mechanical engineering fluid...

MECHANICAL ENGINEERING – FLUID MECHANICS

http://ad.apsalar.com/api/v1/ad?re=0&st=566512895500&h=698b38b3271a5c15b40083940ca1fc2d6823134a


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: AFP /

Units : 2/ mN or Pascal (S.I.) and Dyne/cm2 (C.G.S.)

Dimension : ][][

][

][

][][ 21

2

2

TMLL

MLT

A

FP

Atmospheric pressure: Its value on the surface of the earth at sea level is nearly 25 /10013.1 mN or Pascal in S.I. other practical units of pressure are atmosphere,

bar and torr (mm of Hg)

torr760bar01.11001.11 5 Paatm

Fluid Pressure at a Point: dF

dA

Density ( ρ ):

In a fluid, at a point, density ρ is defined as: dV

dm

V

m

V

0lim

In case of homogenous isotropic substance, it has no directional properties, so is a

scalar.

It has dimensions ][ 3ML and S.I. unit kg/m3 while C.G.S. unit g/cc with

33 /10/1 mkgccg

Density of body = Density of substance

Relative density or specific gravity which is defined as : of water Density

of body DensityRD

If 1m mass of liquid of density 1 and 2m mass of density 2 are mixed, then as

21 mmm and )/()/( 2211 mmV [As /mV ]

)/()/()/( 2211

21

ii

i

pm

m

mm

mm

V

m

If 21 mm ,

21

212

Harmonic mean

If 1V volume of liquid of density 1 and 2V volume of liquid of density 2 are

mixed, then as: 2211 VVm and 21 VVV [As Vm / ]

If VVV 21 2/)( 21 = Arithmetic Mean



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.,

)1()/(

)/(

0

00

00

V

V

V

V

Vm

Vm [As )1(0 VV ]

or

)1(–~)1(

00

With increase in pressure due to decrease in volume, density will increase, i.e.,

V

V

Vm

Vm 0

00 )/(

)/(

[As

V

m ]

By definition of bulk-modulus:V

pVB

0 i.e.,

B

pVV 10

B

p

B

p1~1 0

1

0

Specific Weight ( w ):

It is defined as the weight per unit volume.

Specific weight .

.Weight m g

gVolume Volume

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 H2 or air in case of gas.

.

.

w w w

gs

g

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 1V

m

Inertial force per unit area = A

dtdmv

A

dtdp )/(/ =

A

Avv = 2v



Viscous force per unit area: r

vAF

/

Reynold’s number: area unit per force Viscous

area unit per forceInertial RN

rv

rv

v

/

2

Pascal’s Law: x y zp p p ; where, xp , yp and zp are the pressure at point x,y,z respectively.

Hydrostatic Law:

p

pgz

or dp pg dz

p h

o odp pg dz

p pgh and p

hpg

; 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

liquid by virtue of its height or

position above the surface of

earth or any reference level

taken as zero level.

It is the energy possessed by a

liquid by virtue of its motion or

velocity.

Pressure energy of the liquid PV Potential energy of the liquid

mgh

Kinetic energy of the liquid

mv2/2

Pressure energy per unit mass of

the liquid P/ ρ

Potential energy per unit mass of

the liquid gh

Kinetic energy per unit mass of

the liquid v2/2

Pressure energy per unit volume

of the liquid P

Potential energy per unit volume

of the liquid ρgh

Kinetic energy per unit volume

of the liquid ρ v2/2

Quantities that Satisfy a Balance Equation

Quantit

y

mass x momentum y momentum z

momentum

Energy Species

m mu mv mw E + mV2/2 m

(K)

1 u v w e + V2/2 W

(K)

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,

and m(K)

is the mass of a chemical species, K, W(K)

is the mass fraction of species K.



The other energy term, mV2/2, is the kinetic energy.

zyxtt

zyx

t

m

tStorage

)()()(

xywzxvzyuInflowzyx

xywzxvzyuOutflowzzyyxx

zyxSource S

S

z

ww

y

vv

x

uu

t

zzzyyyxxx

*

S

z

w

y

v

x

u

t

Szyx

Lim

0S*

The Mass Balance Equations:

0

i

i

x

u

t

0

z

w

y

v

x

u

t

0

i

i

i

ix

u

xu

t

0

zw

yv

xu

z

w

y

v

x

u

t

i

ix

utDt

Dor

zw

yv

xu

tDt

D

000

Dt

D

x

u

Dt

D

z

w

y

v

x

u

Dt

D

i

i

00

i

i

x

uor

z

w

y

v

x

u



S

i

i

i

i

xu

x

u

tt

S

i

ix

ut

Momentum Balance Equation:

j

i

ij

j

jjjB

xB

xxxtermsourcedirectionjNet

3

3

2

2

1

1

3,1

jB

xx

uu

t

uj

i

ij

i

jij

For a Newtonian fluid, the stress, σij, is given by the following equation:

ij

i

j

j

i

ijijx

u

x

uP

)

3

2(

3,1)3

2(

jB

x

u

x

uP

xx

uu

t

ujij

i

j

j

i

ij

ii

jij

3,1)3

2(

jB

xx

u

x

u

xx

P

x

uu

t

uj

ji

j

j

i

iji

jij

x

u uu vu wuB

t x y z

2

2 ( )3

P u v u w u

x x x y x y z x z x

Energy Balance Equation:

This directional heat flux is given the symbol qi: i

ix

Tkq

x

qqzy

zyx

qq

VolumeUnit

heatxDirectionNet xxxxxxxxxx

x

q

VolumeUnit

sourceheatxDirectionNet

x

Limitx

0



i

ixyx

x

q

x

q

y

q

x

qRateHeat

Body-force work rate = ρ(uBx + vBy + wBz) =ρuiBi

The work term on each face is given by the following equation:

y-face surface force work = (uσyx + vσyy + wσyz)Δx Δz =uiσiy Δx Δz

y

u

y

wvuWorkForceSurfaceyFaceNet

yiiyzyyyx

)(

j

jiiziiyiixii

x

u

z

u

y

u

x

uWorkForceSurfaceNet

Energy balance equation:

ii

j

jii

i

i

i

i Bux

u

x

q

x

eu

t

e

)2/()2/(22 VV

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:

z

Tk

zy

Tk

yx

Tk

xx

Tk

xx

Tk

xx

q

iiiii

i

j

i

ij

i

j

j

i

ij

iii

i

x

u

x

u

x

uP

x

Tk

xx

eu

t

e

)

3

2(

j

j

j

i

ijx

u

x

u

2)3

2(

j

i

i

j

j

i

iii

i

x

u

x

u

x

uP

x

Tk

xx

eu

t

e

Dissipation to avoid confusion with the general quantity in a balance equation:

2)3

2(

j

i

i

j

j

i

x

u

x

u

x

uDΦ



DΦ

P

x

Tk

xx

eu

t

e

iii

i

The temperature gradient in the Fourier law conduction term may also be written as a gradient

of enthalpy or internal energy:

iT

P

vivi xP

T

cx

e

cx

T

2

111

ip

P

ipi x

P

c

T

x

h

cx

T

11

iT

P

viivii

i

xP

T

cxP

x

e

c

k

xx

eu

t

e

2

11DΦ

Dt

DP

x

P

c

T

xx

h

c

k

xx

hu

t

h

ip

P

iipii

i

1DΦ

Dt

DPT

x

Tk

xx

Tu

t

Tc P

iii

i

p

DΦ



General Balance Equations

φ c Γ(φ)

S(φ)

1 1 0 0

u = ux = u1 1 μ xB

xx

w

zx

v

yx

u

xx

P

)

3

2(

v = uy = u2 1 μ yB

yy

w

zy

v

yy

u

xy

P

)

3

2(

w = uz = u3 1 μ zB

zz

w

zz

v

yz

u

xz

P

)

3

2(

e 1 k/cv

iT

P

vi xP

T

cxP

2

11DΦ

h 1 k/cP

Dt

DP

x

P

c

T

x ip

P

i

1DΦ

T cP k

Dt

DPTPDΦ

T cv k

T

PT

DΦ

W(K)

1 ρDK,Mix r(K)

Momentum equation:

)()( * j

i

j

ii

j

i

j

ii

jijS

x

u

xx

PS

x

u

xx

uu

t

u

General Momentum Equations

φ c Γ(φ)

S*(φ)

1 1 0 0

u = ux = u1 1 μ xB

xx

w

zx

v

yx

u

x

)

3

2(

v = uy = u2 1 μ yB

yy

w

zy

v

yy

u

x

)

3

2(

w = uz = u3 1 μ zB

zz

w

zz

v

yz

u

x

)

3

2(



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 is Bernoulli’s equation:

2

2

1vghP constant

g

vh

g

P

2

2

= constant; g

P

is called pressure head, h is called gravitational head and

g

v

2

2

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:



Relation between coefficient of viscosity and temperature:

Andrade formula = 3/1

/

TCeA

Stoke's Law: F = 6rv

Terminal Velocity:

Weight of the body (W) = mg = (volume × density) × g gr 3

3

4

Upward thrust (T) = weight of the fluid displaced

= (volume × density) of the fluid × g = gr 3

3

4

Viscous force (F) = rv6

When the body attains terminal velocity the net force acting on the body is zero.

W–T–F =0 or F= W – T

grgrrv 33

3

4

3

46 gr )(

3

4 3

Terminal velocity

grv

)(

9

22

Terminal velocity depend on the radius of the sphere so if radius is made n - fold,

terminal velocity will become n2 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 velocity.

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.

Poiseuille’s Formula:

l

rPV

4

or l

rKPV

4

l

rPV

8

4

; where 8

K is the constant of proportionality.

Buoyant Force:

Buoyant force = Weight of fluid displaced by body



Buoyant force on cylinder =Weight of fluid displaced by cylinder

insV Value of immersed part of solid

FB = waterp g Volume of fluid displaced

FB= waterp g Volume of cylinder immersed inside the water

BF mg

FB =2

4wp g d

( w mg pVg )

ins s sV plg V p g

2 2

4 4w cylinderp g d x p g d h

w cylinderp x p h

Relation between B, G and M:

GM Bl

GV

; where l = Least moment of inertia of plane of body at water surface, G =

Centre of gravity, B = Centre of buoyancy, and M = Metacentre.

,min( ),xx yyl l l

3

,12

xx

bdl

3

12yy

bdl

V bdx

Energy Equations:

Fnet = Fg + Fp + Fv + Fc + Ft ; where Gravity force Fg, Pressure force Fp, Viscous force Fv ,

Compressibility force Fc , and Turbulent force Ft.

If fluid is incompressible, then Fc = 0

net g p v tF F F F F ; This is known as Reynolds equation of motion.

If fluid is incompressible and turbulence is negligible, then

0, 0c tF F net g p vF F F F ; This equation is called as Navier-Stokes equation.

If fluid flow is considered ideal then, viscous effect will also be negligible. Then

net g pF F F ; This equation is known as Euler’s equation.

Euler’s equation can be written as: 0dp

gdz vdv



Dimensional analysis:

Quantity Symbol Dimensions

Mass m M

Length l L

Time t T

Temperature T θ

Velocity u LT -1

Acceleration a LT -2

Momentum/Impulse mv MLT -1

Force F MLT -2

Energy - Work W ML 2T

-2

Power P ML 2T

-3

Moment of Force M ML 2T

-2

Angular momentum - ML 2T

-1

Angle η M 0L

0T

0

Angular Velocity ω T -1

Angular acceleration α T -2

Area A L 2

Volume V L 3

First Moment of Area Ar L 3

Second Moment of Area I L 4

Density ρ ML -3

Specific heat-

Constant Pressure

C p L 2 T

-2 θ

-1

Elastic Modulus E ML -1

T -2

Flexural Rigidity EI ML 3T

-2

Shear Modulus G ML -1

T -2

Torsional rigidity GJ ML 3T

-2

Stiffness k MT -2



Angular stiffness T/η ML 2T

-2

Flexibiity 1/k M -1

T 2

Vorticity - T -1

Circulation - L 2T

-1

Viscosity μ ML -1

T -1

Kinematic Viscosity τ L 2T

-1

Diffusivity - L 2T

-1

Friction coefficient f /μ M 0L

0T

0

Restitution coefficient M 0L

0T

0

Specific heat-

Constant volume

C v L 2 T

-2 θ

-1

Boundary layer:

Reynolds number (Re)x

v x v x

v

Displacement Thickness ( *):0

* 1u

dyU

Momentum Thickness ():0

1u u

dyU U

Energy Thickness ( **):2

**

201

u udy

U U

Boundary Conditions for the Velocity Profile: Boundary conditions are as

o (a) 0, 0, 0du

At y udy

; (b) , , 0du

At y u Udy

Turbulent flow:

Shear stress in turbulent flow: v t

du du

dy dy

Turbulent shear stress by Reynold: ' 'u v

Shear stress in turbulent flow due to Prndtle :

2

2 dul

dy


mechanical engineering fluid...

Documents