# Viscosity and the Navier-Stokes equations

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 6 Viscosity and the Navier-Stokes equations 6.1 The Newtonian stress tensor Generally real fluids are not inviscid or ideal. 1 Modifications of Euler s equations, needed to account for real fluid effects at the continuum level, introduce additional forces in the momentum balance equations. There exists a great variety of real fluids which can be treated at the continuum level, differing in what we shall call their rheology. Basically the problem is to identify the forces experienced by a fluid parcel as it is moved about and deformed according to the mathematical description we have developed. Because of the molecular structure of various fluid materials, the nature of these forces can vary considerably and there are many rheological models which attempt to capture the observed properties of fluids under deformation. The simplest of these rheologies is the Newtonian viscous fluid. To understand the assumptions let us restrict attention to the determination of a viscous stress tensor at x, t, which depends only upon the fluid properties within a fluid parcel at that point and time. One could of course imagine fluids where some local average over space determines stress at a point. Also it is easy enough to find fluids with a memory, where the stress at a particular time depends upon the stress history at the point in question. It is reasonable to assume that the forces dues to the rheology of the fluid are developed by the deformation of fluid parcels, and hence could be determined by the velocity field. If we allow only point properties, deformation of parcels must involve more than just the velocity itself; first and higher-order partial derivatives with respect to the spatial coordinates could be important. (The time derivative of velocity has already been taken into consideration in the acceleration terms. A moment s thought shows viscous forces cannot depend 1 In quantum mechanics the superfluid is in many respects an ideal fluid, but the laws governing vorticity, for example, need to be modified. 95

2 96 CHAPTER 6. VISCOSITY AND THE NAVIER-STOKES EQUATIONS u(y y=y A y=y B Figure 6.1: Momentum exchange by molecules between lamina in a shear flow. on velocity. The bulk translation of the fluid with constant velocity produces no force. Thus the deformation of a small fluid parcel must be responsible for the viscous force, and the dominant measure of this deformation should come from the first derivatives of the velocity field, i.e. from the components of the velocity derivative matrix ui x j. The Newtonian viscous fluid is one where the stress tensor is linear in the components of the velocity derivative matrix, with a stress tensor whose specific form will depend on other physical conditions. To see why a linear relation of this might capture the dominant rheology of many fluids consider a flow (u, v = (u(y, 0.Each different plane or lamina of fluid, y= constant, moves with a particular velocity. Now consider the two lamina y = y A, y B as shown in figure 6.1, moving at velocities u B < u A. If a molecule moves from B to A, then it is moving from an environment with velocity u B to an environment with a larger velocity u A. Consequently it must be accelerated to match the new velcity. According to Newton, a force is therefore applied to the lamina y = y A in the direction of negative x. Similarly a molecule moving from y A to y B must slow down, exerting a force on lamina y = y B in the direction of positive x. Thus these exchanges of molecules would tend to reduce the velocity difference between the two lamina. 2 This tendency to reduce the difference in velocities can be thought of as a force applied to each lamina. Thus if we insert a virtual surface at some position y, a force should be exerted on the surface, in the positive x direction if du/dy(y > 0. Generally we expect the gradients of the velocity components to vary on a length scale L comparable to some macroscopic scale- the size of a container, the size of a body around which the fluid flows, etc. On the other hand the scale of the molecular events envisaged above is very small compared to the macroscopic scale. Thus it is reasonable to assume that the force on the 2 Perhaps a more direct analogy would be two boats gliding along on the water on parallel paths, one moving faster than the other. If, at the instant they are side by side, an occupant of the fast boat jumps into the slow boat, the slower boat with speed up, and similarly an occupant of the slow boat can slow up the fast boat by jumping into it.

3 6.1. THE NEWTONIAN STRESS TENSOR 97 y σ 1j n j σ 2j n j σ 2j n j x σ 1j n j z Figure 6.2: Showing why σ 12 = σ 21. The forces are per unit area. The area of each face is 2. lamina is dominated by the first derivative, F(y = µ du dy. (6.1 The constant of proportionality, µ, is called the viscosity, and a fluid obeying this law is called a Newtonian viscous fluid. We have considered so far only a simple planar flow (u(y, 0. In general all of the components of the velocity derivative matrix need to be considered in the construction of the viscous stress tensor. Let us write σ ij = pδ ij + d ij. (6.2 That is, we have simply split off the pressure contribution and exhibited the deviatoric stress tensor d ij, which contains the viscous stress. We first show that d ij, and hence σ ij, must be a symmetric tensor. We can do that by considering figure 6.2. We show a square parcel of fluid of side. We show those forces on each face which exert a torque about the z-axis. We see that the torque is 3 (σ 21 σ 12, since each face has area 2 and each of the four forces considered has a moment of /2 about the z-axis. Now this torque must be balanced by the angular acceleration of the parcel about the z-axis. Now the moment of inertial of the parcel is a multiple of 4. As 0 we see that the angular acceleration must tend to infinity as 1. It follows that the only way to have stability of a parcel is for σ 21 = σ 12. The same argument applies to moments about the other axes. A final requirement we shall place on d ij, so a further condition on the fluids we shall study, is that there should be no preferred direction, the condition of isotropy. The conditions of isotropy of symmetric matrices of second order then imply that d ij can satisfy these while being linear in the components of the velocity derivative matrix only if it has either of two forms: u i x j + u j x i, u k x k δ ij. (6.3

