1. INTRODUCTION TO FLUID MECHANICS 1.1 Definition Of A
1. INTRODUCTION TO FLUID MECHANICS1.1 Definition of a FluidThe difference between solid state and a fluid can be explained under the umbrella of shearstress which is the force on a unit area in which force is parallel to the surface. A solid bodycan resist shear stress, fluids can’t. A fluid starts moving even with the smallest shear stressacting on it. Fluids can be in the form of either liquids or gases, and must be distinguished.Liquids form a free surface under gravity forces, while gases expand until an obstacle stopsthem from further expansion. (Figure 1.1).Figure 1.1 Fluid (left vessel) forms a free surface; Gas (right vessel) expandsEngineers need a more formal and precise definition of a fluid: A fluid is a substance thatdeforms continuously under the application of a shear (tangential) stress no matter how smallthe shear stress may be. Because the fluid motion continues under the application of a shearstress, we can also define a fluid as any substance that cannot sustain a shear stress when atrest.Hence liquids and gases (or vapors) are the forms, or phases, that fluids can take. We wish todistinguish these phases from the solid phase of matter. We can see the difference betweensolid and fluid behaviour in Figure 1.2. If we place a specimen of either substance betweentwo plates (Fig. 1.2a) and then apply a shearing force F, each will initially deform (Fig. 1.2b);however, whereas a solid will then be at rest (assuming the force is not large enough to gobeyond its elastic limit), a fluid will continue to deform (Fig. 1.2c, Fig. 1.2d, etc) as long asthe force is applied.Figure 1.2 Difference in behaviour of a solid and a fluid due to a shear forceEvery flow obeys the three conservation fundamentals of mechanics: Mass conservation (continuity constraint) Momentum conservation (Newton’s 2nd law) Energy conservation (1st law of thermodynamics)-1-
To describe the flow furthermore, thermodynamic equations of state and boundary conditionshave to be considered. The methods, which are available to describe and analyze the flow, canbe subdivided into three basic concepts:Integral approach of a control volume (control room).This method is used for the derivation of conservation equations. A control volume is alimited area with open borders, where mass, momentum and energy can flow through it. Thusit must be possible to draw a balance between incoming, outgoing flows and the variationswithin the control volume. If the details of the flow within the control volume are of nointerests and therefore an integral result is enough, this approach is suggested.Differential approach on a fluid element (infinitesimal system).In order to receive the differential equations of the fluid flow, the conservation equations haveto be derived at an infinitesimal fluid element. To apply this on a defined problem, thedifferential equations have to be mathematically integrated under consideration of therespective boundary condition. It is difficult to receive accurate analytical solutions; this isonly the case for simple geometries and boundary conditions. Therefore numerical methodsare used, in general, to obtain a solution by approximation with the help of computers.Experimental analysis (dimension analysis).In many cases a well thought-out experiment is used to examine a specific flow problem.Especially in those cases, when a mathematical numerical description is not possible, neitherintegral nor differential. For example, when there is no model theory existing for the definedproblem or a numerical approach because of its complex geometries is not workable or is toocostly then an experimental analysis may be necessary.1.2 Method of analysisThe first step in solving a problem is to define the system that you are attempting to analyze.In basic mechanics, we made extensive use of the free-body diagram. We will use a system ora control volume, depending on the problem being studied. In our study of fluid mechanics,we will be most interested in conservation of mass and Newton’s second law of motion and inenergy; in fluid mechanics it will mainly be forces and motion. We must always be aware ofwhether we are using a system or a control volume approach because each leads to differentmathematical expressions of these laws.1.2.1 System and Control VolumeThe observation of flow processes within a defined control volume enables a relatively simplemathematical description of a problem. A control volume may be selected in one of severaldifferent ways. The principle is depicted in Figure 1.3. Water flows into the container shownon the left through two pipes, and flows out through only one pipe. In this case, the containeris the control volume. Application of this control volume can be used to answer questionssuch as how the water level in the container changes dependent on the flow rate in the pipes.-2-
