9.3 Laminar Flat-Plate Boundary Layer: Exact Solution w-19
|
|
- Camilla Harrington
- 6 years ago
- Views:
Transcription
1 9.3 Laminar Flat-Plate Boundary Layer: Exact Solution w-19 Laminar Flat-Plate Boundary Layer: Exact Solution The solution for the laminar boundary layer on a horizontal flat late was obtained by Prtl s student H. Blasius [2] in 198. For two-dimensional, stea, incomressible flow with zero ressure gradient, the goerning equations of motion (Eqs..27) reduce to [3] ð9:3þ ð9:4þ with boundary conditions at y ; u ; at y N; u ð9:þ Equations , with boundary conditions Eq. 9. are a set of nonlinear, couled, artial differential equations for the unknown elocity field u. To sole them, Blasius reasoned that the elocity rofile, u/, should be similar for all alues of x when lotted ersus a nondimensional distance from the wall; the boundary-layer thickness,, was a natural choice for nondimensionalizing the distance from the wall. Thus the solution is of the form u g ðηþ where η ~ y Based on the solution of Stokes [4], Blasius reasoned that ~ η y νx We now introduce the stream function, ψ, @x ffiffiffiffiffiffiffiffiffiffiffiffi νx= set ð9:6þ ð9:7þ ð:4þ satisfies the continuity equation (Eq. 9.3) identically. Substituting for u into Eq. 9.4 reduces the equation to one in which ψ is the single deendent ariable. Defining a dimensionless stream function as f ðηþ ffiffiffiffiffiffiffiffiffi ψ νx ð9:8þ makes f(η) the deendent ariable η the indeendent ariable in Eq With ψ defined by Eq. 9.8 η defined by Eq. 9.7, we can ealuate each of the terms in Eq The elocity comonents are gien ffiffiffiffiffiffiffiffiffi df νx dη νx df dη ð9:9þ ffiffi " ν f 2 ffiffiffiffiffiffiffiffiffi df νx 2 x 1 2 η ffiffi ν f x 2 x
2 w-2 Chater 9 External Incomressible Viscous Flow or 1 ffiffi ν 2 x f By differentiating the elocity comonents, it also can be shown that 2 2x η d2 ffiffiffiffiffiffiffiffiffiffiffiffi d 2 f d 3 f νx dη 3 Substituting these exressions into Eq. 9.4, we obtain with boundary conditions: 2 d3 f dη 3 1 f d2 f ð9:11þ at η ; at η-n; f df dη df dη 1 ð9:12þ The second-order artial differential equations goerning the growth of the laminar boundary layer on a flat late (Eqs ) hae been transformed to a nonlinear, third-order ordinary differential equation (Eq. 9.11) with boundary conditions gien by Eq It is not ossible to sole Eq in closed form; Blasius soled it using a ower series exansion about η matched to an asymtotic exansion for η - N. The same equation later was soled more recisely again using numerical methods by Howarth [], who reorted results to decimal laces. The numerical alues of f, df/dη, d 2 f/ in Table 9.1 were calculated with a ersonal comuter using fourthorder Runge-Kutta numerical integration. The elocity rofile is obtained in dimensionless form by lotting u/ ersus η, using alues from Table 9.1. The resulting rofile is sketched in Fig. 9.3b. Velocity rofiles measured exerimentally are in excellent agreement with the analytical solution. Profiles from all locations on a flat late are similar; they collase to a single rofile when lotted in nondimensional coordinates. From Table 9.1, we see that at η., u/.992. With the boundary-layer thickness,, defined as the alue of y for which u/.99, Eq. 9.7 gies ffiffiffiffiffiffiffiffiffiffiffiffi : :x ffiffiffiffiffiffiffi =νx Re x ð9:13þ The wall shear stress may be exressed as τ w μ μ ffiffiffiffiffiffiffiffiffiffiffiffi d 2 f y η
3 Table Laminar Flat-Plate Boundary Layer: Exact Solution w-21 The Function f(η) for the Laminar Boundary Layer along a Flat Plate at Zero Incidence η y νx f fu u f Then τ w :332 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρμ=x :332ρ2 ffiffiffiffiffiffiffi Re x ð9:14þ the wall shear stress coefficient, C f, is gien by C f 1 2 τ w :664 ffiffiffiffiffiffiffi ρ2 Re x ð9:1þ Each of the results for boundary-layer thickness,, wall shear stress, τ w, skin friction coefficient, C f, Eqs through 9.1, deends on the length Reynolds number, Re x, to the one-half ower. The boundary-layer thickness increases as x 1/2, the wall shear stress skin friction coefficient ary as 1/x 1/2. These results characterize the behaior of the laminar boundary layer on a flat late. Examle 9.2 LAMINAR BONDARY LAYER ON A FLAT PLATE: EXACT SOLTION se the numerical results resented in Table 9.1 to ealuate the following quantities for laminar boundary-layer flow on a flat late: (a) / (for η as η - N). (b) / at the boundary-layer edge. (c) Ratio of the sloe of a at the boundary-layer edge to the sloe of ersus x.