4 98 CHAPTER 6. VISCOSITY AND THE NAVIER-STOKES EQUATIONS For a Newtonian fluid the linearity implies that the most general allowable deviatoric stress has the form ( ui d ij = µ + u j 2 u k δ ij + µ u k δ ij. (6.4 x j x i 3 x k x k Notice that we have divided the two terms so that the first term, proportional to µ, has zero trace. Thus if µ = 0, the deviatoric stress contributes nothing to the normal force on an area element; this is given solely by the pressure force. The possibility of a normal force distinct from the pressure force is allowed by the second term of (6.4. We have attached the term viscosity to µ, so µ is usually called the second viscosity. Often it is taken as zero, an approximation that is generally valid for liquids. The condition µ = 0 is equivalent to what is sometimes called the Stokes relation. In gases in particular µ may be positive, in which case the thermodynamic pressure and the normal stresses are distinct. It should be noted that if we had simply taken d ij to be proportional to the velocity derivative matrix, then the splitting (3.5 would show that only e ij could possibly appear, since otherwise uniform rotation of the fluid would produce a force orthogonal to the rotation axis,, which is never observed. The second term in (6.4 then follows as the only isotropic symmetric tensor linear in the velocity derivative which could be included as a contribution to pressure. In this course we shall be dealing with two special cases of (6.4. The first is an incompressible fluid, in which case σ ij = pδ ij + µ( u i + u j, (incompressible fluid. (6.5 x j x i Note that with the incompressibility σ ij = p + µ 2 u i. (6.6 x j x i The second case is compressible flow in one space dimension. Then u = (u(x, t, 0, 0 and the only non-zero component of the stress tensor is σ 11 = p + µ u x, µ = 4 3 µ + µ, one dimensional gas flow. (6.7 The momentum balance equation in the form ρ Du i Dt = σ ij x j, (6.8 together with the stress tensor given by (6.4, defines the momentum equation for the Navier-Stokes equations. These are the most commonly used equations for the modeling of the rheology of fluids. The have been found to apply to a wide variety of practical problems, but it is important to realize their limitations. First, for highly rarified gases the mean free path of molecules of the gas can become so large that the concept of a fluid parcel, small with respect to the

5 6.2. SOME EXAMPLES OF INCOMPRESSIBLE VISCOUS FLOW 99 macroscopic scale but large with respect to mean free path, becomes untenable. Also, many common fluids, honey being an example, are non-newtonian and can exhibit effects not captured by the Navier-Stokes equations. Finally, whenever a flow involves very small domains of transition, the Navier-Stokes model may break down. Example of this occurs in shock waves in gases, where changes occur over a distance of only several mean free paths, and in the interface between fluids, which can involve transitions over distances comparable to intermolecular scales. In these problems a multi-scale analysis is usually needed, which can bridge the macroscopic-molecular divide. Finally, we point out that the viscosities in this model will generally depend upon temperature, but for simplicity we shall neglect this variation, and in particular for the incompressible case we always take µ to be constant. Also we shall often exhibit the kinematic viscosity ν = µ ρ in place of µ. We remark that ν has dimensions length 2 / time, as can be verified from the momentum equations after division by ρ. 6.2 Some examples of incompressible viscous flow We now take the density and viscosity to be constant and consider several exact solutions of the incompressible Navier-Stokes equations. We shall be dealing with fixed or moving rigid boundaries and we need the following assumption regarding the boundary condition on the velocity in the Navier-Stokes model: Assumption (The non-slip condition: At a rigid boundary the relative motion of fluid and boundary will vanish. Thus at a non-moving rigid wall the velocity of the fluid will be zero, while at any point on a moving boundary the fluid velocity must equal the velocity of that point of the boundary. This condition is valid for gases and fluids in situations where the stress tensor is well approximated by (6.4. It can fail in small domains and in rarified gases, where some slip may occur Couette flow Imagine two rigid planes y = 0, H where the no-slip condition will be applied. The plane y = H moves in the x-direction with constant velocity U, while the plane y = 0 is stationary. The flow is steady, so the velocity field must be a function of y alone. Assuming constant density, u = (u(y, 0 and p x = 0 we obtain a momentum balance if µu yy = 0. (6.9 Thus given that u(0 = 0, u(h = U, we have u = Uy/H. We see that the viscous stress is here constant and equal to µu/h. This is the force per unit area felt by the plane y = 0. No pressure gradient is needed to sustain this stress field. Couette flow is the simplest exact solution of the Navier-Stokes equations with non-zero viscous stress.

6 y 100 CHAPTER 6. VISCOSITY AND THE NAVIER-STOKES EQUATIONS u(y,0 Figure 6.3: The velocity in the Rayleigh problem at t=0 mod 2π, y in units of µ/ω The Rayleigh problem A related unsteady problem results from the time dependent motion in the x- direction with velocity U(tof the plane y = 0. A no-slip condition is applied on this plane. A fluid of constant density occupies the semi-infinite domain y > 0. In this case an exact solution of the Navier-Stokes equations is provided by u = (u(y, t, 0, p = 0, with u t µu yy = 0, u(0 = U(t. (6.10 In the case U(t = U 0 cos ωt we see that u(y, t = R(e iωt f(y where f(y is the complex-valued function of y satisfying We shall also require that u( = 0. Thus iωf µf yy = 0, f(0 = U 0. (6.11 u = RU 0 e iωt (1+iy ω 2ν = U0 cos ( ω ωt 2ν y e y ω 2ν. (6.12 We show the velocity field in figure 6.3. Note that the oscillation dies away extremely rapidly, with barely one reversal before decay is almost complete.

7 6.2. SOME EXAMPLES OF INCOMPRESSIBLE VISCOUS FLOW Poiseuille flow We consider now a flow in a cylindrical geometry. A Newtonian viscous fluid of constant density is in steady motion down a cylindrical tube of radius R and of infinite extent in both directions. Because of viscous stresses at the walls of the tube, we expect there to be a pressure gradient down the tube. Let the axis of the tube be the z-axis, r the radial variable, and u = (u z, u r, u θ = (u z (r, 0, 0 the velocity field in cylindrical polar coordinates. We note here, for future reference, the form of the Navier-Stokes equations in these coordinates: Here u z t + u u z + 1 p ρ z = ν 2 u z, (6.13 u r t + u u r u2 θ r + 1 p ( ρ r = ν u θ t + u u θ + u ru θ + 1 p ( r rρ θ = ν 2 = 2 ( z r u z z + 1 (ru r + 1 u θ r r r θ u = u z z + u r r + u θ r ( r ( r r Lu r 2 u θ r 2 θ Lu θ + 2 u r r 2 θ, (6.14, (6.15 = 0. (6.16 θ, ( ( r 2 θ 2, L = 2 1 r 2. (6.18 For the problem at hand, we set p = Gz+ constant to obtain the following equation for u z (r: ( µ 2 2 u z u z = G = µ r r u z. (6.19 r The no-slip condition applies at r = R, so the relevant solution of (6.19 is u z = G 4µ (R2 r 2. (6.20 Thus the velocity profile is parabolic. The total flux down the tube is Q 2π R 0 ru z dr = πgr4 8µ. (6.21 If a tube of length L is subjected to a pressure difference p at the two ends, then we can expect to drive a total volume flow or flux Q = π pr4 8µL down the tube. The rate W at which work is done to force the fluid down a tube of length L is the pressure difference between the ends of the tube times the volume flow rate Q, i.e. W = πg2 LR 4 (6.22 8µ

