
1 1 T P P 2 xi cv T xi
1 T P P DP c x i Dt p
Φ D
k
Φ D
1 ρDK,Mix
DP
P T
Dt
T P
T
r
Momentum equation:
u j t
ui u j xi
u j P u j S ( j ) S *( j ) xi xi xi xi xi
General Momentum Equations φ
φ 1 u = ux = u1
c 1 1
Γ 0 μ
v = uy = u2
1
μ
w = uz = u3
1
μ
φ
S* 0 u v w 2 ( ) B x x x y x z x x 3 2 u v w ( ) B y 3 x y y y z y y 2 u v w ( ) B z 3 x z y z z z z
MECHANICAL ENGINEERING – FLUID MECHANICS
Bernoulli’s Equation: This equation has four variables: velocity ( ), elevation ( ), pressure ( ), and density ( ). It also has a constant ( ), which is the acceleration due to gravity. Here isBernoulli’s equation:
P gh P
g
v2 2 g
h
1 2 v
v 2 constant
2
2 g
= constant;
P
g
is called pressure head, h is called gravitational head and
is called velocity head.
Factors that influence head loss due to friction are:
Length of the pipe ( ) Effective diameter of the pipe ( ) Velocity of the water in the pipe ( ) Acceleration of gravity ( ) Friction from the surface roughness of the pipe ( )
The head loss due to the pipe is estimated by the following equation:
To estimate the total head loss in a piping system, one adds the head loss from the fittings and the pipe:
Note that the summation symbol ( ) means to add up the losses from all the different sources. A less compact-way to write this equation is:
Combining Bernoulli’s Equation With Head Loss:
MECHANICAL ENGINEERING – FLUID MECHANICS Relation between coefficient of viscosity and temperature: Andrade formula =
A e
C / T
1 / 3
Stoke's Law: F = 6 rv Terminal Velocity:
Weight of the body (W ) = mg = (volume × density) × g
Upward thrust (T ) = weight of the fluid displaced
4 3
= (volume × density) of the fluid × g =
r 3 g
4 3
3
r g
Viscous force ( F ) = 6 rv
When the body attains terminal velocity the net force acting on the body is zero.
W – T – F =0 or F = W – T
6 rv
4 3
r 3 g
4 3
r 3 g
4 3
r 3 ( ) g
2 r ( ) g 2
Terminal velocity v
Terminal velocity depend on the radius of the sphere so if radius is made n - fold, 2 terminal velocity will become n times.
Greater the density of solid greater will be the terminal velocity
Greater the density and viscosity of the fluid lesser will be the terminal vel ocity.
If > then terminal velocity will be positive and hence the spherical body will attain constant velocity in downward direction.
If < then terminal velocity will be negative and hence the spherical body will attain constant velocity in upward direction.
9
Poiseuille’s Formula: 4
V
V
P r
l
or V
P r 4 8 l
KP r 4 l
; where K
8
is the constant of proportionality.
Buoyant Force:
Buoyant force = Weight of fluid displaced by body
MECHANICAL ENGINEERING – FLUID MECHANICS
Buoyant force on cylinder =Weight of fluid displaced by cylinder
V sin Value of immersed part of solid
FB = pwater g Volume of fluid displaced
FB= pwater g Volume of cylinder immersed inside the water
F B mg
FB = pw g
V sin plg Vs ps g
pw g
pw x pcylinder h
4
4
2
d
d 2 x pcylinder g
( w mg pVg )
4
d 2h
Relation between B, G and M:
GM
l
BG ; where l = Least moment of inertia of plane of body at water surface, G = V Centre of gravity, B = Centre of buoyancy, and M = Metacentre.
l min(l xx,l yy ), l xx
V bdx
bd 3 12
, l yy
bd 3 12
Energy Equations:
Fnet = Fg + F p + Fv + Fc + Ft ; where Gravity force Fg, Pressure force F p, Viscous force Fv , Compressibility force Fc , and Turbulent force Ft.
If fluid is incompressible, then Fc = 0 Fnet Fg Fp Fv F t ; This is known as Reynolds equation of motion.
If fluid is incompressible and turbulence is negligible, then
Fc 0, F t 0 Fnet Fg Fp F v ; This equation is called as Navier-Stokes equation .
If fluid flow is considered ideal then, viscous effect will also be negligible. Then Fnet Fg F p ; This equation is known as Euler’s equation .
Euler’s equation can be written as:
dp
gdz vdv 0
MECHANICAL ENGINEERING – FLUID MECHANICS
Dimensional analysis: Quantity
Symbol
Dimensions
Mass
m
M
Length
l
L
Time
t
T
Temperature
T
θ
Velocity
u
LT
-
Acceleration
a
LT
-
mv
MLT
-
Force
F
MLT
-
Energy - Work
W
ML T
-
Power
P
ML T
-
Moment of Force
M
ML T
-
Angular momentum
-
ML T
-
Angle
η
M L T
Angular Velocity
ω
T
-
Angular acceleration
α
T
-
Area
A
L
Volume
V
L
First Moment of Area
Ar
L
Second Moment of Area
I
L
Density
ρ
ML
Momentum/Impulse
Specific heatConstant Pressure
C p
L T
-
-
θ
-1
-2
Elastic Modulus
E
ML T
Flexural Rigidity
EI
ML 3T
Shear Modulus
G
ML T
-2
Torsional rigidity
GJ
ML T
-
k
MT
Stiffness
-1
-
-2
-
MECHANICAL ENGINEERING – FLUID MECHANICS Angular stiffness
T/η
ML 2T
Flexibiity
1/k
M T
Vorticity
-
T
Circulation
-
L T
Viscosity
μ
ML T
Kinematic Viscosity
τ
L 2T -1
Diffusivity
-
L T
f /μ
M L T
Friction coefficient
-1
-2 2
-
-1
0
-1
-
0
M L T
Restitution coefficient
Cv
Specific heatConstant volume
L T
-
θ
0 -
Boundary layer:
Reynolds number
v x
(Re) x
v x v
Displacement Thickness ( *): * 1 0
u dy U
u u 1 dy U U
Momentum Thickness ( ):
Energy Thickness ( **):
Boundary Conditions for the Velocity Profile: Boundary conditions are as du du (a) At y 0, u 0, o 0 ; (b) At y , u U , 0 dy dy
**
0
0
u
2 u
U
U
1
2
dy
Turbulent flow: du
du
Shear stress in turbulent flow: v t
Turbulent shear stress by Reynold:
du Shear stress in turbulent flow due to Prndtle : l dy
dy
dy
u ' v ' 2
2
MECHANICAL ENGINEERING – FLUID MECHANICS
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