Microscopic Momentum Balance Equation (Navier-Stokes)

Size: px
Start display at page:

Download "Microscopic Momentum Balance Equation (Navier-Stokes)"

Transcription

1 CM3110 Transport I Part I: Fluid Mechanics Microscopic Momentum Balance Equation (Naier-Stokes) Professor Faith Morrison Department of Chemical Engineering Michigan Technological Uniersity 1 Microscopic Balances We hae been doing a microscopic control olume balance; these are specific to whateer problem we are soling. We seek equations for microscopic mass, momentum (and energy) balances that are general. equations must not depend on the choice of the control olume, dx d dy equations must capture the appropriate balance 1

2 Arbitrary Control olume in a Flow b `` S ds nˆ V Mass Balance On an arbitrary control olume:.. (details in the book) ate of increase of mass Net conection in (just as we did with the indiidual control olume balance) Microscopic mass balance for any flow

3 Continuity Equation ds S nˆ Microscopic mass balance written on an arbitrarily shaped control olume, V, enclosed by a surface, S V t x x y y x x y y Microscopic mass balance is a scalar equation. Gibbs notation: t 5 Momentum Balance On an arbitrary control olume: (details in the book) V t dv dv g dv ate of increase of momentum (just as we did with the indiidual control olume balance) V Net conection in Microscopic momentum balance for any flow V Force due to graity V dv Viscous forces and pressure forces 3

4 Equation of Motion S ds nˆ V microscopic momentum balance written on an arbitrarily shaped control olume, V, enclosed by a surface, S Gibbs notation: Gibbs notation: P g t P t Naier Stokes Equation g general fluid Newtonian fluid Microscopic momentum balance is a ector equation. 7 Continuity Equation (And Non Newtonian Equation) on the FONT t The one with is for non Newtonian fluids Faith A. Morrison, Michigan 8 4

5 Naier Stokes (Newtonian Fluids Only) is on the back: P t g 9 Problem Soling Procedure soling for elocity and stress fields 1. sketch system. choose coordinate system amended: when using the microscopic balances 3. simplify the continuity equation (mass balance) 4. simplify the 3 components of the equation of motion (momentum balance) (note that for a Newtonian fluid, the equation of motion is the Naier Stokes equation) 5. sole the differential equations for elocity and pressure (if applicable) 6. apply boundary conditions 7. calculate any engineering alues of interest (flow rate, aerage elocity, force on wall) T 10 5

6 EXAMPLE I: Flow of a Newtonian fluid down an inclined plane eisited g x g sin x x x g g cos air H fluid g g g g g x y g sin 0 g cos 11 EXAMPLE I: Flow of a Newtonian fluid down an inclined plane eisited (see hand notes) 1 6

7 As with balance we performed with a control olume we selected, we made modelling assumptions along the way that we can collect and associate with the final result: Model Assumptions: (laminar flow down an incline, Newtonian) 1. no elocity in the x or y directions (laminar flow). well deeloped flow no edge effects in y direction (width) 4. constant density 5. steady state 6. Newtonian fluid 7. no shear stress at interface 8. no slip at wall 13 A r cross-section A: r L (r) EXAMPLE II: Pressure drien flow of a Newtonian fluid in a tube: (Poiseuille flow) steady state constant well deeloped long tube pressure at top pressure at bottom fluid g 14 7

8 15 Naier Stokes:

9 See hand notes 17 List of Common Integrals 014CommonIntegrals.pdf 18 9

10 What is the force on the walls in this flow? Total wetted area force area cross-section A: r (r) L Inside surface of tube?? fluid 19 9 stresses at a point in space y kg m s force kg m / s y area area s area / Momentum Flux f ê y f A( eˆ eˆ yy yx y x eˆ ) y A surface whose unit normal is in the y-direction stress on a y-surface in the y-direction (See discussion of sign conention of stress; this is the tension positie conention) in the -direction y flux of -momentum 0 10

11 What is the shear stress in this flow? Stress on an surface in the direction cross-section A: r L (r) fluid 1 Force on the walls: See hand notes 11