8 102 CHAPTER 6. VISCOSITY AND THE NAVIER-STOKES EQUATIONS Poiseuille flow can be easily observed in the laboratory, particularly in tubes of small radius, and measurements of flow rates through small tubes provides one way of determining a fluid s viscosity. Of course all tubes are finite, the velocity profile (6.20 is not established at once when fluid is introduced into a tube. This entry effect can persist for substantial distances down the tube, depending on the viscosity and the tube radius, and also on the velocity profile at the entrance. Another interesting question concerns the stability of Poiseuille flow in a doubly infinite pipe; this was studied by the engineer Osborne Reynolds in the 1870 s. He observed instability and transition to turbulence in long tubes. An application of Poiseuille flow of some importance is to blood flow; and in the arterial system there are many branches which are too short to escape significant entry effects. A generalization of Poiseuille flow to an arbitrary cylinder, bounded by generators parallel to the z-axis and having a cross section S is easily obtained. The equation for u z is now 2 u z = 2 u z x u z y 2 = G/µ, u z = 0 on S. (6.23 The solution is necessarily 0 for G > 0 and can be found by standard methods for the inhomogeneous Laplace equation Flow down an incline We consider now the flow of a viscous fluid down an incline, see figure 6.4. The velocity has the form (u, v, w = (u(z, 0, 0 and the pressure is a function of z alone. The fluid is forced down the incline by the gravitational body force. The equations to be satisfied are ρg sin α + µ d2 u dz 2 = 0, dp + ρg cosα = 0. (6.24 dz On the free surface z = H the stress must equal the normal stress due to the constant pressure, p 0 say, above the fluid. Thus σ xz = ν du dz = 0 and σ zz = p = p 0 when z = H. Since the no-slip condition applies, we have u(0 = 0. Therefore u = ρg sin α 2µ z(2h z, p = p 0 + ρg(h zcosα. (6.25 The volume flow per unit length in the y-direction is U 0 udz = gh3 sin α. (6.26 3ν

9 6.2. SOME EXAMPLES OF INCOMPRESSIBLE VISCOUS FLOW 103 z g α H x Figure 6.4: Flow of a viscous fluid down an incline Flow with circular streamlines We consider a velocity field in cylindrical polar coordinates of the form (u z, u r, u θ = (0, 0, u θ (r, t, with p = p(r, t. From (6.13-(6.18 the equation for u θ is u θ t ( 2 u θ = ν r + 1 u θ 2 r r u θ, (6.27 r 2 with the equation r r = ρ r u2 θ (6.28 determining the pressure. The vorticity is ω = 1 r From (6.27 we then find an equation for the vorticity ru θ r. (6.29 ω ( 2 t = ν ω r ω = ν 2 ω. (6.30 r r This equation, which is the symmetric form of the heat equation in two space dimensions, may be used to study the decay of a point vortex in two dimensions, see problem The Burgers vortex Th implication of (6.30 is that vorticity confined to circular streamlines in two dimensions will diffuse like heat, never reaching a non-trivial steady state in R 2. We now consider a solution of the Navier-Stokes equations which involves

10 104 CHAPTER 6. VISCOSITY AND THE NAVIER-STOKES EQUATIONS a two-dimensional vorticity field ω = (ω z, ω r, ω θ = (ω(r, 0, 0. The idea is to prevent the vorticity from diffusing by placing it in a steady irrotational flow field of the form (u z, u r, u θ = (αz, αr/2, 0. Thus the full velocity field has the form (u z, u r, u θ = (αz, αr/2, u θ (r, t. (6.31 Now the z-component of the vortiity equation is, with (6.31, ω t α r ω 2 r αω = ν 1 r ( r r ω r, ω = 1 ru θ r r. (6.32 First note that if ν = 0, so that there is no diffusion of ω, we my solve the equation to obtain ω = e αt F(r 2 e αt, (6.33 where F(r 2 is the initial value of ω. This solution exhibits the exponential growth of vorticity coming from the stretching of vortex tubes in the straining flow (αz, αr/2, 0. If now we restore the viscosity, we look for a steady solution of (6.32, representing a vortex in for which diffusion is balanced by the advection of vorticity toward the z-axis. We have 1 r r ( α 2 r2 ω + νr ω = 0. (6.34 r Integrating and enforcing the condition2 that r 2 ω and r ω r vanish when r =, we have α dω rω + ν = 0. ( dr Thus ω(r = Ce αr2 4ν, (6.36 so that u θ = Γ 1 e αr2 4ν, (6.37 2π r where we have redefined the constant to exhibit the total circulation of the vortex. Note that as ν decreases the size of the vortex tubes shrinks. With Γ fixed this would mean that the vorticity of the tube is increased Stagnation-point flow In this example we attempt to modify the two-dimensional stagnation point flow with streamfunction UL 1 xy to a solution in y > 0 of the Navier-Stokes equations with constant density, satisfying the no-slip condition on y = 0. The vorticity will satisfy u ω ( 2 x + v ω y ν ω x + 2 ω = 0. ( y 2