4 w-22 Chater 9 External Incomressible Viscous Flow Gien: Numerical solution for laminar flat-late boundary layer, Table 9.1. Find: (a) / (for η as η - N). (b) / at boundary-layer edge. (c) Ratio of the sloe of a at the boundary-layer edge to the sloe of ersus x. Solution: The dislacement thickness is defined by Eq. 9.1 as Z N 1 2 u Z 1 2 u In order to use the Blasius exact solution to ealuate this integral, ffiffiffiffiffiffiffiffiffiffiffiffi we need to conert ffiffiffiffiffiffiffiffiffiffiffiffi it from one inoling ffiffiffiffiffiffiffiffiffiffiffiffi u y to one inoling fu ( u/) η ariables. From Eq. 9.7, η y =νx; so y η νx= dη νx= Thus, Z ηmax Z νx νx ηmax ð1 2 f uþ dη ð1 2 f uþdη ð1þ Note: Corresonding to the uer limit on y in Eq. 9.1, η max N, orη max. From Eq. 9.13, ffiffiffiffiffiffiffiffiffiffiffiffi =νx so if we diide each side of Eq. 1 by each side of Eq. 9.13, we obtain (with fudf/dη) 1 Z ηmax 1 2 df dη dη Integrating gies Ealuating at η max, we obtain 1 ½η 2 f ðηþšη max 1 ð: 2 3:2833Þ :343 ß The quantity η 2 f(η) becomes constant for η. 7. Ealuating at η max 8 gies ðη Þ 1 ð8: 2 6:2792Þ :344 ß Thus, η-n is.24 ercent larger than η-. From Eq. 9.1, 1 ffiffi ν 2 x f ; so 1 ffi ν 2 x f ðη-nþ 1 2 ffiffiffiffiffiffiffi Re x f Ealuating at the boundary-layer edge (η ), we obtain 1 2 ffiffiffiffiffiffiffi Re ½ð:991Þ 2 3:2833Š :837 ffiffiffiffiffiffiffi :84 ffiffiffiffiffiffiffi x Re x Re x ß ðη Þ Thus is only.84 ercent of at Re x 1 4, only about.12 ercent of at Re x 3 1.
5 9.3 Laminar Flat-Plate Boundary Layer: Exact Solution w-23 The sloe of a at the boundary-layer edge is u :84 ffiffiffiffiffiffiffi Re x The sloe of the boundary-layer edge may be obtained from Eq. 9.13, νx ffiffiffiffiffiffiffiffiffiffiffiffi =νx so Thus, :84 d 2: rffiffiffiffi ffi d ν 1 ν 2 x21=2 2: 2: ffiffiffiffiffiffiffi x Re x :336 d ß This result indicates that the sloe of the s is about 1 3 of the sloe of the boundary layer edge the s enetrate the boundary layer, as sketched below: u This roblem illustrates use of numerical data from the Blasius solution to obtain other information on a flat late laminar boundary layer, including the result that the edge of the boundary layer is not a.
FLUID MECHANICS. Lecture 7 Exact solutions
FLID MECHANICS Lecture 7 Eact solutions 1 Scope o Lecture To present solutions or a ew representative laminar boundary layers where the boundary conditions enable eact analytical solutions to be obtained.
More information7.6 Example von Kármán s Laminar Boundary Layer Problem
CEE 3310 External Flows (Boundary Layers & Drag, Nov. 11, 2016 157 7.5 Review Non-Circular Pipes Laminar: f = 64/Re DH ± 40% Turbulent: f(re DH, ɛ/d H ) Moody chart for f ± 15% Bernoulli-Based Flow Metering
More informationModule 2: External Flows Lecture 6: Exact Solution of Boundary Layer Equation. The Lecture Contains: Blasius Solution. Numerical Approach
The Lecture Contains: Blasius Solution Numerical Approach file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture6/6_1.htm[12/24/2014 5:51:08 PM]
More informationa) Derive general expressions for the stream function Ψ and the velocity potential function φ for the combined flow. [12 Marks]
Question 1 A horizontal irrotational flow system results from the combination of a free vortex, rotating anticlockwise, of strength K=πv θ r, located with its centre at the origin, with a uniform flow
More informationCHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW
CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW 4.1 Introduction Boundary layer concept (Prandtl 1904): Eliminate selected terms in the governing equations Two key questions (1) What are the
More informationLaminar Flow. Chapter ZERO PRESSURE GRADIENT
Chapter 2 Laminar Flow 2.1 ZERO PRESSRE GRADIENT Problem 2.1.1 Consider a uniform flow of velocity over a flat plate of length L of a fluid of kinematic viscosity ν. Assume that the fluid is incompressible
More informationJJMIE Jordan Journal of Mechanical and Industrial Engineering
JJMIE Jordan Journal of Mechanical and Industrial Engineering Volume, Number, Jun. 8 ISSN 995-6665 Pages 7-75 Efficiency of Atkinson Engine at Maximum Power Density using emerature Deendent Secific Heats
More informationA Model Answer for. Problem Set #4 FLUID DYNAMICS
