Fluid Mechanics EBS 189a. Winter quarter, 4 units, CRN Lecture TWRF 12:10-1:00, Chemistry 166; Office hours TH 2-3, WF 4-5; 221 Veihmeyer Hall.

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1 Flui Mechanics EBS 189a. Winter quarter, 4 units, CRN Lecture TWRF 12:10-1:00, Chemistry 166; Office hours TH 2-3, WF 4-5; 221 eihmeyer Hall. Course Description: xioms of flui mechanics, flui statics, kinematics, velocity fiels for one-imensional incompressible flow incluing bounary layers, turbulent flow time averaging, potential flow, imensional analysis, an macroscopic balances to solve a range of practical problems. Concepts: Continuum approach to eforming physical/biological systems, transport theorem integral analysis, stress vector an stress tensor analysis, microscopic an macroscopic analysis of mass as well as linear an angular momentum, ownscaling for information retrieval, application of theory to solve practical problems. Goal: To apply knowlege of mathematics, science an engineering to natural an engineere systems. To use engineering methos to ientify, formulate an solve problems. Prepare for stuy of heat an mass transfer in physical/biological systems. Prerequisites: PHY 9B an MTH 21D (MTH 22 an 22B recommene. Instructor(s: Wes Wallener, Professor, 221 eihmeyer Hall, wwwallener@ucavis.eu, , http: enthusiasm.ucavis.eu (expane course outline. Purnenu Singh, Reaer. Text: Introuction to Flui Mechanics. S. Whitaker. R.E. Kreiger Publishing Co Graing: Two miterms 20% each, final exam 40%, homework 20% (ue class perio after assigne, no creit if late. Brief Course Outline: xioms of Flui Mechanics Mass an Momentum Principles, ector Invariance, Stress ector, Stress Tensor. Statics Fluis at Rest, Forces on Submerge Surfaces. Kinematics Transport Theorems an Mass Conservation, pplication of Macroscopic Mass Balance, Stream Function, Cauchy s First an Secon Equations, iscous Stress, Rate of Strain an orticity Tensors, Physical Interpretation of the Rate of Strain an the orticity Tensors, elocity Potential an Stream Function, Newton s Law of iscosity, the Equations of Motion, Navier Stokes Equation, pplications. Empiricism Dimensional nalysis, Transition an Turbulent Flow, Time verage Continuity an Navier Stokes Equation, Physical Interpretation of Turbulence, Ey iscosity an Prantl s Mixing Length Theory, pplication to Turbulent Pipe Flow. Macroscopic Balances an Downscaling Bernoulli s Equation, Moving Control olumes an Inertial Frames, Mechanical Energy Balance, pplications, Turbulent Flow in Pipes, Friction Factors, Pipeline Design. Prepare by Wes Wallener

2 Expane Course Outline, Winter Date Lecture/Topic Stuy Homework Mass an momentum principles. continuous material boy an the Euler Cut are introuce an the stress vector is shown to be a irection epenent vector. Mass, linear momentum an angular momentum conservation principles are reviewe. Relations between Newton s laws to Euler s laws are evelope. Note that for ector invariance. Reference an inertial frames are efine. Transformations of base vectors an components of vectors arise from invariance. The summation an free inex notation as well as the Kronecker elta are powerful tools use in mechanics Stress vector. Stress is a oubly irecte quantity an Cauchy s lemma reveals its nature an supports the evelopment of Cauchy s Funamental Theorem, evelope in the next lecture. The projecte area theorem as well as the projection operator (tensor, with which you are alreay familiar, are tools necessary to unerstan the evelopment of Cauchy s Funamental Theorem Stress tensor. Cauchy s Funamental Theorem provies the relationship between the stress tensor an the stress vector. Sec. 1.3 an pages an rea Ch. 1. Review Sec. 4.3 an Ch. 13 in Calculus an nalytic Geometry by Stein an Barcellos, Sec. 1.6 Sec. 4.2 an pages Stein an Barcellos ch. 12. Sec. 4.2 pply Euler s Secon Law to a two particle system using Euler cuts aroun particle I, particle II an particles I an II. t rˆxρvˆ = rˆ rˆ t (rˆ cm cm x m( ρvˆ Use, rˆ = vˆ t â bˆ (âxbˆ = xbˆ + âx t t t Show that Euler s Secon Law is restricte to the strong form of Newton s Thir Law. 1. Using vector invariance, show how prime basis vectors can be transforme into unprime basis vectors. 2. Show how to fin the transformation of vector components from the prime to the unprime coorinate system. (Hint: use orthogonality conition shown in class. 3. Using vector invariance, show how unprime basis vectors can be transforme into prime basis vectors. Fin another orthogonality conition starting with the prime basis vectors. Show that a B = B T a, in which a is a first orer tensor (a vector an B is a secon orer tensor, using mixe notation. t 1