11 6.3. DYNAMICAL SIMILARITY fprime eta Figure 6.5: f versus η for the viscous stagnatioin point flow. If we set ψ = UL 1 xf(y, then ω = UL 1 yf. Insertion in (6.38 gives F F FF Re 1 F, (6.39 where Re = UL/ν. The boundary conditions are that F(0 = F (0 = 0 to make ψ, u, v vanish on the wall y = 0, and F y as y, so that we obtain the irrotational stagnation point flow at y =. One integration of (6.39 can be carried out to obtain With F = Re 1/2 f(η, η = Re 1/2 y, (6.40 becomes F 2 FF Re 1 F = 1. (6.40 f 2 ff f = 1, (6.41 with conditions f ( = 1, f(0 = f (0 = 0. We show in figure 6.5 the solution f (η of this ODE problem. This represents a gradual transition through a layer of thickness of order UL/nu between the null velocity on the boundary and the velocity U(x/L which u has at the wall in the irrotational stagnation point flow. We shall be returning to a discussion of such transition layers in chapter 7, where we take up the study of boundary layers. 6.3 Dynamical similarity In the stagnation point example just considered, the dimensional combination Re = U L/ν has occurred as a parameter. This parameter, called the Reynolds

12 106 CHAPTER 6. VISCOSITY AND THE NAVIER-STOKES EQUATIONS number in honor of Osborne Reynolds, arose because we chose to exhibit the problem in a dimensionless notation. Consider now the Navier-Stokes equations with constant density it their dimensional form: u t + u u + 1 ρ p ν 2 u = 0, u = 0. (6.42 We may define dimensionless (starred variables as follows: u = u/u,x = x/l, p = p/ρu 2. (6.43 Here U, L are assumed to be a velocity and length characteristic of the problem being studied. In the case of flow past a body, L might be a body diameter and U the flow speed at infinity. In these starred variables it is easily checked that the equations become u t + u u + p 1 Re 2 u = 0, u = 0. (6.44 Thus Re survives as the only dimensionless parameter in the equations. For a given value of Re a given problem will have a solution or solutions which are fully determined by the value of Re. 3 Nevertheless the set of solutions is fully determined by Re and Re alone. Thus we are able to make a correspondence between various problems having different U and L but the same value of Re. We call this correspondence dynamical self-similarity. Two flows which are self-similar in this respect become identical which expressed in the starred, dimensionless variables (6.43. In a sense the statement the viscosity ν is small conveys no dynamical information, although the intended implication might be that Re 1. If L is also small, then it could well be that Re = 1 or e 1. The only meaningful way to state that a fluid is almost inviscid is through the Reynolds number, Re 1. If we want to consider fluids whose viscosity is dominant compared to inertial forces, we should require Re 1. These remarks underline the oft-repeated definition of Re as the ratio of inertial to viscous forces. This is because ρu u µ 2 u = Reu u 2 u Re (6.45 since we regard all starred variables as of order unity. example 6.1: The drag D per unit length of a circular cylinder of radius L in a two-dimensional uniform flow of speed U must satisfy D = ρu 2 LF(Re for some function F. Note that we are assuming here that cylinders are fully determined by their radius. In experiments other factors, such as surface material or roughness, slight ellipticity, etc. must be considered. Problem set 6 3 It is not always the case that well-formulated boundary-value problems for the Navier- Stokes equations have unique solutions. See the example of viscous flow in a diverging channel, page 79 of Landau an Lifshitz.

13 6.3. DYNAMICAL SIMILARITY 107 Q H θ θ 2L Figure 6.6: Bifurcating Poiseuill flow. Assume a parabolic profile in each section. 1. Consider the following optimzation problem: A Newtonian viscous fluid of constant density flows through a cylindrical tube of radius R 1, which then bifurcates into two straight tubes of radius R 2, see the figure. A volume flow Q is introduced into the upper tube, which divides into flows of equal flux Q/2 at the bifurcation. Because of the material composition of the tubes, it is desirable that the wall stress µdu/dr, evaluated at the wall, be the same in both tubes. If L and H are given and fixed, what is the angle θ which minimizes the rate of working required to sustain the flow Q?. Be sure to verify that you have a true minimum. 2. Look for a solution of (6.30 of the form ω = t 1 F(r/ t, satisfying ω(, t = 0, 2π rω(r, tdr = 1, t >. Show, by computing u 0 θ with u θ (, t = 0, that this represents the decay of a point vortex of unit strength in a vbiscous fluid, i.e. lim u θ(r, t = 1, r > 0. (6.46 t 0+ 2πr 3. A Navier-Stokes fluid has constant ρ, µ, no body forces. Consider a motion in a fixed bounded domain V with no-slip condition on its rigid boundary. Show that de/dt = Φ, E = ρ u 2 /2dV, Φ = µ ( u 2 dv. V This shows that for such a fluid kinetic energy is converted into heat at a rate Φ(t. This last function of time gives the net viscous dissipation for the fluid contained in V. (Hint: ( u = ( u 2 u. Also (A B = A B B A. V

14 108 CHAPTER 6. VISCOSITY AND THE NAVIER-STOKES EQUATIONS 4. In two dimensions, with streamfunction ψ, where (u, v = (ψ y, ψ x, show that the incompressible Navier-Stokes equations without body forces for a fluid of constant ρ, µ reduce to t 2 ψ ( (ψ, 2 ψ ν 4 ψ = 0. (x, y In terms of ψ, what are the boundary conditions on a rigid boundary if the no-slip condition is satisfied there? 5. Find the time-periodic 2D flow in a channel H < y < H, filled with viscous incompressible fluid, given that the pressure gradient is dp/dx = A + B cos(ωt, where A, B, ω are constants. This is an oscillating 2D Poiseuille flow. You may assume that u(y, t is even in y and periodic in t with period 2π/ω. 6. verify ( The plane z = 0 is rotating about the z-axis with an angular velocity Ω. A Newtonian viscous fluid of constant density and viscosity occupies z > 0 and the fluid satisfies the no-slip condition on the plane, i.e. at z = 0 the fluid rotates with the plane. By centrifugal effect we expect the fluid near the plane to be thrown out radially and a compensating flow of fluid downward toward the plane. Using cylindrical polar coordinates, look for a steady solution of the Navier- Stokes equations of the form (u z, u r, u θ = (f(z, rg(z, rh(z. (6.47 We assume that the velocity component u θ vanishes as z. Show that then p ρ = ν df dz 1 2 f2 + F, (6.48 where F is a function of r alone. Now argue that, if h( = 0, i.e. no rotation at infinity, then F must in fact be a constant. From the r and θ component of the momentum equation together with u = 0, find equations for f, g, h and justify the following conditions: f = df dz = 0, h = Ω, z = 0; f, h 0, z. (6.49 (The solution of these equations is discussed on pp of L&L and of Batchelor.

### Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

### 1 The basic equations of fluid dynamics

1 The basic equations of fluid dynamics The main task in fluid dynamics is to find the velocity field describing the flow in a given domain. To do this, one uses the basic equations of fluid flow, which

### Chapter 1. Governing Equations of Fluid Flow and Heat Transfer

Chapter 1 Governing Equations of Fluid Flow and Heat Transfer Following fundamental laws can be used to derive governing differential equations that are solved in a Computational Fluid Dynamics (CFD) study

### UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH College of Informatics and Electronics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2004-05 MODULE TITLE: Fluid mechanics DURATION OF EXAMINATION:

### Basic Principles in Microfluidics

Basic Principles in Microfluidics 1 Newton s Second Law for Fluidics Newton s 2 nd Law (F= ma) : Time rate of change of momentum of a system equal to net force acting on system!f = dp dt Sum of forces

### INTRODUCTION TO FLUID MECHANICS

INTRODUCTION TO FLUID MECHANICS SIXTH EDITION ROBERT W. FOX Purdue University ALAN T. MCDONALD Purdue University PHILIP J. PRITCHARD Manhattan College JOHN WILEY & SONS, INC. CONTENTS CHAPTER 1 INTRODUCTION

### 1. Fluids Mechanics and Fluid Properties. 1.1 Objectives of this section. 1.2 Fluids

1. Fluids Mechanics and Fluid Properties What is fluid mechanics? As its name suggests it is the branch of applied mechanics concerned with the statics and dynamics of fluids - both liquids and gases.

### Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture No. # 36 Pipe Flow Systems

Fluid Mechanics Prof. T. I. Eldho Department of Civil Engineering Indian Institute of Technology, Bombay Lecture No. # 36 Pipe Flow Systems Welcome back to the video course on Fluid Mechanics. In today

### Fundamentals of Fluid Mechanics

Sixth Edition. Fundamentals of Fluid Mechanics International Student Version BRUCE R. MUNSON DONALD F. YOUNG Department of Aerospace Engineering and Engineering Mechanics THEODORE H. OKIISHI Department

### Lecture 4 Classification of Flows. Applied Computational Fluid Dynamics

Lecture 4 Classification of Flows Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (00-006) Fluent Inc. (00) 1 Classification: fluid flow vs. granular flow

### Fluids and Solids: Fundamentals

Fluids and Solids: Fundamentals We normally recognize three states of matter: solid; liquid and gas. However, liquid and gas are both fluids: in contrast to solids they lack the ability to resist deformation.

### FLUID DYNAMICS. Intrinsic properties of fluids. Fluids behavior under various conditions

FLUID DYNAMICS Intrinsic properties of fluids Fluids behavior under various conditions Methods by which we can manipulate and utilize the fluids to produce desired results TYPES OF FLUID FLOW Laminar or

### Boundary Conditions in Fluid Mechanics

Boundary Conditions in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University The governing equations for the velocity and pressure fields are partial

### Basic Equations, Boundary Conditions and Dimensionless Parameters

Chapter 2 Basic Equations, Boundary Conditions and Dimensionless Parameters In the foregoing chapter, many basic concepts related to the present investigation and the associated literature survey were

### EXAMPLE: Water Flow in a Pipe

EXAMPLE: Water Flow in a Pipe P 1 > P 2 Velocity profile is parabolic (we will learn why it is parabolic later, but since friction comes from walls the shape is intuitive) The pressure drops linearly along

### CBE 6333, R. Levicky 1 Differential Balance Equations

CBE 6333, R. Levicky 1 Differential Balance Equations We have previously derived integral balances for mass, momentum, and energy for a control volume. The control volume was assumed to be some large object,

### du u U 0 U dy y b 0 b

BASIC CONCEPTS/DEFINITIONS OF FLUID MECHANICS (by Marios M. Fyrillas) 1. Density (πυκνότητα) Symbol: 3 Units of measure: kg / m Equation: m ( m mass, V volume) V. Pressure (πίεση) Alternative definition:

### Distinguished Professor George Washington University. Graw Hill

Mechanics of Fluids Fourth Edition Irving H. Shames Distinguished Professor George Washington University Graw Hill Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok

### Fundamentals of Transport Processes Prof. Kumaran Department of Chemical Engineering Indian Institute of Science, Bangalore

Fundamentals of Transport Processes Prof. Kumaran Department of Chemical Engineering Indian Institute of Science, Bangalore Module No. # 02 Lecture No. # 06 Mechanisms of diffusion-1 Welcome to the sixth

### CBE 6333, R. Levicky 1. Potential Flow

CBE 6333, R. Levicky Part I. Theoretical Background. Potential Flow Potential Flow. Potential flow is irrotational flow. Irrotational flows are often characterized by negligible viscosity effects. Viscous

### Fluid Mechanics: Static s Kinematics Dynamics Fluid

Fluid Mechanics: Fluid mechanics may be defined as that branch of engineering science that deals with the behavior of fluid under the condition of rest and motion Fluid mechanics may be divided into three

### Blasius solution. Chapter 19. 19.1 Boundary layer over a semi-infinite flat plate

Chapter 19 Blasius solution 191 Boundary layer over a semi-infinite flat plate Let us consider a uniform and stationary flow impinging tangentially upon a vertical flat plate of semi-infinite length Fig

### Introduction to COMSOL. The Navier-Stokes Equations

Flow Between Parallel Plates Modified from the COMSOL ChE Library module rev 10/13/08 Modified by Robert P. Hesketh, Chemical Engineering, Rowan University Fall 2008 Introduction to COMSOL The following

### Chapter 8 Steady Incompressible Flow in Pressure Conduits

Chapter 8 Steady Incompressible Flow in Pressure Conduits Outline 8.1 Laminar Flow and turbulent flow Reynolds Experiment 8.2 Reynolds number 8.3 Hydraulic Radius 8.4 Friction Head Loss in Conduits of

### ENSC 283 Introduction and Properties of Fluids

ENSC 283 Introduction and Properties of Fluids Spring 2009 Prepared by: M. Bahrami Mechatronics System Engineering, School of Engineering and Sciences, SFU 1 Pressure Pressure is the (compression) force

### Stokes flow. Chapter 7

Chapter 7 Stokes flow We have seen in section 6.3 that the dimensionless form of the Navier-Stokes equations for a Newtonian viscous fluid of constant density and constant viscosity is, now dropping the

### Chapter 3 Ideal Fluid Flow

2.20 - Marine Hydrodynamics, Spring 2005 Lecture 7 2.20 - Marine Hydrodynamics Lecture 7 Chapter 3 Ideal Fluid Flow The structure of Lecture 7 has as follows: In paragraph 3.0 we introduce the concept

### Physics of the Atmosphere I

Physics of the Atmosphere I WS 2008/09 Ulrich Platt Institut f. Umweltphysik R. 424 Ulrich.Platt@iup.uni-heidelberg.de heidelberg.de Last week The conservation of mass implies the continuity equation:

### Ch 2 Properties of Fluids - II. Ideal Fluids. Real Fluids. Viscosity (1) Viscosity (3) Viscosity (2)

Ch 2 Properties of Fluids - II Ideal Fluids 1 Prepared for CEE 3500 CEE Fluid Mechanics by Gilberto E. Urroz, August 2005 2 Ideal fluid: a fluid with no friction Also referred to as an inviscid (zero viscosity)

### Viscous flow in pipe

Viscous flow in pipe Henryk Kudela Contents 1 Laminar or turbulent flow 1 2 Balance of Momentum - Navier-Stokes Equation 2 3 Laminar flow in pipe 2 3.1 Friction factor for laminar flow...........................

### Scalars, Vectors and Tensors

Scalars, Vectors and Tensors A scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. Examples are hydrostatic pressure and temperature. A vector

### OUTLINE Introduction to Laminar and Turbulent Flow Reynolds Number Laminar Flow

REAL FLUIDS OUTLINE Introduction to Laminar and Turbulent Flow Reynolds Number Laminar Flow Flow in Circular Pipes Hagen Poiseuille Equation Flow between parallel plates REAL FLUIDS Turbulent Flow About

### Lecture 5 Hemodynamics. Description of fluid flow. The equation of continuity

1 Lecture 5 Hemodynamics Description of fluid flow Hydrodynamics is the part of physics, which studies the motion of fluids. It is based on the laws of mechanics. Hemodynamics studies the motion of blood

### Chapter 4 Rotating Coordinate Systems and the Equations of Motion

Chapter 4 Rotating Coordinate Systems and the Equations of Motion 1. Rates of change of vectors We have derived the Navier Stokes equations in an inertial (non accelerating frame of reference) for which

### Lecture L5 - Other Coordinate Systems

S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates

### 11 Navier-Stokes equations and turbulence

11 Navier-Stokes equations and turbulence So far, we have considered ideal gas dynamics governed by the Euler equations, where internal friction in the gas is assumed to be absent. Real fluids have internal

### Lecture 8 - Turbulence. Applied Computational Fluid Dynamics

Lecture 8 - Turbulence Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (2002-2006) Fluent Inc. (2002) 1 Turbulence What is turbulence? Effect of turbulence

### Strain and deformation

Outline Strain and deformation a global overview Mark van Kraaij Seminar on Continuum Mechanics Outline Continuum mechanics Continuum mechanics Continuum mechanics is a branch of mechanics concerned with

### Min-218 Fundamentals of Fluid Flow

Excerpt from "Chap 3: Principles of Airflow," Practical Mine Ventilation Engineerg to be Pubished by Intertec Micromedia Publishing Company, Chicago, IL in March 1999. 1. Definition of A Fluid A fluid

### Contents. Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 1

Contents 1. Introduction 2. Fluids 3. Physics of Microfluidic Systems 4. Microfabrication Technologies 5. Flow Control 6. Micropumps 7. Sensors 8. Ink-Jet Technology 9. Liquid Handling 10.Microarrays 11.Microreactors

### AA200 Chapter 9 - Viscous flow along a wall

AA200 Chapter 9 - Viscous flow along a wall 9.1 The no-slip condition 9.2 The equations of motion 9.3 Plane, Compressible Couette Flow (Review) 9.4 The viscous boundary layer on a wall 9.5 The laminar

### Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows

Lecture 3 Fluid Dynamics and Balance Equa6ons for Reac6ng Flows 3.- 1 Basics: equations of continuum mechanics - balance equations for mass and momentum - balance equations for the energy and the chemical

### CBE 6333, R. Levicky 1 Review of Fluid Mechanics Terminology

CBE 6333, R. Levicky 1 Review of Fluid Mechanics Terminology The Continuum Hypothesis: We will regard macroscopic behavior of fluids as if the fluids are perfectly continuous in structure. In reality,

### Chapter (1) Fluids and their Properties

Chapter (1) Fluids and their Properties Fluids (Liquids or gases) which a substance deforms continuously, or flows, when subjected to shearing forces. If a fluid is at rest, there are no shearing forces

### The Navier Stokes Equations

1 The Navier Stokes Equations Remark 1.1. Basic principles and variables. The basic equations of fluid dynamics are called Navier Stokes equations. In the case of an isothermal flow, a flow at constant

### Lecture L22-2D Rigid Body Dynamics: Work and Energy

J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

### Dimensional Analysis

Dimensional Analysis An Important Example from Fluid Mechanics: Viscous Shear Forces V d t / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Ƭ = F/A = μ V/d More generally, the viscous