A Model Answer for Problem Set #4 FLUID DYNAMICS Problem. Some elocity measurements in a threedimensional incomressible flow field indicate that u = 6xy and = -4y z. There is some conflicting data for
More information6.7 Thermal wind in pressure coordinates
176 CHAPTER 6. THE EQUATIONS OF FLUID MOTION 6.7 Thermal wind in ressure coordinates The thermal wind relation aroriate to the atmoshere is untidy when exressed with height as a vertical coordinate (because
More informationFLUID MECHANICS. Chapter 9 Flow over Immersed Bodies
FLUID MECHANICS Chapter 9 Flow over Immersed Bodies CHAP 9. FLOW OVER IMMERSED BODIES CONTENTS 9.1 General External Flow Characteristics 9.3 Drag 9.4 Lift 9.1 General External Flow Characteristics 9.1.1
More informationMathCAD solutions for problem of laminar boundary-layer flow
MathCAD solutions for problem of laminar boundary-layer flow DANIELA CÂRSTEA Industrial High-School Group of Railways Transport, Craiova, ROMANIA danacrst@yahoo.com Abstract: -The problem of laminar boundary-layer
More informationInternational Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)
International Association of ientific Innovation and Research (IASIR) (An Association Unifying the iences, Engineering, and Alied Research) International Journal of Emerging Technologies in Comutational
More informationChapter 2 Thermal Analysis of Friction Welding
Chater 2 Thermal Analysis of Friction Welding Abstract Thermal energy is generated during the friction welding rocess. In this case, the solid surfaces rub against each other and heat is generated as a
More informationSimplifications to Conservation Equations
Chater 5 Simlifications to Conservation Equations 5.1 Steady Flow If fluid roerties at a oint in a field do not change with time, then they are a function of sace only. They are reresented by: ϕ = ϕq 1,
More informationBoundary Layer (Reorganization of the Lecture Notes from Professor Anthony Jacobi and Professor Nenad Miljkoic) Consider a steady flow of a Newtonian, Fourier-Biot fluid oer a flat surface with constant
More informationProblem 4.3. Problem 4.4
Problem 4.3 Problem 4.4 Problem 4.5 Problem 4.6 Problem 4.7 This is forced convection flow over a streamlined body. Viscous (velocity) boundary layer approximations can be made if the Reynolds number Re
More informationInternational Journal of Mathematics Trends and Technology- Volume3 Issue4-2012
Effect of Hall current on Unsteady Flow of a Dusty Conducting Fluid through orous medium between Parallel Porous Plates with Temerature Deendent Viscosity and Thermal Radiation Harshbardhan Singh and Dr.
More informationThe Second Law: The Machinery
The Second Law: The Machinery Chater 5 of Atkins: The Second Law: The Concets Sections 3.7-3.9 8th Ed, 3.3 9th Ed; 3.4 10 Ed.; 3E 11th Ed. Combining First and Second Laws Proerties of the Internal Energy
More informationChapter 5 Mass, Momentum, and Energy Equations
57:00 Mechanics of Fluids and Transort Processes Chater 5 Professor Fred Stern Fall 006 Chater 5 Mass, Momentum, and Energy Equations Flow Rate and Conservation of Mass. cross-sectional area oriented normal
More informationCOMPUTER SIMULATION OF A LABORATORY HYDRAULIC SYSTEM WITH MATLAB-SIMULINK
DVNCED ENGINEERING 4(20101, ISSN 1846-5900 COMPUTER SIMULTION OF LORTORY HYDRULIC SYSTEM WITH MTL-SIMULINK Grego, G. & Siminiati, D. bstract: The article resents some selected roblems related to modeling
More informationEffect of Hall current and rotation on heat transfer in MHD flow of oscillating dusty fluid in a porous channel
Indian Journal of Pure & Alied Physics Vol. 5 October 03. 669-68 Effect of Hall current and rotation on heat transfer in MHD flow of oscillating dusty fluid in a orous channel Khem Chand K D Singh & Shavnam
More informationTHE EFFECT OF LONGITUDINAL VIBRATION ON LAMINAR FORCED CONVECTION HEAT TRANSFER IN A HORIZONTAL TUBE
mber 3 Volume 3 September 26 Manal H. AL-Hafidh Ali Mohsen Rishem Ass. Prof. /Uniersity of Baghdad Mechanical Engineer ABSTRACT The effect of longitudinal ibration on the laminar forced conection heat
More informationPressure-sensitivity Effects on Toughness Measurements of Compact Tension Specimens for Strain-hardening Solids