3 Static Flui. Flui uner no shear stress is static an the normal to the surface an the stress vector are collinear. The graient, ivergence an Stokes theorems are the relations use to fin the point or fiel equations of mass an momentum conservation from the integral equations. These equations are integrate for arbitrary control volumes to provie the ensity, pressure an velocity fiels Forces on Submerge Surfaces. Euler s first equation is integrate to calculate the forces on curve surfaces an to erive rchemies Principle. The projecte area theory simplifies the calculations for complex geometries Kinematics. Material an spatial escriptions of moving particles are the founation for etermining their position, velocity an acceleration uring eformation. When the ientifie particles in the observe system o not change, the erivative is efine as the material erivative. If ifferent particles are consiere over time, the erivative is efine as the general erivative. Streamlines, path lines an streak lines iffer accoring to the particles observe an tracke Transport Theorems an Mass Conservation. Time erivatives of material volume integrals have alreay appeare in Euler s first an secon laws as well as the mass conservation equation. To evelop the microscopic point equations of motion an mass conservation, the orer of ifferentiation an ifferentiation must be reverse, an in so oing time erivatives of volume integrals can be forme as integrals of erivatives. Furthermore, in macroscopic analysis presente later in the course, erivatives of volume integrals which are not material, will appear Sec. 2.2 an Use the ivergence theorem to show that rˆxsnˆ = 2. Problem 2.1. Sec Problem 2.8 Pages an Secs. 3.4 an 3.5 rˆx S Problems 3.1 an 3.2 Show the evelopment the Special form of the Reynol s Transport Theorem (Hint: let S=?v. an Leibnitz rule will be use reverse the operations pplication of Macroscopic Mass Balance. Sec. 7.1 Flow in veins an arteries is a transient process in which elastic conuits expan an contract. Consier an artery having a raial velocity at the inner raius of cm/s. The length L of the artery is 13 cm an the volumetric flow rate at the entrance of the artery is 0.3 cm 3 /s. t some instant of time, the inner raius of the artery is 0.15 cm. t that particular moment, what is the volumetric flow rate at the artery exit? (hint: =r?l in which is cross sectional area, r is the raius an? is the angle in raians. 2

4 Stream Function. The velocity fiel is calculate as the graient of a scalar fiel for the special case of steay incompressible flow. The resulting scalar stream function satisfies the continuity equation exactly Monay classes meet on this Friay. No Class Cauchy s First an Secon Equations. The special form of the Reynols transport theorem applie to Euler s first law transforms the integral equation into the point or fiel equation of linear momentum incluing the stress tensor. Representing the stress vector in Euler s first law as a function of the stress tensor allowe the transformation of the area integral into a volume integral. The ivergence theorem, Reynols transport theorem, Cauchy s lemma an Cauchy s first equation transform Euler s secon law, an integral equation, into the point or fiel equation showing the symmetry of the stress tensor. This is Cauchy s secon equation iscous Stress, Rate of Strain an orticity Tensors. The viscous stress tensor is efine as a function of pressure an the rate of eformation of the flui via the strain tensor. The velocity graient tensor is ecompose into the symmetric rate of strain tensor an the vorticity tensor which oes not contribute to the rate of eformation of the flui Physical Interpretation of the Rate of Strain an the orticity Tensors. From the velocity graient calculate the rate of stretching of a line element, the rate of angle change between material line elements an the rate of rotation (rigi boy rotation. Relate the vorticity tensor an vorticity vector elocity Potential an Stream Function. Stokes theorem is use to show that irrotationality an simple topological connectivity are the conitions for the velocity to be etermine as the graient of a three imensional scalar fiel f. The ifferential of the scalar fiel f is exact an conservative. Because the stream function? an f are harmonic an they satisfy the Cauchy-Riemann equations, the lines of constant? an f are orthogonal. Pages Pages Sec. 5.5 If a stream function exists for a velocity fiel of 2 2 v = a(x y v x y = 2axy v z = 0 in which a is parameter, fin the stream function?. Use the special form of the Reynols transport theorem to show that t rˆxρvˆ = ρ D Dt (rˆxvˆ For the plane Couette Flow illustrate in Figure 5.3-1, fin the rate of strain in the? irection,? that maximizes rate of strain, the rate of ecrease of the angle between unit vectors initially in the x an y irections an the components of the vorticity vector. If a velocity potential function exists for a velocity fiel of 2 2 v = a(x y v x y = 2axy v = 0 z in which a is parameter, fin the velocity potential function f. 3