### ME19b. SOLUTIONS. Feb. 11, 2010. Due Feb. 18

ME19b. SOLTIONS. Feb. 11, 21. Due Feb. 18 PROBLEM B14 Consider the long thin racing boats used in competitive rowing events. Assume that the major component of resistance to motion is the skin friction

### Basic Fluid Dynamics

Basic Fluid Dynamics Yue-Kin Tsang February 9, 2011 1 ontinuum hypothesis In the continuum model of fluids, physical quantities are considered to be varying continuously in space, for example, we may speak

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### 11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

### Chapter 5. Microfluidic Dynamics

Chapter 5 Thermofluid Engineering and Microsystems Microfluidic Dynamics Navier-Stokes equation 1. The momentum equation 2. Interpretation of the NSequation 3. Characteristics of flows in microfluidics

### Chapter 2. Derivation of the Equations of Open Channel Flow. 2.1 General Considerations

Chapter 2. Derivation of the Equations of Open Channel Flow 2.1 General Considerations Of interest is water flowing in a channel with a free surface, which is usually referred to as open channel flow.

### 1.4 Review. 1.5 Thermodynamic Properties. CEE 3310 Thermodynamic Properties, Aug. 26,

CEE 3310 Thermodynamic Properties, Aug. 26, 2011 11 1.4 Review A fluid is a substance that can not support a shear stress. Liquids differ from gasses in that liquids that do not completely fill a container

### Viscous flow through pipes of various cross-sections

IOP PUBLISHING Eur. J. Phys. 28 (2007 521 527 EUROPEAN JOURNAL OF PHYSICS doi:10.1088/0143-0807/28/3/014 Viscous flow through pipes of various cross-sections John Lekner School of Chemical and Physical

### Chapter 4. Dimensionless expressions. 4.1 Dimensional analysis

Chapter 4 Dimensionless expressions Dimensionless numbers occur in several contexts. Without the need for dynamical equations, one can draw a list (real or tentative) of physically relevant parameters,

### Notes on Polymer Rheology Outline

1 Why is rheology important? Examples of its importance Summary of important variables Description of the flow equations Flow regimes - laminar vs. turbulent - Reynolds number - definition of viscosity

### Lecture 3: Simple Flows

Lecture 3: Simple Flows E. J. Hinch In this lecture, we will study some simple flow phenomena for the non-newtonian fluid. From analyzing these simple phenomena, we will find that non-newtonian fluid has

### When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.

Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs

### Turbulence: The manifestation of eddies and their role in conservation laws

Turbulence: The manifestation of eddies and their role in conservation laws George E. Hrabovsky MAST Presentation Given to the Chaos and Complex Systems Seminar University of Wisconsin - Madison 26 February,

### TYPES OF FLUID FLOW. Laminar or streamline flow. Turbulent flow

FLUID DYNAMICS We will deal with Intrinsic properties of fluids Fluids behavior under various conditions Methods by which we can manipulate and utilize the fluids to produce desired results TYPES OF FLUID

### Fluent Software Training TRN Boundary Conditions. Fluent Inc. 2/20/01

Boundary Conditions C1 Overview Inlet and Outlet Boundaries Velocity Outline Profiles Turbulence Parameters Pressure Boundaries and others... Wall, Symmetry, Periodic and Axis Boundaries Internal Cell

### Lecture 6 - Boundary Conditions. Applied Computational Fluid Dynamics

Lecture 6 - Boundary Conditions Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (2002-2006) Fluent Inc. (2002) 1 Outline Overview. Inlet and outlet boundaries.

### Lecture 24 - Surface tension, viscous flow, thermodynamics

Lecture 24 - Surface tension, viscous flow, thermodynamics Surface tension, surface energy The atoms at the surface of a solid or liquid are not happy. Their bonding is less ideal than the bonding of atoms

### HWR 431 / 531 HYDROGEOLOGY LAB SECTION LABORATORY 2 DARCY S LAW

HWR 431 / 531 HYDROGEOLOGY LAB SECTION LABORATORY 2 DARCY S LAW Introduction In 1856, Henry Darcy, a French hydraulic engineer, published a report in which he described a series of experiments he had performed

### High Speed Aerodynamics Prof. K. P. Sinhamahapatra Department of Aerospace Engineering Indian Institute of Technology, Kharagpur

High Speed Aerodynamics Prof. K. P. Sinhamahapatra Department of Aerospace Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 06 One-dimensional Gas Dynamics (Contd.) We

### Introduction to Microfluidics. Date: 2013/04/26. Dr. Yi-Chung Tung. Outline

Introduction to Microfluidics Date: 2013/04/26 Dr. Yi-Chung Tung Outline Introduction to Microfluidics Basic Fluid Mechanics Concepts Equivalent Fluidic Circuit Model Conclusion What is Microfluidics Microfluidics

### NUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES

Vol. XX 2012 No. 4 28 34 J. ŠIMIČEK O. HUBOVÁ NUMERICAL ANALYSIS OF THE EFFECTS OF WIND ON BUILDING STRUCTURES Jozef ŠIMIČEK email: jozef.simicek@stuba.sk Research field: Statics and Dynamics Fluids mechanics

### SIZE OF A MOLECULE FROM A VISCOSITY MEASUREMENT

Experiment 8, page 1 Version of April 25, 216 Experiment 446.8 SIZE OF A MOLECULE FROM A VISCOSITY MEASUREMENT Theory Viscous Flow. Fluids attempt to minimize flow gradients by exerting a frictional force,

### CE 3500 Fluid Mechanics / Fall 2014 / City College of New York

1 Drag Coefficient The force ( F ) of the wind blowing against a building is given by F=C D ρu 2 A/2, where U is the wind speed, ρ is density of the air, A the cross-sectional area of the building, and

### Dimensionless form of equations

Dimensionless form of equations Motivation: sometimes equations are normalized in order to facilitate the scale-up of obtained results to real flow conditions avoid round-off due to manipulations with

### Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

### Physics-biophysics 1

Physics-biophysics 1 Flow of fluids Péter Makra University of Szeged 2011-2012. autumn semester Version: 1.1 Latest update: 29th September 2011 Péter Makra (University of Szeged) Physics-biophysics 1 2011-2012.

### Transport Phenomena I

Transport Phenomena I Andrew Rosen December 14, 013 Contents 1 Dimensional Analysis and Scale-Up 4 1.1 Procedure............................................... 4 1. Example................................................

### Chapter 7. Potential flow theory. 7.1 Basic concepts. 7.1.1 Velocity potential

Chapter 7 Potential flow theory Flows past immersed bodies generate boundary layers within which the presence of the body is felt through viscous effects. Outside these boundary layers, the flow is typically

### ATM 316: Dynamic Meteorology I Final Review, December 2014

ATM 316: Dynamic Meteorology I Final Review, December 2014 Scalars and Vectors Scalar: magnitude, without reference to coordinate system Vector: magnitude + direction, with reference to coordinate system

### For Water to Move a driving force is needed

RECALL FIRST CLASS: Q K Head Difference Area Distance between Heads Q 0.01 cm 0.19 m 6cm 0.75cm 1 liter 86400sec 1.17 liter ~ 1 liter sec 0.63 m 1000cm 3 day day day constant head 0.4 m 0.1 m FINE SAND

### Let s first see how precession works in quantitative detail. The system is illustrated below: ...

lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

### Abaqus/CFD Sample Problems. Abaqus 6.10

Abaqus/CFD Sample Problems Abaqus 6.10 Contents 1. Oscillatory Laminar Plane Poiseuille Flow 2. Flow in Shear Driven Cavities 3. Buoyancy Driven Flow in Cavities 4. Turbulent Flow in a Rectangular Channel

### LECTURE 5: Fluid jets. We consider here the form and stability of fluid jets falling under the influence of gravity.

LECTURE 5: Fluid jets We consider here the form and stability of fluid jets falling under the influence of gravity. 5.1 The shape of a falling fluid jet Consider a circular orifice of radius a ejecting

### CBE 6333, R. Levicky 1 Fluid Kinematics and Constitutive Laws

CBE 6333, R. Levicky 1 Fluid Kinematics and Constitutive Laws In using the differential balance equations derived previously, it will be insightful to better understand the basic types of motion that a

### Practice Problems on Viscosity. free surface. water. y x. Answer(s): base: free surface: 0

viscosity_01 Determine the magnitude and direction of the shear stress that the water applies: a. to the base b. to the free surface free surface U y x h u water u U y y 2 h h 2 2U base: yx y 0 h free

### Practice Problems on the Navier-Stokes Equations

ns_0 A viscous, incompressible, Newtonian liquid flows in stead, laminar, planar flow down a vertical wall. The thickness,, of the liquid film remains constant. Since the liquid free surface is eposed

### A fundamental study of the flow past a circular cylinder using Abaqus/CFD

A fundamental study of the flow past a circular cylinder using Abaqus/CFD Masami Sato, and Takaya Kobayashi Mechanical Design & Analysis Corporation Abstract: The latest release of Abaqus version 6.10

### CE 204 FLUID MECHANICS

CE 204 FLUID MECHANICS Onur AKAY Assistant Professor Okan University Department of Civil Engineering Akfırat Campus 34959 Tuzla-Istanbul/TURKEY Phone: +90-216-677-1630 ext.1974 Fax: +90-216-677-1486 E-mail:

### The Viscosity of Fluids

Experiment #11 The Viscosity of Fluids References: 1. Your first year physics textbook. 2. D. Tabor, Gases, Liquids and Solids: and Other States of Matter (Cambridge Press, 1991). 3. J.R. Van Wazer et

### Lecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2)

Lecture 3. Turbulent fluxes and TKE budgets (Garratt, Ch 2) In this lecture How does turbulence affect the ensemble-mean equations of fluid motion/transport? Force balance in a quasi-steady turbulent boundary

### Chapter 6 Circular Motion

Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example

### 4 Microscopic dynamics

4 Microscopic dynamics In this section we will look at the first model that people came up with when they started to model polymers from the microscopic level. It s called the Oldroyd B model. We will

### Gravity waves on water

Phys374, Spring 2006, Prof. Ted Jacobson Department of Physics, University of Maryland Gravity waves on water Waves on the surface of water can arise from the restoring force of gravity or of surface tension,

### Chapter 5 MASS, BERNOULLI AND ENERGY EQUATIONS

Fluid Mechanics: Fundamentals and Applications, 2nd Edition Yunus A. Cengel, John M. Cimbala McGraw-Hill, 2010 Chapter 5 MASS, BERNOULLI AND ENERGY EQUATIONS Lecture slides by Hasan Hacışevki Copyright

### Laminar to Turbulent Transition in Cylindrical Pipes

Course I: Fluid Mechanics & Energy Conversion Laminar to Turbulent Transition in Cylindrical Pipes By, Sai Sandeep Tallam IIT Roorkee Mentors: Dr- Ing. Buelent Unsal Ms. Mina Nishi Indo German Winter Academy

### CE 6303 MECHANICS OF FLUIDS L T P C QUESTION BANK PART - A

CE 6303 MECHANICS OF FLUIDS L T P C QUESTION BANK 3 0 0 3 UNIT I FLUID PROPERTIES AND FLUID STATICS PART - A 1. Define fluid and fluid mechanics. 2. Define real and ideal fluids. 3. Define mass density

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### Physics 1A Lecture 10C

Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

### Viscous Flow in Pipes

Viscous Flow in Pipes Excerpted from supplemental materials of Prof. Kuang-An Chang, Dept. of Civil Engin., Texas A&M Univ., for his spring 2008 course CVEN 311, Fluid Dynamics. (See a related handout

### Chemistry 223: Collisions, Reactions, and Transport David Ronis McGill University

Chemistry 223: Collisions, Reactions, and Transport David Ronis McGill University. Effusion, Surface Collisions and Reactions In the previous section, we found the parameter b by computing the average

### Perfect Fluidity in Cold Atomic Gases?

Perfect Fluidity in Cold Atomic Gases? Thomas Schaefer North Carolina State University 1 Hydrodynamics Long-wavelength, low-frequency dynamics of conserved or spontaneoulsy broken symmetry variables τ