American Journal of Alied Sciences (9): 19-195, 5 ISSN 1546-939 5 Science Publications Pressure-sensitivity Effects on Toughness Measurements of Comact Tension Secimens for Strain-hardening Solids Abdulhamid
More informationDetermination of Pressure Losses in Hydraulic Pipeline Systems by Considering Temperature and Pressure
Paer received: 7.10.008 UDC 61.64 Paer acceted: 0.04.009 Determination of Pressure Losses in Hydraulic Pieline Systems by Considering Temerature and Pressure Vladimir Savi 1,* - Darko Kneževi - Darko Lovrec
More informationChapter 10: Flow Flow in in Conduits Conduits Dr Ali Jawarneh
Chater 10: Flow in Conduits By Dr Ali Jawarneh Hashemite University 1 Outline In this chater we will: Analyse the shear stress distribution across a ie section. Discuss and analyse the case of laminar
More informationDAY 19: Boundary Layer
DAY 19: Boundary Layer flat plate : let us neglect the shape of the leading edge for now flat plate boundary layer: in blue we highlight the region of the flow where velocity is influenced by the presence
More informationA note on variational representation for singular values of matrix
Alied Mathematics and Comutation 43 (2003) 559 563 www.elsevier.com/locate/amc A note on variational reresentation for singular values of matrix Zhi-Hao Cao *, Li-Hong Feng Deartment of Mathematics and
More informationMODELING OF UNSTEADY AERODYNAMIC CHARACTERISTCS OF DELTA WINGS.
IAS00 ONGRESS MODEING OF UNSTEADY AERODYNAMI HARATERISTS OF DETA WINGS. Jouannet hristoher, rus Petter inköings Uniersity eywords: Delta wings, Unsteady, Modeling, Preliminary design, Aerodynamic coefficient.
More informationLecture 5. Differential Analysis of Fluid Flow Navier-Stockes equation
Lectre 5 Differential Analsis of Flid Flo Naier-Stockes eqation Differential analsis of Flid Flo The aim: to rodce differential eqation describing the motion of flid in detail Flid Element Kinematics An
More informationMasters in Mechanical Engineering Aerodynamics 1 st Semester 2015/16
Masters in Mechanical Engineering Aerodynamics st Semester 05/6 Exam st season, 8 January 06 Name : Time : 8:30 Number: Duration : 3 hours st Part : No textbooks/notes allowed nd Part : Textbooks allowed
More informationBoundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity
Boundary layer flows The logarithmic law of the wall Mixing length model for turbulent viscosity Tobias Knopp D 23. November 28 Reynolds averaged Navier-Stokes equations Consider the RANS equations with
More informationApplications of parabolized stability equation for predicting transition position in boundary layers
Appl. Math. Mech. -Engl. Ed., 33(6), 679 686 (2012) DOI 10.1007/s10483-012-1579-7 c Shanghai University and Springer-Verlag Berlin Heidelberg 2012 Applied Mathematics and Mechanics (English Edition) Applications
More informationarxiv: v1 [math-ph] 21 Dec 2007
Dynamic Phase ransitions in PV Systems ian Ma Deartment of Mathematics, Sichuan Uniersity, Chengdu, P. R. China Shouhong Wang Deartment of Mathematics, Indiana Uniersity, Bloomington, IN 4745 (Dated: February
More informationNode-voltage method using virtual current sources technique for special cases
Node-oltage method using irtual current sources technique for secial cases George E. Chatzarakis and Marina D. Tortoreli Electrical and Electronics Engineering Deartments, School of Pedagogical and Technological
More informationFluid Mechanics. Chapter 9 Surface Resistance. Dr. Amer Khalil Ababneh
Fluid Mechanics Chapter 9 Surface Resistance Dr. Amer Khalil Ababneh Wind tunnel used for testing flow over models. Introduction Resistances exerted by surfaces are a result of viscous stresses which create
More informationResearch Article Comparison of HPM and PEM for the Flow of a Non-newtonian Fluid between Heated Parallel Plates
Research Journal of Alied Sciences, Engineering and Technology 7(): 46-434, 4 DOI:.96/rjaset.7.793 ISSN: 4-7459; e-issn: 4-7467 4 Maxwell Scientific Publication Cor. Submitted: November, 3 Acceted: January
More informationHIEMENZ MAGNETIC FLOW BY DIFFERENTIAL TRANSFORMATION METHOD AND PADE APPROXIMANT
ACTA TECHNICA CORVINIENSIS Bulletin of Engineering Tome IX [6], Fascicule [January arch] ISSN: 67 389..K. NAYAK,. G.C. DASH HIEENZ AGNETIC FLOW BY DIFFERENTIAL TRANSFORATION ETHOD AND PADE APPROXIANT.
More informationViscosity and Fluid Suction/Injection Effects on Free Convection Flow from a Vertical Plate in a Porous Medium Saturated with a Pseudoplastic Fluid
ISSN 749-3889 (print), 749-3897 (online) International Journal of Nonlinear Science Vol.8(4) No.,pp.7-38 Viscosity and Fluid Suction/Injection Effects on Free Convection Flow from a Vertical Plate in a
More informationSupplementary Information Microfluidic quadrupole and floating concentration gradient Mohammad A. Qasaimeh, Thomas Gervais, and David Juncker
Mohammad A. Qasaimeh, Thomas Gerais, and Daid Juncker Supplementary Figure S1 The microfluidic quadrupole (MQ is modeled as two source (Q inj and two drain (Q asp points arranged in the classical quardupolar
More informationNUMERICAL ANALYSIS OF THE IMPACT OF THE INLET AND OUTLET JETS FOR THE THERMAL STRATIFICATION INSIDE A STORAGE TANK
NUMERICAL ANALYSIS OF HE IMAC OF HE INLE AND OULE JES FOR HE HERMAL SRAIFICAION INSIDE A SORAGE ANK A. Zachár I. Farkas F. Szlivka Deartment of Comuter Science Szent IstvÆn University Æter K. u.. G d llı
More informationBoundary Layer Theory. v = 0, ( v)v = p + 1 Re 2 v. Consider a cylinder of length L kept in a inviscid irrotational flow.
1 Boundary Layer Theory 1 Introduction Consider the steady flow of a viscous fluid. The governing equations based on length scale L and velocity scale U is given by v =, ( v)v = p + 1 Re 2 v For small
More informationM E 320 Professor John M. Cimbala Lecture 38
M E 320 Professor John M. Cimbala Lecture 38 Today, we will: Discuss displacement thickness in a laminar boundary layer Discuss the turbulent boundary layer on a flat plate, and compare with laminar flow
More informationAE301 Aerodynamics I UNIT A: Fundamental Concepts
AE301 Aerodynamics I UNIT A: Fundamental Concets ROAD MAP... A-1: Engineering Fundamentals Reiew A-: Standard Atmoshere A-3: Goerning Equations of Aerodynamics A-4: Airseed Measurements A-5: Aerodynamic
More information(British) (SI) British Metric L T [V] = L T. [a] = 2 [F] = F = 2 T
Hydraulics ecture # CWR 40 age () ecture # Outline: Review of terminology in fluid mechanics: Energy or work Hydraulic head Bernoulli s aw, Conductivity (examle) ransient & turbulent Friction head loss
More informationUnsteady Flow of a Dusty Conducting Fluid through porous medium between Parallel Porous Plates with Temperature Dependent Viscosity and Heat Source
Volume Issue3 3- June www.ijsret.org ISSN 78-88 Unsteady Flow of a Dusty Conducting Fluid through orous medium between Parallel Porous Plates with Temerature Deendent Viscosity and Heat Source Shalini
More informationChapter 6 Momentum Transfer in an External Laminar Boundary Layer
6. Similarit Soltions Chapter 6 Momentm Transfer in an Eternal Laminar Bondar Laer Consider a laminar incompressible bondar laer with constant properties. Assme the flow is stead and two-dimensional aligned
More informationScaling laws in granular continuous avalanches in a rotating drum
Physica A 356 (25) 178 183 www.elsevier.com/locate/hysa Scaling laws in granular continuous avalanches in a rotating drum N. Seu lveda, G. Krstulovic, S. Rica Deartamento de Física, Facultad de Ciencias
More informationUNIVERSITY OF OSLO. Faculty of Mathematics and Natural Sciences. MEK4300/9300 Viscous flow og turbulence
UNIVERSITY OF OSLO Faculty of Mathematics and Natural Sciences Examination in: Day of examination: Friday 15. June 212 Examination hours: 9. 13. This problem set consists of 5 pages. Appendices: Permitted
More information6.2 Governing Equations for Natural Convection
6. Governing Equations for Natural Convection 6..1 Generalized Governing Equations The governing equations for natural convection are special cases of the generalized governing equations that were discussed
More informationLecture 7: Thermodynamic Potentials
Lecture 7: Thermodynamic Potentials Chater II. Thermodynamic Quantities A.G. Petukhov, PHY 743 etember 27, 2017 Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember
More informationChapter 9 Flow over Immersed Bodies
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 009 1 Chapter 9 Flow over Immersed Bodies Fluid flows are broadly categorized: 1. Internal flows such as ducts/pipes,
More informationExternal Flows. Dye streak. turbulent. laminar transition
Eternal Flos An internal flo is surrounded by solid boundaries that can restrict the development of its boundary layer, for eample, a pipe flo. An eternal flo, on the other hand, are flos over bodies immersed
More information2, where dp is the constant, R is the radius of
Dynamics of Viscous Flows (Lectures 8 to ) Q. Choose the correct answer (i) The average velocity of a one-dimensional incompressible fully developed viscous flow between two fixed parallel plates is m/s.