5 Newton s Law of iscosity an the Equations of Motion. The linear tensor equation of viscosity is limite to isotropic fluis. Substituting this form into the stress equations of motions provies the Navier- Stokes Equation. Pages 14-16, Sec MIDTERM (Through Lecture pplication. Uniformly accelerate flow. Pages Problem pplication. One-imensional laminar flow. Pages Problem pplication. Transient flow an the suenly accelerate flat plat. The von Karman-Pohlhausen integral metho provies an estimate of the propagation of a isturbance at a bounary as well as the velocity profile pplication. Laminar bounary layer equations. Several restrictions are mae to the conservation an state equations which allow for an approximate solution pplication. The von Karman-Pohlhausen integral metho provies an estimate of the velocity profile for laminar bounary layer flow Dimensional nalysis. The governing equations are mae imensionless to reuce the number of experiments neee to solve flow problems that are not susceptible to analysis Transition an Turbulent Flow. Small isturbances in the laminar flow region create velocity variations in time. In the laminar bounary layer flui parcels follow a straight path, eform an rotate while in the transition region the path is curvilinear an parcels oscillate. In the turbulent region, the path is unefine an the parcel rotates unpreictably. elocity is ecompose into time average an turbulent fluctuation terms. Because the time scales for each term are isparate, the time average of the time average velocity is equal to the time average velocity Time verage Continuity an Navier Stokes Equations. Leibnitz rule is use to begin eriving the time average equations of incompressible flow. Sec Problem Class notes Class notes Sec. 5.5 Sec. 6.1 an Class notes Starting with the integral motion equation y=δh y =δ H vx vxx u vxx x = ν y Sec. 6.2 Problem 6.2 y = 0 y= 0 y= 0, represent the velocity component as a thir orer polynomial in y an δ H show that δ v = x H u Work the racing sloop exa mple problem which starts on page 163. Show all the steps, o not just copy what is in the text. Discuss geometric an ynamic similarity. Problem 6.1 4

6 Time verage Continuity an Navier Stokes Equations. The time-average equations of motion inicates that turbulent flow can be treate in the same way as laminar flows provie the pressure an velocity are replace by the time-average quantities an the viscous stress tensor is replace by the total time-average stress tensor which is the sum of the viscous an turbulent stress tensors Physical Interpretation of Turbulence. Turbulence is generate near the tube wall an the intensity falls off towar the center of the tube. In the central region the generating force (shear eformation ecreases an viscous forces ten to reuce turbulence Ey iscosity an Prantl s Mixing Length Theory. s an analog to laminar flow, a turbulent or Ey viscosity was evelope by Prantl through a simplifie interpretation of turbulent momentum transfer pplication to Turbulent Pipe Flow elocity Profiles. Mixing length theory is applie to turbulent pipe flow to calculate the time average velocity profile Macroscopic Momentum Balance. To solve more complex problems that are not subject to microscopic analysis, we supplement information from intuition an experimentation an fin solutions which are correct on the average. The governing equations are satisfie for a control volume rather than point-wise. Information lost through integration must be replace by intuition, experiment or analysis at smaller length scales pplication. Jets an Plates. Force exerte by the flui on a plate is calculate using the macroscopic momentum conservation principle. ssumptions that must be mae in orer to arrive at the simple solution will be ientifie Bernoulli s Equation. Bernoulli s is obtaine by first extracting the component of the Navier-Stokes equation tangent to a streamline, simplifying that result by neglecting the local acceleration an viscous effects, an then integrating along a streamline. Sec. 6.2 Sec. 6.3 Sec. 6.4 Sec. 6.5 Secs. 7.1 an 7.2 Secs. 7.1 an 7.8 Pages Derive Euler s First Law t ρvˆ = ρbˆ + m( t tˆ (nˆ starting from the following axiomatic statement of the linear momentum principle t a( ρvˆ + ρbˆ + t a ( t a ( t tˆ ρvˆ(vˆ ŵ nˆ = m ( t (nˆ Problem 7-6. Ignore the comments about the energy equation an use the momentum balance to solve this problem. Microscopic scale information lost by integration over area is recovere or justifie by the statement that viscous surface forces can be neglecte. Problems 7-5, 7-7 (Use Torricelli s equation with C = 1 so that this problem can be solve using a macroscopic momentum balance analysis, an Torricelli s equation: Q = C o 2gh(t 5

7 Moving Control olumes an Inertial Frames. Because the mass an the linear an angular momentum balance equations are vali in inertial frames, careful selection of the inertial frame can simplify problem solving. The problem of fining the force exerte by a plane jet impinging on a curve vane is illustrative. Sec. 7.9 but ignore the energy balance iscussion Mechanical Energy Balance. By forming the scalar prouct with the velocity vector the necessity of evaluating terms in the macroscopic momentum balance equation at soli surfaces is eliminate but a viscous issipation term arises which must be evaluate experimentally. Keep in min that the momentum an mechanical energy balances come fro m the same physical principle but the assumptions in making approximate solutions are ifferent pplication. Suen expansion in a pipeline. Sec. 7.5 an pages Turbulent flow in pipes. We begin eveloping a Pages consistent metho of interpreting experimental ata MIDTERM (Through lecture Friction Factors. Experimental ata is interprete to generalize application of the momentum balance equation to non-circular conuits as well as flows aroun spheres an cyliners Pipeline Design. The macroscopic mechanical energy balance equation simplifies calculation of healoss in a pipeline. The energy an momentum balance equations are combine to arrive at the simplifie approach Review FINL EXM (Comprehensive 4-6 pm, Chem 166 Problems 7-4, 7-16 an Note that problem 7-17 shoul rea: Does a converging nozzle on a garen hose place the hose (at the junction between the hose an the nozzle in tension or compression? Sec. 7.3 Problems 7-3 an 7-20 Problems 7-9 an Sec. 8.2 Problems 8-3 an 8-4. Sec. 8.3 Problem