The pipes themselves are called “control cross-sections”, as a change in the control volumecan be induced only through inflow or outflow across these cross-sections. In the illustrationon the right, the control volume travels along with the moving, swimming duck. In this case itis only reasonable that the velocity of the control volume and of the duck are the same.Figure 1.3 Control volumes; stationary (left) and moving (right).A control volume is an arbitrary volume in space through which fluid flows. The geometricboundary of the control volume is called the control surface. The control surface may be realor imaginary; it may be at rest or in motion. Figure 1.4 shows flow through a pipe junction,with a control surface drawn on it. Note that some regions of the surface correspond tophysical boundaries (the walls of the pipe) and others (at locations 1, 2, and 3) are parts of thesurface that are imaginary (inlets or outlets). For the control volume defined by this surface,we could write equations for the basic laws and obtain results such as the flow rate at outlet 3given the flow rates at inlet and outlet 2, the force required to hold the junction in place, andso on. It is always important to take care in selecting a control volume, as the choice has a bigeffect on the mathematical form of the basic laws.Figure 1.4 Fluid flow through a pipe junction.1.3 Eulerian and Lagrangian ApproachEuler was born in Switzerland, in the town of Basel, on the 15th of April 1707. At that time,Basel was one of the main centres of mathematics in Europe. At the age of 7, Euler startedschool while his father hired a private mathematics tutor for him. At 13, Euler was alreadyattending lectures at the local university, and in 1723 gained his masters degree, with a-3-
dissertation comparing the natural philosophy systems of Newton and Descartes. On hisfather's wishes, Euler furthered his education by enrolling in the theological faculty, butdevoted all his spare time to studying mathematics. He wrote two articles on reverse trajectorywhich were highly valued by his teacher Bernoulli. In 1727 Euler applied for a position asphysics professor at Basel university, but was turned down.At this time a new centre of science had appeared in Europe - the Petersburg Academy ofSciences. As Russia had few scientists of its own, many foreigners were invited to work atthis centre - among them Euler. On the 24th of May 1727 Euler arrived in Petersburg. Hisgreat talents were soon recognised. Among the areas he worked in include his theory of theproduction of the human voice, the theory of sound and music, the mechanics of vision, andhis work on telescopic and microscopic perception. On the basis of this last work, notpublished until 1779, the construction of telescopes and microscopes was made possible.Lagrange (1736-1813) was an Italian-French Mathematician and Astronomer. By the age of18 he was teaching geometry at the Rotal Artillery School of Turin, where he organizeddiscussion group that became the Turin Academy of Sciences. In 1755, Lagrange sent Euler aletter in which he discussed the Calculus of Variations. Euler was deeply impressed byLagrange's work, and he held back his own work on the subject to let Lagrange publish first.When Euler left Berlin for St. Petersburg, he recommended that Lagrange succeed him as thedirector of the Berlin Academy. Over the course of a long and celebrated career (he would belionized by Marie Antoinette, and made a count by Napoleon before his death), Lagrangepublished a systemization of mechanics using his calculus of variations (Traité de mécaniqueanalytique), and did significant work on the three-body problem and astronomicalperturbations.The studies carried out and shared by Euler and Lagrange can be seen from a web page spondents/Lagrange.htmlFluid mechanics problems can be regarded from two different points of view. First, theLagrangian approach, which tracks the individual particles (molecules) within the current.The covered distance of such a particle is viewed as a function of the time. The Lagrangianmethod is mainly used in the particle mechanics. Consider, for example, the application ofNewton’s second law to a particle of fixed mass. Mathematically, we can write Newton’ssecond law for a system of mass m as F ma m dvd 2r m 2dtdtWe could use this Lagrangian approach to analyze a fluid flow by assuming the fluid to becomposed of a very large number of particles whose motion must be described. However,keeping track of the motion of each fluid particle would become a horrendous bookkeepingproblem. Consequently, a particle description becomes unmanageable.Often we find it convenient to use a different type of description. Particularly with control-4-
volume analyses, it is convenient to use the field, or Eulerian, method of description, whichfocuses attention on the properties of a flow at a given point in space as a function of time. Inthe Eulerian method of description, the properties of a flow field are described as functions ofspace coordinates and time. Thus, fluid dynamics measurements can be better accomplishedusing the Eulerian approach. For example, to measure the velocity or the pressure in a tube(flow), a measurement system is installed at a specific point (x,y,z) within it. The measuringresults then show a pressure field p(x,y,z) or a velocity field v(x,y,z). The velocity field is themost important parameter of a current because other attributes are derivatives of it, e.g. thetransport of water, such as acceleration (a dv/dt).Figure 1.5 Streaklines over an automobiles.1.4 Systems of DimensionsAny valid equation that relates physical quantities must be dimensionally homogeneous; eachterm in the equation must have the same dimensions. We recognize that Newton’s second law( F m a) relates the four dimensions, F, M, L, and t. Thus force and mass cannot both beselected as primary dimensions without introducing a constant of proportionality that hasdimensions (and units). Length and time are primary dimensions in all dimensional systems incommon use. In some systems, mass is taken as a primary dimension. In others, force isselected as a primary dimension.-5-
Question 1-1Determine the dimensions and units of the quantities given below?Area, volume, velocity, angular velocity, acceleration, pressure, stress, energy, power,density, dynamic viscosity and kinematic viscosity.Solution 1-1There are only four primary dimensions in fluid mechanics from which all other dimensionscan be derived. These are mass [M], length [L], time [T], temperature [Θ]. Secondarydimensions are those derived as combinations of the primary dimensions, for example:velocity, acceleration and force.Newton’s Second Law:F m x a1 Newton Force (1 kg) (m/sec2)so dimensionally we can see that force has the units of mass times length divided by timesquared:[F] [MLT-2]SecondaryDimensionAreaVolumeVelocityAngular velocityAccelerationPressure or StressEnergy, Heat, WorkPowerDensityDynamic ViscosityKinematic ][ML-1T-2][ML2T-2][ML2T-3][ML-3][ML-1T-1][L2T-1]SI Unitm2m3m/s1/sm/s2Pa kg/(ms2)J NmW J/skg/m3kg/(ms)m2/sIn engineering, we strive to make equations and formulas have consistent dimensions. That is,each term in an equation, and obviously both sides of the equation, should be reducible to thesame dimensions. For example, a very important equation we will derive later on is theBernoulli equationP1 V12P2 V22 z1 z2 g 2 g g 2 gwhich relates the pressure P, velocity V, and elevation z between points 1 and 2 along astreamline for a steady, frictionless incompressible flow (density ρ). This equation isdimensionally consistent because each term in the equation can be reduced to dimension of[L]Almost all equations you are likely to encounter will be dimensionally consistent. However,you should be alert to some still commonly used equations that are not; these are often-6-
“engineering” equations derived many years ago, or are empirical (based on experiment ratherthan theory). For example, civil engineers often use the semi-empirical Manning equation:R 2 3 S o1 2V n1.5 SummaryIn this chapter we introduced or reviewed a number of basic concepts and definitions,including: How fluids are defined, and the no-slip condition System/control volume concepts Lagrangian and Eulerian descriptions Units and dimensionsQuestions to solve:1 For each quantity listed, indicate dimensions using mass as a primary dimension, and givetypical SI units:(a) Power(b) Pressure(c) Modulus of elasticity(d) Angular velocity(e) Energy(f) Moment of a force(g) Momentum(h) Shear stress(i) Strain(j) Angular momentum2 For each quantity listed, indicate dimensions using force as a primary dimension, and givetypical SI units:(a) Power(b) Pressure(c) Modulus of elasticity(d) Angular velocity(e) Energy(f) Momentum(g) Shear stress(h) Angular momentum3 The drag force FD on a body is given by1FD V 2 AC D2Hence the drag depends on speed V, fluid density ρ, and body size (indicated by frontal areaA) and shape (indicated by drag coefficient CD). What are the dimensions of CD?-7-
2. FUNDAMENTAL CONCEPTS IN FLUID MECHANICS2.1 Fluid as a ContinuumAs a consequence of the continuum assumption, each fluid property is assumed to have adefinite value at every point in space. Thus, the fluid properties such as density, temperature,velocity, and so on are considered to be continuous functions of position and time. Forexample, if density was measured simultaneously at an infinite number of points in the fluidsystem, we would obtain an expression for the density distribution as a function of the spacecoordinates at a given instant,(2.1) ( x, y, z)The density at a point may also vary with time (as a result of work done on or by the fluidand/or heat transfer to the fluid). Thus the complete representation of density (the fieldrepresentation) is given by(2.2) ( x, y, z, t )Since density is a scalar quantity, requiring only the specification of a magnitude for acomplete description, the field represented by Equation (2.2) is a scalar field. An alternativeway of expressing the density of a substance (solid or fluid) is to compare it to an acceptedreference value, typically the maximum density of water, water (1000 kg/m3 at 4oC). Thus,the specific gravity, SG, of a substance is expressed asSG (2.3) waterFor example, the SG of mercury is typically 13.6—mercury is 13.6 times as dense as water.Appendix A contains specific gravity data for selected engineering materials. The specificgravity of liquids is a function of temperature; for most liquids specific gravity decreases withincreasing temperature. The specific weight, γ, of a substance is another useful materialproperty. It is defined as the weight of a substance per unit volume and given asmg (2.4)VFor example, the specific weight of water is approximately 9.81 kN/m3.2.2 Velocity FieldA very important property defined by a flow is the velocity field, given byV V ( x, y , z , t )(2.5)Velocity is a vector quantity, requiring a magnitude and direction for a complete description.The velocity vector, V, also can be written in terms of its three scalar components.V ui vj wk(2.6)In general, each component, u, v, and w, will be a function of x, y, z, and t. Equation (2.5)indicates the velocity of a fluid particle that is passing through the point x, y, z at time instantt, in the Eulerian sense. We can keep measuring the velocity at the same point or choose anyother point x, y, z at the next time instant.-8-
If properties at every point in a flow field do not change with time, the flow is termed steady.Stated mathematically, the definition of steady flow is 0 twhere η represents any fluid property. Hence, for steady flow, 0or ( x, y, z) tand V 0 tV V ( x, y , z )orIn steady flow, any property may vary from point to point in the field, but all propertiesremain constant with time at every point.2.2.1 One, two and three dimensional flowsAlthough most flow fields are inherently three-dimensional, analysis based on fewerdimensions is frequently meaningful. Consider, for example, the steady flow through a longstraight pipe that has a divergent section, as shown in Fig. 2.1 In this example, we are usingcylindrical coordinates (r, θ, x). We will learn in the next chapters that the velocitydistribution in this pipe may be described as r 2 u u max 1 R (2.7)This is shown on the left of Fig. 2.1. The velocity u(r) is a function of only one coordinate,and so the flow is one-dimensional. On the other hand, in the diverging section, the velocitydecreases in the x direction, and the flow becomes two-dimensional: u u(r.x)The term uniform flow field is used to describe a flow in which the velocity is constant. i.e.,independent of all space coordinates, throughout the entire flow field.Figure 2.1 Examples of one and two dimensional flows-9-
2.3 Timelines, Pathlines, Streaklines and Streamlines.If a number of adjacent fluid particles in a flow field are marked at a given instant, they forma line in the fluid at that instant; this line is called a timeline. Subsequent observations of theline may provide information about the flow field.A pathline is the path or trajectory traced out by a moving fluid particle. To make a pathlinevisible we might identify a fluid particle at a given instant.Streamlines are lines drawn in the flow field so that at a given instant they are tangent to thedirection of flow at every point in the flow field. Since the streamlines are tangent to thevelocity vector at every point in the flow field, there can be no flow across a streamline.Streamlines are the most commonly used visualization technique.In steady flow the velocity at each point in the flow field remains constant with time and,consequently the streamline shapes do not vary from one instant to the next. This implies thata particle located on a given streamline will always move along the same streamline.Furthermore, consecutive particles passing through a fixed point in space will be on the samestreamline and, subsequently, will remain on this streamline. Thus, in a steady flow, pathlines,streaklines and streamlines are identical lines in the flow field.Figure 2.2 shows a photograph of five streaklines for flow over an automobile in a windtunnel. A streakline is the line produced in a flow when all particles moving through a fixedpoint are marked in some way. We can also define them as streamlines. These are lines drawnin the flow field so that at a given instant they are tangent to the direction of flow at everypoint in the flow field. Since the streamlines are tangent to the velocity vector at every pointin the flow field, there is no flow across a streamline. Pathlines are, as the name implies,showing the paths of individual particles. Finally, timelines are created by marking a line in aflow and watching how it evolves over time.Figure 2.2 Streaklines over an automobile in a wind tunnelFigure 2.2 shows streaklines, but in fact the pattern shown also represents streamlines andpahtlines. This is due to the conclusion that for steady flow streaklines, streamlines andpathlines are identical.- 10 -
We can use the velocity field to derive the shapes of streaklines, pathlines, and streamlines.Starting with streamlines: Because the streamlines are parallel to the velocity vector, we canwrite (for 2D)dy v ( x, y ) (2.8) dx streamline u ( x, y )Note that streamlines are obtained at an instant in time; if the flow is unsteady, time t is heldconstant in Eq. 2.8. Solution of this equation gives the equation y y(x), with an undeterminedintegration constant, the value of which determines the particular streamline.For pathlines (again considering 2D), we let x xp(t) and y yp(t) where xp(t) and yp(t) are theinstantaneous coordinates of a specific particle. We then getdx u ( x, y , t )(2.9) dt particledy v ( x, y , t ) dt particle(2.10)The simultaneous solution of these equations gives the path of a particle in parametric formxp(t), yp(t).Question 2-1A velocity field is given by V Axi Ayj ; the units of velocity are m/s; x and y are given inmeters; A 0.3s-1(a) Obtain an equation for the streamlines in the xy plane.(b) Plot the streamline passing through the point (x0, y0) (2, 8).(c) Determine the velocity of a particle at the point (2, 8).(d) If the particle passing through the point (x0, y0) (2, 8) is marked at time t 0, determine thelocation of the particle at time t 6 s.(e) What is the velocity of this particle at time t 6 sec(f) Show that the equation of the particle path is the same as the equation of the streamline.Solution 2-1(a)Streamlines are lines drawn in the flow field such that, at a given instant, they are tangentto the direction of flow at every point. Consequently,dy v( x, y ) Ay y dx streamline u ( x, y )AxxSeparating variables and integrating, we obtaindydx y xorln y ln x c1- 11 -
This can be written as xy c(b) For the streamline passing through the point (x0, y0) (2, 8) the constant, c, has a value of16 and the equation of the streamline through the point (2, 8) isxy x0 y 0 16m 2The plot is as sketched below.(c) The velocity field is V Axi Ayj . At the point (2, 8) the velocity isV A( xi yj ) 0.3s 1 (2i 8 j )m 0.6i 2.4 j m/sec(d) A particle moving in the flow field will have velocity given byV Axi AyjThusdydx Ay Ax and v p dtdtSeparating variables and integrating (in each equation) givesup xtdx x x 0 Adtoandytdy y y 0 AdtoThenlnxy At and ln Atxoyoorx x o e Atand y y o e At- 12 -
At t 6 sec,x (2)e (0.3)6 12.1mAt t 6 sec, the particle is at (12.1, 1.32)mandy (8)e (0.3)6 1.31m(e) At the point (12.1, 1.32) m,V A( xi yj ) 0.3s 1 (12 .1i 1.32 j )m 3.63i 0.396 j m/sec(f) To determine the equation of the pathline, we use the parametric Equationsx x o e Atand y y o e Atand eliminate t. Solving for eAt from both equationsyxe At o yxoThereforexy xo y o 16m 22.4 Stress Field.In our study of fluid mechanics, we will need to understand what kinds of forces act on fluidparticles. Each fluid particle can experience: surface forces (pressure, friction) that aregenerated by contact with other particles or a solid surface; and body forces (such as gravityand electromagnetic) that are experienced throughout the particle. The gravitational bodyforce acting on an element of volume, dV, is given by ρgdV where ρ is the density (mass perunit volume) and g is the local gravitational acceleration. Thus the gravitational body forceper unit volume is ρg and the gravitational body force per unit mass is g.Surface forces on a fluid particle lead to stresses. The concept of stress is useful for describinghow forces acting on the boundaries of a medium (fluid or solid) are transmitted throughoutthe medium. The difference between a fluid and a solid is, as we’ve seen, that stresses in afluid are mostly generated by motion rather than by deflection.Imagine the surface of a fluid particle in contact with other fluid particles, and consider thecontact force being generated between the particles. We consider the stress on the elementδAx, whose outwardly drawn normal is in the x direction. The force, δF, has been resolvedinto components along each of the coordinate directions. Dividing the magnitude of eachforce component by the area, δAx, and taking the limit as δAx approaches zero, we define thethree stress components shown in Fig. 2.3.Referring to the infinitesimal element as shown in Fig. 2.4, we see that there are six planes(two x planes, two y planes, and two z planes) on which stresses may act. In order todesignate the plane of interest, we could use terms like front and back, top and bottom, or leftand right. However, it is more logical to name the planes in terms of the coordinate axes. Theplanes are named and denoted as positive or negative according to the direction of the- 13 -
outwardly drawn normal to the plane.Figure 2.3 Force and stress components on the element of area δAxThus the top plane, for example, is a positive y plane and the back plane is a negative z plane.It is also necessary to adopt a sign convention for stress. A stress component is positive whenthe direction of the stress component and the plane on which it acts are both positive or bothnegative. Thus τyx 3.5N/cm2 represents a shear stress on a positive y plane in the positive xdirection or a shear stress on a negative y plane in the negative x direction. In Fig. 2.4 allstresses have been drawn as positive stresses. Stress components are negative when thedirection of the stress component and the plane on which it acts are of opposite sign.Figure 2.4 Notation for stresses2.5 Viscosity.We can easily say that solids are elastic, and fluids are viscous (and it’s interesting to note thatmany biological tissues are viscoelastic, meaning they combine features of a solid and afluid). For a fluid at rest, there will be no shear stresses.In solid bodies the shear stress yx is a function of the deforming angle of a mass unit (SeeFig.2.5a) dependent on the material law. In contrast to solid bodies, the shear stress in fluids is- 14 -
a function of the angular deforming speed d dt (Fig. 2.5b). The dynamic viscosity μ of afluid is defined as the ratio of shear stress divided by the velocity gradient. Therefore thedimension is (force)(time)/(distance)2 or Pa·s. Thus the shear stress is proportional to thevelocity gradient du dy , and the proportional factor is the dynamic viscosity:du(2.11)dyThe kinematic viscosity (in m2/s) is the ratio of dynamic viscosity and the mass density: yx (2.12)Figure 2.5 Shear stress in a solid body (a) and in a viscous fluid (b) is in this case only kinematic, because the mass units are eliminated. The viscosity ofliquids and gases depends on the temperature. The pressure in viscosity of fluids usually playsan insignificant role, and thus, it can be neglected.Thus, the fluid element of Fig. 2.5, when subjected to shear stress τyx, experiences a rate ofdeformation (shear rate) given by du/dy. We have established that any fluid that experiences ashear stress will flow. Fluids in which shear stress is directly proportional to rate ofdeformation are Newtonian fluids. The term non-Newtonian is used to classify all fluids inwhich shear stress is not directly proportional to shear rate.Figure 2.6 Flow behaviors of Newtonian and other fluids- 15 -
Note that, since the dimensions of τ are [F/L2] and the dimensions of du/dy are [1/t], μ hasdimensions [Ft/L2]. Since the dimensions of force, F, mass,M, length, L, and time, t, arerelated by Newton’s second law of motion, the dimensions of μ can also be expressed as[M/Lt]. The calculation of viscous shear stress is illustrated in the following question.Question 2-2An infinite plate is moved over a second plate on a layer of liquid as shown. For small gapwidth, d, we assume a linear velocity distribution in the liquid. The liquid viscosity is 0.0065g/cm.s and its specific gravity is 0.88.Determine:(a) The viscosity of the liquid, in N s/m2.(b) The kinematic viscosity of the liquid, in m2/s.(c) The shear stress on the upper plate, in N/m2.(d) The shear stress on the lower plate, in Pa.(e) The direction of each shear stress calculated in parts (c) and (d).Solution 2-2Given: Linear velocity profile in the liquid between infinite parallel plates are as shown.μ 0:0065 g/cm.sSG 0:88Find:(a) μ in units of N s/m2(b) ν in units of m2/s.(c) τ on upper plate in units of N/m2(d) τ on lower plate in units of Pa.(e) Direction of stresses in parts (c) and (d). duin which dy(1) Linear velocity distribution (given)(2) Steady flow(3) μ constantGoverning equation: Assumptions:- 16 -
0.0065g1kg100cm cm.s 1000 gm 6.5 10 4 kg / m.secConvert the kilograms into Newton by using the Newtons Second Law1Newton1kg m / sec2kg1N1 0.00065 2m.s m / sec kg 6.5 10 4 N .s / m 2 SG H SG H 2O2O 46.5 10 N .s / m 2(0.88)1000 kg m3 7.39 10 7 m 2 / s du dy y dSince u varies linearly with respect to y,du u U 0 U dy y d 0 dm1mm 1000 1000s 1s 0.3mmmU1000 6.5 10 4 N .s / m 2 0.65 N / m 2ds 0.3 upperUPa.m 2 0.65 N / m 2 0.65PadNDirections of shear stresses on upper and lower plates are as shown in the figure below. Theupper plate is a negative y surface; so positive τ acts in the negative x direction. The lowerplate is a positive y surface; so positive τ acts in the positive x direction. lower - 17 -
2.6 Description and Classification of Fluid Motions.Fluid mechanics is a huge discipline; it covers everything from the aerodynamics of asupersonic transport vehicle to the lubrication of human joints by fluid. We need to breakfluid mechanics down into manageable proportions. It turns out that the two most difficultaspects of a fluid mechanics analysis to deal with are: (1) the fluid’s viscous natur
INTRODUCTION TO FLUID MECHANICS 1.1 Definition of a Fluid . Engineers need a more formal and precise definition of a fluid: A fluid is a substance that deforms continuously under the application of a shear (tangential) stress no matter how small the shear stress may be. Because the fluid
Fluid Mechanics Fluid Engineers basic tools Experimental testing Computational Fluid Theoretical estimates Dynamics Fluid Mechanics, SG2214 Fluid Mechanics Definition of fluid F solid F fluid A fluid deforms continuously under the action of a s
Fluid Mechanics 63 Chapter 6 Fluid Mechanics _ 6.0 Introduction Fluid mechanics is a branch of applied mechanics concerned with the static and dynamics of fluid - both liquids and gases. . Solution The relative density of fluid is defined as the rate of its density to the density of water. Thus, the relative density of oil is 850/1000 0.85.
Applied Fluid Mechanics 1. The Nature of Fluid and the Study of Fluid Mechanics 2. Viscosity of Fluid 3. Pressure Measurement 4. Forces Due to Static Fluid 5. Buoyancy and Stability 6. Flow of Fluid and Bernoulli's Equation 7. General Energy Equation 8. Reynolds Number, Laminar Flow, Turbulent Flow and Energy Losses Due to Friction
Fundamentals of Fluid Mechanics. 1 F. UNDAMENTALS OF . F. LUID . M. ECHANICS . 1.1 A. SSUMPTIONS . 1. Fluid is a continuum 2. Fluid is inviscid 3. Fluid is adiabatic 4. Fluid is a perfect gas 5. Fluid is a constant-density fluid 6. Discontinuities (shocks, waves, vortex sheets) are treated as separate and serve as boundaries for continuous .
Chapter 06 Fluid Mechanics _ 6.0 Introduction Fluid mechanics is a branch of applied mechanics concerned with the static and dynamics of fluid - both liquids and gases. The analysis of the behavior of fluids is based on the fundamental laws of mechanics, which relate continuity of
L M A B CVT Revision: December 2006 2007 Sentra CVT FLUID PFP:KLE50 Checking CVT Fluid UCS005XN FLUID LEVEL CHECK Fluid level should be checked with the fluid warmed up to 50 to 80 C (122 to 176 F). 1. Check for fluid leakage. 2. With the engine warmed up, drive the vehicle to warm up the CVT fluid. When ambient temperature is 20 C (68 F .
An Introduction to Fluid Mechanics School of Civil Engineering, University of Leeds. CIVE1400 FLUID MECHANICS Dr Andrew Sleigh May 2001 Table of Contents 0. CONTENTS OF THE MODULE 3 0.1 Objectives: 3 0.2 Consists of: 3 0.3 Specific Elements: 4 0.4 Books: 4 0.5 Other Teaching Resources. 5 0.6 Civil Engineering Fluid Mechanics 6 0.7 System of units 7
Motion of a Fluid ElementMotion of a Fluid Element 1. 1. Fluid Fluid Translation: The element moves from one point to another. 3. 3. Fluid Fluid Rotation: The element rotates about any or all of the x,y,z axes. Fluid Deformation: 4. 4. Angular Deformation:The element's angles between the sides Angular Deformation:The element's angles between the sides