More informationMIXED CONVECTION SLIP FLOW WITH TEMPERATURE JUMP ALONG A MOVING PLATE IN PRESENCE OF FREE STREAM
THERMAL SCIENCE, Year 015, Vol. 19, No. 1, pp. 119-18 119 MIXED CONVECTION SLIP FLOW WITH TEMPERATURE JUMP ALONG A MOVING PLATE IN PRESENCE OF FREE STREAM by Gurminder SINGH *a and Oluwole Daniel MAKINDE
More informationHigh speed wind tunnels 2.0 Definition of high speed. 2.1 Types of high speed wind tunnels
Module Lectures 6 to 1 High Seed Wind Tunnels Keywords: Blow down wind tunnels, Indraft wind tunnels, suersonic wind tunnels, c-d nozzles, second throat diffuser, shocks, condensation in wind tunnels,
More informationChapter 8 Internal Forced Convection
Chater 8 Internal Forced Convection 8.1 Hydrodynamic Considerations 8.1.1 Flow Conditions may be determined exerimentally, as shown in Figs. 7.1-7.2. Re D ρumd μ where u m is the mean fluid velocity over
More informationFinite Element Analysis of V-Bending of Polypropylene Using Hydrostatic-Pressure-Dependent Plastic Constitutive Equation*
Materials Transactions, Vol. 48, No. 1 (7). 6 to 664 #7 The Jaan Society for Technology of Plasticity Finite Element Analysis of V-Bending of Polyroylene Using Hydrostatic-Pressure-Deendent Plastic onstitutive
More informationEE 508 Lecture 13. Statistical Characterization of Filter Characteristics
EE 508 Lecture 3 Statistical Characterization of Filter Characteristics Comonents used to build filters are not recisely redictable R L C Temerature Variations Manufacturing Variations Aging Model variations
More informationMasters in Mechanical Engineering. Problems of incompressible viscous flow. 2µ dx y(y h)+ U h y 0 < y < h,
Masters in Mechanical Engineering Problems of incompressible viscous flow 1. Consider the laminar Couette flow between two infinite flat plates (lower plate (y = 0) with no velocity and top plate (y =
More informationHeat and Mass Transfer
1 Comments on six papers published by S.P. Anjali Devi and R. Kandasamy in Heat and Mass Transfer, ZAMM, Mechanics Research Communications, International Communications in Heat and Mass Transfer, Communications
More informationSELF-SIMILAR FLOW UNDER THE ACTION OF MONOCHROMATIC RADIATION BEHIND A STRONG CYLINDRICAL SHOCK WAVE IN A NON-IDEAL GAS
SELF-SIMILAR FLOW UNDER THE ACTION OF MONOCHROMATIC RADIATION BEHIND A STRONG CYLINDRICAL SHOCK WAVE IN A NON-IDEAL GAS *J. P. Vishwakarma and Vijay Kumar Pandey Deartment of Mathematics & Statistics,
More informationCHAPTER-III GENERAL GROUP THEORETIC TRANSFORMATIONS FROM BOUNDARY VALUE TO INITIAL VALUE PROBLEMS
CHAPTER-III GENERAL GROUP THEORETIC TRANSFORMATIONS FROM BOUNDARY VALUE TO INITIAL VALUE PROBLEMS 3.1 Introduction: The present chapter treats one of the most important applications of the concept of continuous
More informationFlying characteristics in the free molecular region (influence of accommodation coefficients)
Microsyst Technol (25) 11: 85 811 DOI 1.17/s542-5-538- TECHNICAL PAPER Shigehisa Fukui Æ Hidekazu Shimada Æ Kiyomi Yamane Hiroshige Matsuoka Flying characteristics in the free molecular region (influence
More informationFE FORMULATIONS FOR PLASTICITY
G These slides are designed based on the book: Finite Elements in Plasticity Theory and Practice, D.R.J. Owen and E. Hinton, 1970, Pineridge Press Ltd., Swansea, UK. 1 Course Content: A INTRODUCTION AND
More information3.4 Design Methods for Fractional Delay Allpass Filters
Chater 3. Fractional Delay Filters 15 3.4 Design Methods for Fractional Delay Allass Filters Above we have studied the design of FIR filters for fractional delay aroximation. ow we show how recursive or
More informationLagrangian analysis of the laminar flat plate boundary layer
Lagrangian analysis of the laminar flat plate boundary layer Mohammad Gabr* EgyptAir Maintenance & Engineering, Cairo, Egypt 1 The flow properties at the leading edge of a flat plate represent a singularity
More informationEE 508 Lecture 13. Statistical Characterization of Filter Characteristics
EE 508 Lecture 3 Statistical Characterization of Filter Characteristics Comonents used to build filters are not recisely redictable L C Temerature Variations Manufacturing Variations Aging Model variations
More information1. Introduction, tensors, kinematics
1. Introduction, tensors, kinematics Content: Introduction to fluids, Cartesian tensors, vector algebra using tensor notation, operators in tensor form, Eulerian and Lagrangian description of scalar and
More informationFree convection modeling over a vertical flat plate embedded in saturated porous medium with a variable heat source and radiation flux
ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 9 (2013) No. 3, pp. 163-172 Free convection modeling over a vertical flat plate embedded in saturated porous medium with a variable
More informationLower bound solutions for bearing capacity of jointed rock
Comuters and Geotechnics 31 (2004) 23 36 www.elsevier.com/locate/comgeo Lower bound solutions for bearing caacity of jointed rock D.J. Sutcliffe a, H.S. Yu b, *, S.W. Sloan c a Deartment of Civil, Surveying
More informationE o e associated with a light field (both the real part and the. ikr t. under the assumptions that J free
Reiew Problems Chaters 1-5 True and False Questions E1. T or F: The otical index of any material aries with frequency. E2. T or F: The frequency of light can change as it enters a crystal. E3. T or F:
More informationINTEGRAL ANALYSIS OF LAMINAR INDIRECT FREE CONVECTION BOUNDARY LAYERS WITH WEAK BLOWING FOR SCHMIDT NO. 1
INTEGRA ANAYSIS OF AMINAR INDIRECT FREE CONVECTION BOUNDARY AYERS WITH WEAK BOWING FOR SCHMIDT NO. Baburaj A.Puthenveettil and Jaywant H.Arakeri Department of Mechanical Engineering, Indian Institute of
More informationFinite difference solution of the mixed convection flow of MHD micropolar fluid past a moving surface with radiation effect
Finite difference solution of the mixed convection flo of MHD micropolar fluid past a moving surface ith radiation effect LOKENDRA KUMAR, G. SWAPNA, BANI SINGH Department of Mathematics Jaypee Institute
More informationWuhan, , P.R. China Published online: 25 Jan 2007.
This article was downloaded by: [Huazhong University of Science & Technology ] On: 26 December 2014, At: 20:25 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number:
More informationChapter 9 Flow over Immersed Bodies
57:00 Mechanics of Fluids and Transport Processes Chapter 9 Professor Fred Stern Fall 014 1 Chapter 9 Flow over Immersed Bodies Fluid flows are broadly categorized: 1. Internal flows such as ducts/pipes,
More informationINDIAN INSTITUTE OF TECHNOOGY, KHARAGPUR Date: -- AN No. of Students: 5 Sub. No.: ME64/ME64 Time: Hours Full Marks: 6 Mid Autumn Semester Examination Sub. Name: Convective Heat and Mass Transfer Instructions:
More informationSTATIC, STAGNATION, AND DYNAMIC PRESSURES
STATIC, STAGNATION, AND DYNAMIC PRESSURES Bernolli eqation is g constant In this eqation is called static ressre, becase it is the ressre that wold be measred by an instrment that is static with resect
More informationUNIVERSITY OF EAST ANGLIA
UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2007 2008 FLUIDS DYNAMICS WITH ADVANCED TOPICS Time allowed: 3 hours Attempt question ONE and FOUR other questions. Candidates must
More informationQuasi-Three-Dimensional Simulation of Viscoelastic Flow through a Straight Channel with a Square Cross Section
Article Nihon Reoroji Gakkaishi Vol.34, No.2, 105~113 (Journal of the Society of Rheology, Jaan) 2006 The Society of Rheology, Jaan Quasi-Three-Dimensional Simulation of Viscoelastic Flow through a Straight
More informationEngineering Fracture Mechanics
Engineering Fracture Mechanics 90 (2012) 172 179 Contents lists available at SciVerse ScienceDirect Engineering Fracture Mechanics journal homeage: www.elsevier.com/locate/engfracmech Discussion of: Mode
More informationNonlinear Radiation Effects on Hydromagnetic Boundary Layer Flow and Heat Transfer over a Shrinking Surface
Journal of Applied Fluid Mechanics, Vol. 8, No. 3, pp. 613-61, 015. Available online at www.jafmonline.net, ISSN 1735-357, EISSN 1735-3645. DOI: 10.18869/acadpub.jafm.73.38.636 Nonlinear Radiation Effects
More informationUNIT IV BOUNDARY LAYER AND FLOW THROUGH PIPES Definition of boundary layer Thickness and classification Displacement and momentum thickness Development of laminar and turbulent flows in circular pipes
More informationGuide to Exercises Hints A in Chapters 3 6
Guide to Exercises Hints A in Chaters 3 6 HINT A1 Exercise 3.1 What comonent casting methods, which give high accuracy and good surface finish, can be considered? If you have no suggestion, look f recision
More informationEffect of Variable Viscosity on Hydro Magnetic Flow and Heat Transfer Over a Stretching Surface with Variable Temperature
37 Effect of Variable Viscosity on Hydro Magnetic Flow and Heat Transfer Over a Stretching Surface with Variable Temperature M. Y. Akl Department of Basic Science, Faculty of Engineering (Shopra Branch),
