Fluid Mechanics Theory I

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1 Fluid Mechanics Theory I Last Class: 1. Introduction 2. MicroTAS or Lab on a Chip 3. Microfluidics Length Scale 4. Fundamentals 5. Different Aspects of Microfluidcs Today s Contents: 1. Introduction to Fluids 2. Gas and Liquid Flows 3. Governing Equations for Gas and Liquid Flows 4. Boundary Conditions 5. Low Reynolds Flows 6. Bernoulli s Equation 1

2 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. 2

3 Intermolecular Forces No interactions Attraction Equilibrium Repulsion 3

4 Lennard Jones Potential Model σ is the characteristic length scale ε Repulsion Attraction (Van Der Waals) 4

5 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. 5

6 Fluid : Gas Flow and Liquid Flow 6

7 Fluid Modeling Family Tree 7

8 Continuum Assumptions the number of molecules mass of a single molecule A rule of thumb for discontinuity length scale 8

9 Continuum Fluid Mechanics at Small Scale General Governing Equations COM COLM COE 9

10 Gas Flow : Kinetic Gas Theory The equation for a dilute gas is the ideal gas law: or where p is the pressure, is the density of the gas, R is the specific gas constant for the gas being evaluated, n is the number density of the gas, K is Boltzmann s constant (K = J/K), and T is the absolute temperature. Using the second form of the ideal gas law, it is possible to calculate that at standard conditions (273.15K; 101, 625Pa), the number density of any gas is n = m 3 10

11 Dilute Gas Mean free path (intermolecular spacing) Molecular diameter Gases for which δ/d >> 1 are said to be dilute gases, while those not meeting this condition are said to be dense gases. For dilute gases, the most common mode of intermolecular interaction is binary collisions. Simultaneous multiple molecule collisions are unlikely. Practically, values of δ/d greater than 7 are considered to be dilute. 11

12 Dimensionless Numbers : Mach # 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 c s, and is given by: 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. 12

13 Dimensionless Numbers : Knudsen # 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. 13

14 Kundsen Number Regimes (for gas) 14

15 Dimensionless Numbers : Reynolds # 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: In channel flow, the actual numbers turn out to be Re<2300 for laminar flow and Re> 4100 for turbulent flow. 15

16 Dimensionless Numbers : Stokes # In experimental fluid dynamics, the Stokes number (S t ) 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 S t >>1, particles will detach from a flow especially where the flow decelerates abruptly. For S t <<1, particles follow fluid streamlines closely. If S t <<0.1, tracing accuracy errors are below 1% Where τp is the particle response time, and τf is the fluid response time. crashed! S t <<1 S t >>1 16

17 Dimensionless Numbers : Prandtle # The Prandtl number Pr is a dimensionless number; the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. where ν is kinematic viscosity, and α is thermal diffusivity. 17

18 Dimensionless Numbers : Peclet # 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. Pe = τ d /τ a = U/lD Where τ a is the hydrodynamic transport time, τ d is the molecular diffusion time, l is the characteristic scale and D is the diffusion coefficient. 18

19 Dimensionless Numbers : Womersley # Womersley number (α) is a dimensionless number in biofluid mechanics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscous effects. The Womersley number is important in keeping dynamic similarity when scaling 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 19

20 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 velocity scale D is the diameter, R is the radius of curvature of the path of the channel. 20

21 Example: Gas Flow Calculation Calculate all the significant parameters to describe a flow of diatomic nitrogen N 2 at 350K and 200 kpa at a speed of 100 m/s through a channel measuring 10 μmin diameter. 21

22 Example: Gas Flow Calculation 22

23 Liquid Fluid : Governing Eqs When 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 equation can be eliminated altogether (or at least decoupled from conservation of mass and momentum), and the conservation of mass and momentum equations can be simplified to: 23

24 Boundary Conditions 24

25 Boundary Conditions (Cont.) 25

26 BCs for Gas Flows 26

27 Gases Flowing Through Channels 27

28 BCs for Liquid Flows 28

29 BCs for Liquid Flows (Cont.) 29

30 Parallel Flows 30

31 Analytical Solutions of Liquid Flows: Circular Cross Section 31

32 Analytical Solutions of Liquid Flows: Rectangular Cross Section 32

33 Analytical Solutions of Liquid Flows: Rectangular Cross Section 33

34 Hydraulic Diameter 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 D h. The hydraulic diameter is given by: 34

35 Example: Hydraulic Diameter Calculations of Different Geometries 35

36 Example: (Cont.) 36

37 Low Reynolds Number Flows Many microfluidic devices operate in regimes where the flow moves slowly at least by macroscopic standards. To determine whether a flow is slow relative to its length scale, we need to scale the original dimensional variables to determine their relative size. Consider a flow in some geometry whose characteristic size is represented by D and average velocity u. Wecan scale the spatial coordinates with D and the velocity field with u according to the relations with the inverse scaling: 37

38 Dimensionless Forms 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 Inverse of Froude # 38

39 Dimensionless Forms (Cont.) If the Reynolds number is very small, Re << 1, the entire left side of the last equation becomes negligible, leaving only: 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). 39

40 Examples 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: = 10 3 kg/m 3 Length scale: 10 μm= 10 5 m Velocity scale: 1 mm/s = 10 3 m/s Then: 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 = 10 3 kg/m3 Length scale: 100 μm= 10 4 m Velocity scale: 10 m/s Then: Re = 10 3 Clearly not low Reynolds number behavior borderline turbulent behavior 40

41 Entrance Effects Since the geometry through which the flow is happening is independent of the z position, it might be tempting to consider the flow to be two dimensional. However, the aspect ratio (ratio between the height of channels and the lateral feature size) is usually near 1, meaning that the flow must definitely be considered three dimensional. One example where this situation can be important is in considering the entrance length effect. For macroscopic flows where the Reynolds number is usually assumed to be relatively high, the entrance length Le can be accurately predicted by: 41

42 Entrance Effects 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: Experiment is much shorter than the prediction!! 42

43 Bernoulli s Equation Y x 2 p 2 X m v 2 A 2 time 2 p 1 x 1 y 2 A 1 m v 1 y 1 time 1 Pressure energy density arising from internal forces within moving fluid (similar to energy stored in a spring) for any point along a steady flow or streamline p + ½ v 2 + g y = constant KE of bulk motion of fluid PE for location of fluid 43

44 Mass element m moves from (1) to (2) Derivation of BE m = A 1 x 1 = A 2 x 2 = V where V = A 1 x 1 = A 2 x 2 Equation of continuity A V = constant A 1 v 1 = A 2 v 2 A 1 > A 2 v 1 < v 2 Since v 1 < v 2 the mass element has been accelerated by the net force F 1 F 2 = p 1 A 1 p 2 A 2 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.

45 Derivation of BE (Cont.) K = ½ m v 22 ½ m v 12 = ½ Vv 22 ½ V v 1 2 U = m g y 2 m g y 1 = V g y 2 = V g y 1 W net = F 1 x 1 F 2 x 2 = p 1 A 1 x 1 p 2 A 2 x 2 W net = p 1 V p 2 V = K + U p 1 V p 2 V = ½ Vv 22 ½ V v 12 + Vg y 2 Vg y 1 Rearranging p 1 + ½ v 12 + g y 1 = p 2 + ½ v 22 + g y 2 Notice: applies only to an ideal fluid (zero viscosity)

46 force high speed low pressure force What happens when two ships or trucks pass alongside each other? Have you noticed this effect in driving across the Sydney Harbour Bridge?

47 artery Flow speeds up at constriction Pressure is lower Internal force acting on artery wall is reduced External forces causes artery to collapse Arteriosclerosis and vascular flutter

48 Ideal fluid Real fluid

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