Fluid Mechanics Theory I - myweb.ncku.edu.twmyweb.ncku.edu.tw/~oswaldchuang/Courses/FY2013...
Transcript of Fluid Mechanics Theory I - myweb.ncku.edu.twmyweb.ncku.edu.tw/~oswaldchuang/Courses/FY2013...
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Fluid Mechanics Theory I
Today’s Contents:1. Introduction to Fluids2. Gas and Liquid Flows3. Governing Equations for Gas and Liquid Flows4. Boundary Conditions5. Low Reynolds Flows6. Bernoulli’s Equation
Last Class:1. Introduction2. MicroTAS or Lab on a Chip3. Microfluidics Length Scale4. Fundamentals5. Different Aspects of Microfluidcs
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Introduction to Fluids
Flowing fluids can be characterized by the properties of both the fluid and the flow. These can be organized into four main categories:
• Kinematic properties such as linear and angular velocity, vorticity, acceleration, and strain rate;
• Transport properties such as viscosity, thermal conductivity, and diffusivity;• Thermodynamic properties such as pressure, temperature, and density;• Miscellaneous properties such as surface tension, vapor pressure, and surface accommodation coefficients.
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Intermolecular Forces
No interactions
Attraction
Equilibrium
Repulsion
http://chemwiki.ucdavis.edu/
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Lennard‐Jones Potential Model
σ is the characteristic length scale
ε
Repulsion Attraction (Van Der Waals)
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Three States of Matter
As the solid is heated up to and beyond its melting temperature, the average molecular thermal energy becomes high enough that the molecules are able to vibrate freely from one set of neighbors to another. The material is then called a liquid. The molecules of a liquid are still relatively close together (still approximately σ).
If the temperature of the liquid is raised, the vibration of the molecules increases still further. Eventually, the amplitude of vibration is great enough that, at the boiling temperature, the molecules jump energetically away from each other and assume a mean spacing of approximately 10σ (at standard conditions). The material is now called a gas.
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Fluid : Gas Flow and Liquid Flow
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Fluid Modeling Family Tree
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Continuum Assumptions
mass of a single molecule
the number ofmolecules
length scaleA rule of thumb for discontinuity
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Continuum Fluid Mechanics at Small Scale
COM
COLM
COE
General Governing Equations
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Gas Flow : Kinetic Gas Theory
where p is the pressure, is the density of the gas, R is the specific gasconstant for the gas being evaluated, n is the number density of the gas, K isBoltzmann’s constant (K = 1.3805×10−23 J/K), and T is the absolutetemperature. Using the second form of the ideal gas law, it is possible tocalculate that at standard conditions (273.15K; 101, 625Pa), the numberdensity of any gas is n = 2.70×1025m−3
The equation for a dilute gas is the ideal gas law:
or
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Gases for which δ/d >> 1 are said to be dilute gases, while those notmeeting this condition are said to be dense gases. For dilute gases, the mostcommon mode of intermolecular interaction is binary collisions.Simultaneous multiple molecule collisions are unlikely. Practically, values ofδ/d greater than 7 are considered to be dilute.
Dilute Gas
Mean free path (intermolecular spacing)
Molecular diameter
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In addition to these parameters, there are several dimensionless groups of parameters that are very important in assessing the state of a fluid in motion. These are the Mach number Ma, the Knudsen number Kn, and the Reynolds number Re. The Mach number is the ratio between the flow velocity u and the speed of sound cs, and is given by:
Dimensionless Numbers : Mach #
The Mach number is a measure of the compressibility of a gas and can be thought of as the ratio of inertial forces to elastic forces. Flows for which Ma < 1 are called subsonic and flows for which Ma > 1 are called supersonic. When Ma = 1, the flow is said to be sonic.
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The Knudsen number has tremendous importance in gas dynamics. It provides a measure of how rarefied a flow is, or how low the density is, relative to the length scale of the flow. The Knudsen number is given by:
where is the mean free path given in and L is some length scale characteristic of the flow.
Dimensionless Numbers : Knudsen #
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Kundsen Number Regimes (for gas)
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The physical significance of the Reynolds number is that it is a measure of the ratio between inertial forces and viscous forces in a particular flow, which is given by:
where u is some velocity characteristic of the flow, L a length scale characteristic of the flow, is the density, is the dynamic viscosity, and is the kinematic viscosity. The different regimes of behavior are:
Dimensionless Numbers : Reynolds #
In channel flow, the actual numbers turn out to be Re<2300 for laminar flow and Re> 4100 for turbulent flow.
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Dimensionless Numbers : Stokes #In experimental fluid dynamics, the Stokes number (St) is a measure of flow tracer fidelity in particle image velocimetry (PIV) experiments where very small particles are entrained in turbulent flows and optically observed to determine the speed and direction of fluid movement. for St>>1, particles will detach from a flow especially where the flow decelerates abruptly. For St<<1, particles follow fluid streamlines closely. If St<<0.1, tracing accuracy errors are below 1%
Where τp is the particle response time, and τf is the fluid response time.
St>>1St<<1
crashed!
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Dimensionless Numbers : Prandtle #The Prandtl number Pr is a dimensionless number; the ratio of momentumdiffusivity (kinematic viscosity) to thermal diffusivity.
where ν is kinematic viscosity, and α is thermal diffusivity.
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Dimensionless Numbers : Peclet #
Pe = τd/τa= U/lD
Where τa is the hydrodynamic transport time, τd is the molecular diffusion time, l is the characteristic scale and D is the diffusion coefficient.
A dimensionless number relevant in the study of transport phenomena in fluid flows. It is defined to be the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient.
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Dimensionless Numbers : Womersley #Womersley number (α) is a dimensionless number in biofluid mechanics. It is adimensionless expression of the pulsatile flow frequency in relation to viscouseffects. The Womersley number is important in keeping dynamic similarity whenscaling an experiment.
where R is an appropriate length scale (for example the radius of a pipe), ω is the angular frequency of the oscillations, and ν is the kinematic viscosity
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Dimensionless Numbers : Dean #Dean number is a dimensionless group in fluid mechanics, which occurs in the study of flow in curved pipes and channels.
where ρ is the density of the fluid, μ is the dynamic viscosity, V is the axial velocityscale D is the diameter, R is the radius of curvature of the path of the channel.
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Example: Gas Flow CalculationCalculate all the significant parameters to describe a flow of diatomic nitrogen N2 at 350K and 200 kPa at a speed of 100 m/s through a channel measuring 10 μm in diameter.
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Example: Gas Flow Calculation
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Liquid Fluid : Governing EqsWhen considering the flow of an incompressible, Newtonian, isotropic fluid, the previous equations can be simplified considerably to:
Assuming that the flow remains at a constant temperature. The energy equationcan be eliminated altogether (or at least decoupled from conservation of mass andmomentum), and the conservation of mass and momentum equations can besimplified to:
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Boundary Conditions
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Boundary Conditions (Cont.)
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BCs for Gas Flows
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Gases Flowing Through Channels
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BCs for Liquid Flows
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BCs for Liquid Flows (Cont.)
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Parallel Flows
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Analytical Solutions of Liquid Flows:Circular Cross Section
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Analytical Solutions of Liquid Flows:Rectangular Cross Section
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Analytical Solutions of Liquid Flows:Rectangular Cross Section
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Clearly, there are several shapes missing from this list that can be significant in microfluidics, as well as many others that the reader may chance to encounter less frequently. Two examples of these shapes are the trapezoidal cross section created by anisotropic wet etches in silicon, and the rectangular cross section with rounded corners often created by isotropic wet etches of amorphous materials. One method for approximating the flows through these geometries is using a concept known as the hydraulic diameter Dh. The hydraulic diameter is given by:
Hydraulic Diameter
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Example: Hydraulic Diameter Calculations of Different Geometries
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Example: (Cont.)
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Low Reynolds Number FlowsMany microfluidic devices operate in regimes where the flow moves slowly—at least by macroscopic standards. To determine whether a flow is slowrelative to its length scale, we need to scale the original dimensionalvariables to determine their relative size. Consider a flow in some geometrywhose characteristic size is represented by D and average velocity u. We canscale the spatial coordinates with D and the velocity field with u according tothe relations
with the inverse scaling:
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Assuming an isothermal flow of a Newtonian, isotropic fluid, the conservation of mass and momentum equations can be simplified to:
substitute Into the above equation
Dimensionless Forms
Inverse of Froude #
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If the Reynolds number is very small, Re << 1, the entire left side of the last equation becomes negligible, leaving only:
Dimensionless Forms (Cont.)
The equation is linear.
If the pressure gradient is increased by some constant factor A, the entire flow field is increased by the same factor; that is, the velocity at every point in the flow is multiplied by A.
Time does not appear explicitly in the equation. Consequently, low Reynolds number flows are completely reversible (except for diffusion effects that are not included in the equation).
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To get an idea of what kind of systems might exhibit low Reynolds behavior, it is useful to perform a sample calculation. A typical biomedical microdevice might exhibit the following behavior:
Fluid properties similar to water: = 103 kg/m3
Length scale: 10 μm = 10−5 mVelocity scale: 1 mm/s = 10−3 m/sThen: Re = 10−2
Clearly low Reynolds number behavior.
As another example, consider the flow in a microchannel heat exchanger:Fluid properties similar to water: μ = 103 kg/m3Length scale: 100 μm = 10−4mVelocity scale: 10 m/sThen: Re = 103
Clearly not low Reynolds number behavior—borderline turbulent behavior
Examples
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Since the geometry through which the flow is happening is independent ofthe z position, it might be tempting to consider the flow to be twodimensional. However, the aspect ratio (ratio between the height of channelsand the lateral feature size) is usually near 1, meaning that the flow mustdefinitely be considered three‐dimensional. One example where thissituation can be important is in considering the entrance length effect.
Entrance Effects
For macroscopic flows where the Reynolds number is usually assumed to be relatively high, the entrance length Le can be accurately predicted by:
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small‐scale systems often exhibit low Reynolds number behavior. As the Reynolds number tends towards zero, the entrance length also tends towards zero, in disagreement with low Reynolds number experiments. A better expression for low Reynolds numbers is given by:
Entrance Effects
Experiment is much shorter than the prediction!!
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Bernoulli’s Equation
y1
y2
x1
x2 p2
A2
A1 v1
v2
p1
X
Y
time 1
time 2
m
m
for any point along a steady flow or streamline
p + ½ v2 + g y = constant
KE of bulk motion of fluid
PE for location of fluid
Pressure energy density arising from internal forces within moving fluid (similar to energy stored in a spring)
Mass element mmoves from (1) to (2)
m = A1 x1 = A2 x2 = V where V = A1 x1 = A2 x2
Equation of continuity A V = constant
A1 v1 = A2 v2 A1 > A2 v1 < v2
Since v1 < v2 the mass element has been accelerated by the net force
F1 – F2 = p1 A1 – p2 A2
Conservation of energy
A pressurized fluid must contain energy by the virtue that work must be done to establish the pressure.A fluid that undergoes a pressure change undergoes an energy change.
Derivation of BE
K = ½ m v22 ‐ ½ m v12 = ½ V v22 ‐ ½ V v12
U = m g y2 –m g y1 = V g y2 = V g y1
Wnet = F1 x1 – F2 x2 = p1 A1 x1 – p2 A2 x2
Wnet = p1 V – p2 V = K + U
p1 V – p2 V = ½ V v22 ‐ ½ V v12 + V g y2 ‐ V g y1
Rearranging
p1 + ½ v12 + g y1 = p2 + ½ v22 + g y2
Notice: applies only to an ideal fluid (zero viscosity)
Derivation of BE (Cont.)
high speedlow pressure
force
force
What happens when two ships or trucks pass alongside each other?Have you noticed this effect in driving across the Sydney Harbour Bridge?
artery
External forces causesartery to collapse
Flow speeds up at constrictionPressure is lowerInternal force acting on artery wall is reduced
Arteriosclerosis and vascular flutter
Ideal fluid
Real fluid