More informationContents of today s lecture
Contents of today s lecture Blasius solution for laminar flat-plate boundary layer where external velocity is constant ( ) About solution methods for laminar boundary layers Thwaites method as an example
More informationChapter 7: Natural Convection
7-1 Introduction 7- The Grashof Number 7-3 Natural Convection over Surfaces 7-4 Natural Convection Inside Enclosures 7-5 Similarity Solution 7-6 Integral Method 7-7 Combined Natural and Forced Convection
More informationPreconditioning techniques for Newton s method for the incompressible Navier Stokes equations
Preconditioning techniques for Newton s method for the incomressible Navier Stokes equations H. C. ELMAN 1, D. LOGHIN 2 and A. J. WATHEN 3 1 Deartment of Comuter Science, University of Maryland, College
More informationHomogeneous and Inhomogeneous Model for Flow and Heat Transfer in Porous Materials as High Temperature Solar Air Receivers
Excert from the roceedings of the COMSOL Conference 1 aris Homogeneous and Inhomogeneous Model for Flow and Heat ransfer in orous Materials as High emerature Solar Air Receivers Olena Smirnova 1 *, homas
More informationAdiabatic Shear Bands in Simple and Dipolar Plastic Materials
Adiabatic Shear Bands in Simle and Diolar Plastic Materials T W \-1RIGHT us Army Ballistic Research Laboratory Aberdeen Proving Ground, MD 215 R C BATRA University of Missouri-Rolla Rolla, Missouri 6541
More informationFURTHER SOLUTIONS OF THE FALKNER-SKAN EQUATION
FURTHER SOLUTIONS OF THE FALKNER-SKAN EQUATION LAZHAR BOUGOFFA a, RUBAYYI T. ALQAHTANI b Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU),
More informationFinite-Sample Bias Propagation in the Yule-Walker Method of Autoregressive Estimation
Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 008 Finite-Samle Bias Proagation in the Yule-Walker Method of Autoregressie Estimation Piet
More informationNonlinear Analysis: Modelling and Control, 2008, Vol. 13, No. 4,
Nonlinear Analysis: Modelling and Control, 2008, Vol. 13, No. 4, 513 524 Effects of Temperature Dependent Thermal Conductivity on Magnetohydrodynamic (MHD) Free Convection Flow along a Vertical Flat Plate
More informationTHE full Navier-Stokes equations are difficult or impossible
Unsteady Reversed Stagnation-Point Flow over a Flat Plate Vai Kuong Sin, Member, ASME; Fellow, MIEME, and Chon Kit Chio arxiv:130.997v1 [physics.flu-dyn] 13 Feb 013 Abstract This paper investigates the
More informationLaminar and Turbulent developing flow with/without heat transfer over a flat plate
Laminar and Turbulent developing flow with/without heat transfer over a flat plate Introduction The purpose of the project was to use the FLOLAB software to model the laminar and turbulent flow over a
More informationHigher order theory for analytic saddle point approximations to the Ρ Ρ and Ρ Ś reflected arrivals at a solid/solid interface
Higher order theory for analytic saddle oint aroximations to the Ρ Ρ and Ρ Ś reflected arrivals at a solid/solid interface P.F Daley ABSTACT The high frequency solution to the roblem of a Ρ P and Ρ S reflected
More informationYoung s Modulus Measurement Using a Simplified Transparent Indenter Measurement Technique
Exerimental Mechanics (008) 48:9 5 DOI 0.007/s340-007-9074-4 Young s Modulus Measurement Using a Simlified Transarent Indenter Measurement Technique C. Feng & B.S. Kang Received: October 006 /Acceted:
More informationMultiscale Surfaces and Amontons Law of Friction
Tribol Lett (23) 49:539 543 DOI.7/s249-2-94-6 ORIGINAL PAPER Multiscale Surfaces and Amontons Law of Friction J. R. Barber Received: 4 December 22 / Acceted: 24 December 22 / Published online: 26 January
More informationu y
VO., NO., FEBRUARY 8 ISSN 89-668 6-8 Asian Research Publishing Network (ARPN). All rights reserved. NON-NEWTONIAN EFFECTS OF OAD CARRYING CAPACITY AND FRICTIONA FORCE USING RABINOWITS FUID ON TE PERFORMANCE
More informationChapter 9. Flow over Immersed Bodies
Chapter 9 Flow over Immersed Bodies We consider flows over bodies that are immersed in a fluid and the flows are termed external flows. We are interested in the fluid force (lift and drag) over the bodies.
More informationExact Solutions in Finite Compressible Elasticity via the Complementary Energy Function
Exact Solutions in Finite Comressible Elasticity via the Comlementary Energy Function Francis Rooney Deartment of Mathematics University of Wisconsin Madison, USA Sean Eberhard Mathematical Institute,
More information