12 4 1 3 Engineering Quantities of Interest (tube flow) aerage elocity olumetric flow rate component of force on the wall Q rdr d rdr d rdr d Must work these out for each problem in the coordinate system in use; see inside back coer of book. 4 1

13 Engineering Quantities of Interest (any flow) olumetric flow rate aerage elocity component of force on the wall For more complex flows, we use the Gibbs notation ersions (will discuss soon) Hagen Poiseuille Equation** 6 13

14 a 0 0,max r rdrd Lg P 4L o Lg P 8L o P L P L r rdrd 7 a 1.5 p p 0 p p 0 L L Velocity maximum is twice the aerage (for incline it was.5 the aerage) r 8 14

15 EXAMPLE II: Pressure drien flow of a Newtonian fluid in a tube: Poiseuille flow /<> Bullet shaped; flow down an incline was parabola, but a sheet. 9 Can this modeling method work for complex flows? Answer: yes. (with some qualifiers) 30 15

Microscopic Momentum Balance Equation (Navier-Stokes)

Microscopic Momentum Balance Equation (Navier-Stokes) CM3110 Transport I Part I: Fluid Mechanics Microscopic Momentum Balance Equation (Navier-Stokes) Professor Faith Morrison Department of Chemical Engineering Michigan Technological University 1 Microscopic

More information

ds nˆ v x v t v y Microscopic Balances 10/3/2011

ds nˆ v x v t v y Microscopic Balances 10/3/2011 Microscopic Balances We hae been doing microscopic shell hllbl balances that t are specific to whateer problem we are soling. We seek equations for microscopic mass, momentum (and energy) balances that

More information

Microscopic Momentum Balances

Microscopic Momentum Balances 013 Fluids ectue 6 7 Moison CM3110 10//013 CM3110 Tanspot I Pat I: Fluid Mechanics Micoscopic Momentum Balances Pofesso Faith Moison Depatment of Chemical Engineeing Michigan Technological Uniesity 1 Micoscopic

More information

What we know about Fluid Mechanics. What we know about Fluid Mechanics

What we know about Fluid Mechanics. What we know about Fluid Mechanics What we know about Fluid Mechanics 1. Survey says. 3. Image from: www.axs.com 4. 5. 6. 1 What we know about Fluid Mechanics 1. MEB (single input, single output, steady, incompressible, no rxn, no phase

More information

Chapter 3: Newtonian Fluids

Chapter 3: Newtonian Fluids Chapter 3: Newtonian Fluids CM4650 Michigan Tech Navier-Stokes Equation v vv p t 2 v g 1 Chapter 3: Newtonian Fluid TWO GOALS Derive governing equations (mass and momentum balances Solve governing equations

More information

Shell Balances in Fluid Mechanics

Shell Balances in Fluid Mechanics Shell Balances in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University When fluid flow occurs in a single direction everywhere in a system, shell

More information

Chapter 8 Laminar Flows with Dependence on One Dimension

Chapter 8 Laminar Flows with Dependence on One Dimension Chapter 8 Laminar Flows with Dependence on One Dimension Couette low Planar Couette low Cylindrical Couette low Planer rotational Couette low Hele-Shaw low Poiseuille low Friction actor and Reynolds number

More information

Fluid Mechanics II Viscosity and shear stresses

Fluid Mechanics II Viscosity and shear stresses Fluid Mechanics II Viscosity and shear stresses Shear stresses in a Newtonian fluid A fluid at rest can not resist shearing forces. Under the action of such forces it deforms continuously, however small

More information

Flux - definition: (same format for all types of transport, momentum, energy, mass)

Flux - definition: (same format for all types of transport, momentum, energy, mass) Fundamentals of Transport Flu - definition: (same format for all types of transport, momentum, energy, mass) flu in a given direction Quantity of property being transferred ( time)( area) More can be transported

More information

PHY121 Physics for the Life Sciences I

PHY121 Physics for the Life Sciences I PHY Physics for the Life Sciences I Lecture 0. Fluid flow: kinematics describing the motion. Fluid flow: dynamics causes and effects, Bernoulli s Equation 3. Viscosity and Poiseuille s Law for narrow tubes

More information

Basic Fluid Mechanics

Basic Fluid Mechanics Basic Fluid Mechanics Chapter 3B: Conservation of Mass C3B: Conservation of Mass 1 3.2 Governing Equations There are two basic types of governing equations that we will encounter in this course Differential

More information

Chapter 5. The Differential Forms of the Fundamental Laws

Chapter 5. The Differential Forms of the Fundamental Laws Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations

More information

Getting started: CFD notation

Getting started: CFD notation PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =

More information

AE/ME 339. K. M. Isaac Professor of Aerospace Engineering. 12/21/01 topic7_ns_equations 1

AE/ME 339. K. M. Isaac Professor of Aerospace Engineering. 12/21/01 topic7_ns_equations 1 AE/ME 339 Professor of Aerospace Engineering 12/21/01 topic7_ns_equations 1 Continuity equation Governing equation summary Non-conservation form D Dt. V 0.(2.29) Conservation form ( V ) 0...(2.33) t 12/21/01

More information

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control

More information

Chapter 9: Differential Analysis of Fluid Flow

Chapter 9: Differential Analysis of Fluid Flow of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known

More information

Macroscopic Momentum Balances

Macroscopic Momentum Balances Lecture 13 F. Morrson CM3110 2013 10/22/2013 CM3110 Transport I Part I: Flud Mechancs Macroscopc Momentum Balances Professor Fath Morrson Department of Chemcal Engneerng Mchgan Technologcal Unersty 1 Macroscopc

More information

Boundary Conditions in Fluid Mechanics

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

More information

Viscosity and Polymer Melt Flow. Rheology-Processing / Chapter 2 1

Viscosity and Polymer Melt Flow. Rheology-Processing / Chapter 2 1 Viscosity and Polymer Melt Flow Rheology-Processing / Chapter 2 1 Viscosity: a fluid property resistance to flow (a more technical definition resistance to shearing) Remember that: τ μ du dy shear stress

More information

150A Review Session 2/13/2014 Fluid Statics. Pressure acts in all directions, normal to the surrounding surfaces

150A Review Session 2/13/2014 Fluid Statics. Pressure acts in all directions, normal to the surrounding surfaces Fluid Statics Pressure acts in all directions, normal to the surrounding surfaces or Whenever a pressure difference is the driving force, use gauge pressure o Bernoulli equation o Momentum balance with

More information

Chapter 2: 1D Kinematics Tuesday January 13th

Chapter 2: 1D Kinematics Tuesday January 13th Chapter : D Kinematics Tuesday January 3th Motion in a straight line (D Kinematics) Aerage elocity and aerage speed Instantaneous elocity and speed Acceleration Short summary Constant acceleration a special

More information

Chapter 6: Momentum Analysis

Chapter 6: Momentum Analysis 6-1 Introduction 6-2Newton s Law and Conservation of Momentum 6-3 Choosing a Control Volume 6-4 Forces Acting on a Control Volume 6-5Linear Momentum Equation 6-6 Angular Momentum 6-7 The Second Law of

More information

VISUAL PHYSICS ONLINE RECTLINEAR MOTION: UNIFORM ACCELERATION

VISUAL PHYSICS ONLINE RECTLINEAR MOTION: UNIFORM ACCELERATION VISUAL PHYSICS ONLINE RECTLINEAR MOTION: UNIFORM ACCELERATION Predict Obsere Explain Exercise 1 Take an A4 sheet of paper and a heay object (cricket ball, basketball, brick, book, etc). Predict what will

More information

FORMULA SHEET. General formulas:

FORMULA SHEET. General formulas: FORMULA SHEET You may use this formula sheet during the Advanced Transport Phenomena course and it should contain all formulas you need during this course. Note that the weeks are numbered from 1.1 to

More information

Fluid Mechanics Qualifying Examination Sample Exam 2

Fluid Mechanics Qualifying Examination Sample Exam 2 Fluid Mechanics Qualifying Examination Sample Exam 2 Allotted Time: 3 Hours The exam is closed book and closed notes. Students are allowed one (double-sided) formula sheet. There are five questions on

More information

SLIP MODEL PERFORMANCE FOR MICRO-SCALE GAS FLOWS

SLIP MODEL PERFORMANCE FOR MICRO-SCALE GAS FLOWS 3th AIAA Thermophysics Conference 3- June 3, Orlando, Florida AIAA 3-5 SLIP MODEL PERFORMANCE FOR MICRO-SCALE GAS FLOWS Matthew J. McNenly* Department of Aerospace Engineering Uniersity of Michigan, Ann

More information

UNDERSTAND MOTION IN ONE AND TWO DIMENSIONS

UNDERSTAND MOTION IN ONE AND TWO DIMENSIONS SUBAREA I. COMPETENCY 1.0 UNDERSTAND MOTION IN ONE AND TWO DIMENSIONS MECHANICS Skill 1.1 Calculating displacement, aerage elocity, instantaneous elocity, and acceleration in a gien frame of reference

More information

Differential relations for fluid flow

Differential relations for fluid flow Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow

More information

Chapter 3: Newtonian Fluid Mechanics QUICK START W

Chapter 3: Newtonian Fluid Mechanics QUICK START W Chapter 3: Newtonian Fluid Mechanics TWO GOAL Derive governing equations (mass and momentum balances olve governing equations or velocity and stress ields QUICK TART W First, beore we get deep into derivation,

More information

Section 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow

Section 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow Section 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow Anderson: Chapter 2 pp. 41-54 1 Equation of State: Section 1 Review p = R g T " > R g = R u M w - R u = 8314.4126

More information

vs. Chapter 4: Standard Flows Chapter 4: Standard Flows for Rheology shear elongation 2/1/2016 CM4650 Lectures 1-3: Intro, Mathematical Review

vs. Chapter 4: Standard Flows Chapter 4: Standard Flows for Rheology shear elongation 2/1/2016 CM4650 Lectures 1-3: Intro, Mathematical Review CM465 Lectures -3: Intro, Mathematical //6 Chapter 4: Standard Flows CM465 Polymer Rheology Michigan Tech Newtonian fluids: vs. non-newtonian fluids: How can we investigate non-newtonian behavior? CONSTANT

More information

ESCI 485 Air/sea Interaction Lesson 3 The Surface Layer

ESCI 485 Air/sea Interaction Lesson 3 The Surface Layer ESCI 485 Air/sea Interaction Lesson 3 he Surface Layer References: Air-sea Interaction: Laws and Mechanisms, Csanady Structure of the Atmospheric Boundary Layer, Sorbjan HE PLANEARY BOUNDARY LAYER he atmospheric

More information

Supplementary Information Microfluidic quadrupole and floating concentration gradient Mohammad A. Qasaimeh, Thomas Gervais, and David Juncker

Supplementary 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 information

MA3D1 Fluid Dynamics Support Class 5 - Shear Flows and Blunt Bodies

MA3D1 Fluid Dynamics Support Class 5 - Shear Flows and Blunt Bodies MA3D1 Fluid Dynamics Support Class 5 - Shear Flows and Blunt Bodies 13th February 2015 Jorge Lindley email: J.V.M.Lindley@warwick.ac.uk 1 2D Flows - Shear flows Example 1. Flow over an inclined plane A

More information

REE Internal Fluid Flow Sheet 2 - Solution Fundamentals of Fluid Mechanics

REE Internal Fluid Flow Sheet 2 - Solution Fundamentals of Fluid Mechanics REE 307 - Internal Fluid Flow Sheet 2 - Solution Fundamentals of Fluid Mechanics 1. Is the following flows physically possible, that is, satisfy the continuity equation? Substitute the expressions for

More information

The Kinetic Theory of Gases

The Kinetic Theory of Gases 978-1-107-1788-3 Classical and Quantum Thermal Physics The Kinetic Theory of Gases CHAPTER 1 1.0 Kinetic Theory, Classical and Quantum Thermodynamics Two important components of the unierse are: the matter

More information

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

More information

CJ57.P.003 REASONING AND SOLUTION According to the impulse-momentum theorem (see Equation 7.4), F t = mv

CJ57.P.003 REASONING AND SOLUTION According to the impulse-momentum theorem (see Equation 7.4), F t = mv Solution to HW#7 CJ57.CQ.003. RASONNG AND SOLUTON a. Yes. Momentum is a ector, and the two objects hae the same momentum. This means that the direction o each object s momentum is the same. Momentum is

More information

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept. AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says

More information

Chapter 2: Fluid Dynamics Review

Chapter 2: Fluid Dynamics Review 7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading

More information

(a) During the first part of the motion, the displacement is x 1 = 40 km and the time interval is t 1 (30 km / h) (80 km) 40 km/h. t. (2.

(a) During the first part of the motion, the displacement is x 1 = 40 km and the time interval is t 1 (30 km / h) (80 km) 40 km/h. t. (2. Chapter 3. Since the trip consists of two parts, let the displacements during first and second parts of the motion be x and x, and the corresponding time interals be t and t, respectiely. Now, because

More information

0 a 3 a 2 a 3 0 a 1 a 2 a 1 0

0 a 3 a 2 a 3 0 a 1 a 2 a 1 0 Chapter Flow kinematics Vector and tensor formulae This introductory section presents a brief account of different definitions of ector and tensor analysis that will be used in the following chapters.

More information

PHYS 1443 Section 004 Lecture #4 Thursday, Sept. 4, 2014

PHYS 1443 Section 004 Lecture #4 Thursday, Sept. 4, 2014 PHYS 1443 Section 004 Lecture #4 Thursday, Sept. 4, 014 One Dimensional Motion Motion under constant acceleration One dimensional Kinematic Equations How do we sole kinematic problems? Falling motions

More information

CHAPTER 8 ENTROPY GENERATION AND TRANSPORT

CHAPTER 8 ENTROPY GENERATION AND TRANSPORT CHAPTER 8 ENTROPY GENERATION AND TRANSPORT 8.1 CONVECTIVE FORM OF THE GIBBS EQUATION In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?

More information

BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW

BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW Proceedings of,, BOUNDARY LAYER ANALYSIS WITH NAVIER-STOKES EQUATION IN 2D CHANNEL FLOW Yunho Jang Department of Mechanical and Industrial Engineering University of Massachusetts Amherst, MA 01002 Email:

More information

r t t x t y t z t, y t are zero, then construct a table for all four functions. dy dx 0 and 0 dt dt horizontal tangent vertical tangent

r t t x t y t z t, y t are zero, then construct a table for all four functions. dy dx 0 and 0 dt dt horizontal tangent vertical tangent 3. uggestions for the Formula heets Below are some suggestions for many more formulae than can be placed easily on both sides of the two standard 8½"" sheets of paper for the final examination. It is strongly

More information

Lesson 10 Steady Electric Currents

Lesson 10 Steady Electric Currents Lesson Steady lectric Currents 楊尚達 Shang-Da Yang Institute of Photonics Technologies Department of lectrical ngineering National Tsing Hua Uniersity, Taiwan Outline Current density Current laws Boundary

More information

Chapter 6: Momentum Analysis of Flow Systems

Chapter 6: Momentum Analysis of Flow Systems Chapter 6: Momentum Analysis of Flow Systems Introduction Fluid flow problems can be analyzed using one of three basic approaches: differential, experimental, and integral (or control volume). In Chap.

More information

AA210A Fundamentals of Compressible Flow. Chapter 5 -The conservation equations

AA210A Fundamentals of Compressible Flow. Chapter 5 -The conservation equations AA210A Fundamentals of Compressible Flow Chapter 5 -The conservation equations 1 5.1 Leibniz rule for differentiation of integrals Differentiation under the integral sign. According to the fundamental

More information

ME 509, Spring 2016, Final Exam, Solutions

ME 509, Spring 2016, Final Exam, Solutions ME 509, Spring 2016, Final Exam, Solutions 05/03/2016 DON T BEGIN UNTIL YOU RE TOLD TO! Instructions: This exam is to be done independently in 120 minutes. You may use 2 pieces of letter-sized (8.5 11

More information

Feb 6, 2013 PHYSICS I Lecture 5

Feb 6, 2013 PHYSICS I Lecture 5 95.141 Feb 6, 213 PHYSICS I Lecture 5 Course website: faculty.uml.edu/pchowdhury/95.141/ www.masteringphysics.com Course: UML95141SPRING213 Lecture Capture h"p://echo36.uml.edu/chowdhury213/physics1spring.html

More information

ENGI Gradient, Divergence, Curl Page 5.01

ENGI Gradient, Divergence, Curl Page 5.01 ENGI 94 5. - Gradient, Divergence, Curl Page 5. 5. The Gradient Operator A brief review is provided here for the gradient operator in both Cartesian and orthogonal non-cartesian coordinate systems. Sections

More information

CM3110 Transport Processes and Unit Operations I

CM3110 Transport Processes and Unit Operations I CM3110 Transport Processes and Unit Operations I Professor Faith Morrison Department of Chemical Engineering Michigan Technological University CM3110 - Momentum and Heat Transport CM310 Heat and Mass Transport

More information

CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer

CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer You are assigned to design a fallingcylinder viscometer to measure the viscosity of Newtonian liquids. A schematic

More information

Chapter 2 Motion Along a Straight Line

Chapter 2 Motion Along a Straight Line Chapter Motion Along a Straight Line In this chapter we will study how objects moe along a straight line The following parameters will be defined: (1) Displacement () Aerage elocity (3) Aerage speed (4)

More information

CHAPTER 4 ANALYTICAL SOLUTIONS OF COUPLE STRESS FLUID FLOWS THROUGH POROUS MEDIUM BETWEEN PARALLEL PLATES WITH SLIP BOUNDARY CONDITIONS

CHAPTER 4 ANALYTICAL SOLUTIONS OF COUPLE STRESS FLUID FLOWS THROUGH POROUS MEDIUM BETWEEN PARALLEL PLATES WITH SLIP BOUNDARY CONDITIONS CHAPTER 4 ANALYTICAL SOLUTIONS OF COUPLE STRESS FLUID FLOWS THROUGH POROUS MEDIUM BETWEEN PARALLEL PLATES WITH SLIP BOUNDARY CONDITIONS Introduction: The objective of this chapter is to establish analytical

More information

Part 2: CM3110 Transport Processes and Unit Operations I. Professor Faith Morrison. CM2110/CM Review. Concerned now with rates of heat transfer

Part 2: CM3110 Transport Processes and Unit Operations I. Professor Faith Morrison. CM2110/CM Review. Concerned now with rates of heat transfer CM30 anspot Pocesses and Unit Opeations I Pat : Pofesso Fait Moison Depatment of Cemical Engineeing Micigan ecnological Uniesity CM30 - Momentum and Heat anspot CM30 Heat and Mass anspot www.cem.mtu.edu/~fmoiso/cm30/cm30.tml

More information

THE EFFECT OF LONGITUDINAL VIBRATION ON LAMINAR FORCED CONVECTION HEAT TRANSFER IN A HORIZONTAL TUBE

THE 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 information

Ph.D. Qualifying Exam in Fluid Mechanics

Ph.D. Qualifying Exam in Fluid Mechanics Student ID Department of Mechanical Engineering Michigan State University East Lansing, Michigan Ph.D. Qualifying Exam in Fluid Mechanics Closed book and Notes, Some basic equations are provided on an

More information

AE/ME 339. K. M. Isaac. 9/22/2005 Topic 6 FluidFlowEquations_Introduction. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

AE/ME 339. K. M. Isaac. 9/22/2005 Topic 6 FluidFlowEquations_Introduction. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept. AE/ME 339 Computational Fluid Dynamics (CFD) 1...in the phrase computational fluid dynamics the word computational is simply an adjective to fluid dynamics.... -John D. Anderson 2 1 Equations of Fluid

More information

- Marine Hydrodynamics. Lecture 4. Knowns Equations # