Fluid Mechanics - APS

五维生物

2015-11-30

APS, ME

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Continuum : meaning the matter in the body is continuously distributed and fills the entire region of space it occupies.
Fluid Particle :
- macro small
- micro big enough
- have basic physical properties

1 Physical properties

1.1 Density

$$\rho=\dfrac{m}{V}$$

kg/m3
kg
m3

Relative Density : $d=\dfrac{\rho}{\rho_w}$

to water 4 degree
mercury : 12.59
oil : 0.85-0.9

1.2 Specific volume

$$v=\dfrac{V}{m}=\dfrac{1}{\rho}$$

m3/kg

1.3 Compressibility & Swelling

1.3.1 Equation of state

$$pv=R_gT$$

p Pa
v specific volume
$R_g=\dfrac{pV}{nT}=\dfrac{8314}{M}$ J/(kg K) : gas constant
T K

1.3.2 Coefficient of expansion

$$\alpha_V=\dfrac{1}{V}\dfrac{dV}{dt}$$

for ideal gas : $\alpha_V=\dfrac{1}{T}$
Unit : $K^{-1}$

1.3.3 Compression ratio

$$\kappa_T=-\dfrac{1}{V}\dfrac{dV}{dp}$$

Unit : $Pa^{-1}$

Bulk modulus

$$K=\dfrac{1}{\kappa_T}$$
Incompressible fluid : $\kappa_T$

1.4 Viscosity

1.4.1 Newtons’s inner friction law

1.4.2 Different fluid

plastic
pseudoplastic
newtonian
dilatant
ideal

1.4.3 dynamic viscosity

$$\mu=\dfrac{\tau}{dv/dy}$$

Unit : $Pa\cdot s$

1.4.4 kinematic viscosity

$$v=\dfrac{\mu}{\rho}$$

Unit : m2/s

ideal fluid : no viscosity

2 Hydrostatics

2.1 Force on the balanced fluid

2.1.1 Mass force

$\propto m$

unit mass force : $\vec{f}=f_x\vec i+f_y\vec j+f_z\vec k\=\dfrac{F_x}{m}\vec i+\dfrac{F_y}{m}\vec j+\dfrac{F_z}{m}\vec k$

2.1.2 Surface force

have something to do with S

Surface Stress

normal stress : $p=\dfrac{dP}{dA}$
shear stress : $\tau=\dfrac{T}{A}$

2.2 Differential Equation

2.2.1 Euler equations

$$\begin{cases}f_x-\dfrac{1}{\rho}\dfrac{\partial P}{\partial x}&=0\f_y-\dfrac{1}{\rho}\dfrac{\partial P}{\partial y}&=0\f_z-\dfrac{1}{\rho}\dfrac{\partial P}{\partial z}&=0\end{cases}$$

2.2.2 Potential function

$$dp=-\rho dW$$

2.2.3 Isobaric Surface

$$f_xdx+f_ydy+f_zdz=0$$

constant potential surface : $dW=0$
perpendicular to the mass force : $\vec{a_m}\cdot d\vec{s}=0$
two insoluble fluid interface is isobaric surface

2.3 Static pressure

Incompressible fliud

$dp=-\rho dW=-\rho gdz$

$\downarrow \rho=C$

$dz+\dfrac{dp}{\rho g}=0$

$$z+\dfrac{p}{\rho g}=c$$

Calculation

Absolute pressure : $p$
Vacuum pressure / Measuring pressure: $p_m$
Local pressure : $p_a$

$p>p_a,\quad p=p_a+p_m\p<p_a,\quad p=p_a-p_m$

manometer

2.4 Force to the Wall surface

plane
cylinder

3 Fluid Dynamics

3.1 Lagrange

describe each particle path

3.2 Euler

describe all the particle transient parameter at same time, adapt to fluid parcel

3.2.1 Field

velocity
pressure
density
temperature

3.2.2 Type

Constant field : with time no change
Uniform field : with position no change

3.2.3 Controlled body

a space has constant position in coordinate, with any shape

3.3 Fluid motion

3.3.1 Particle derivative

relationship between all physical quantities and time

3.3.2 Path line & Stream line

P : lagrange particle moving path
S : fluid filed $\rightarrow$velocity field

$$\dfrac{dx}{v_x}=\dfrac{dy}{v_y}=\dfrac{dz}{v_z}=t$$
- Properties
  - constant shape, particle moving path same with stream line
  - dont converge except at station and odd point
- stream tube

3.3.3 Flow rate

unit time through some controlled surface

$$dq_v=vdA$$

net flow rate : closed surface

$$q_v=\oiint\limits_A\vec v\vec n dA$$

3.3.4 Average velocity & Kinetic energy & Momentum

$\bar u=\dfrac{q_v}{A}$
laminar : $\alpha=2,\beta=\frac{4}{3}$
turbulent : $\alpha=1.06\approx1,\beta=1.02 \approx1$

3.4 Continuous equation

law of conservation of mass

one dimensional flow

3.5 Ideal fluid dynamic differential equation– Euler equation

$$f-\dfrac{1}{\rho}\bigtriangledown p=\dfrac{\partial u}{\partial t}+(u\bigtriangledown)u$$

constant flow : $\dfrac{\partial u_x}{\partial t}=0$, same in y z
still : $u_x=u_y=u_z=0$

3.6 Real fluid dynamic differential equation – N-S equation

with viscosity
Navier Stokes equation

3.7 Bernoulli equation

3.7.1 Ideal

under N-S

incompressible ideal stable flow
along stream line intergal
mass force only gravity

$\rho$ is constant

$$z+\dfrac{p}{\rho g}+\dfrac{v^2}{2g}=C$$

z : elevation head
$\dfrac{p}{\rho g}$ : pressure head
$\dfrac{v^2}{2g}$ : velocity head
sum : total head

3.7.2 Real

$h_v$ : head loss

$h_f=\lambda\dfrac{l}{d}\dfrac{U^2}{2g}$ : frictional head loss
- $\lambda$ : frictional loss factor
- U : average velocity at each section
$h_j=\zeta\dfrac{U^2}{2g}$ : local head loss

$\sum h_v=\sum h_f+\sum h_j$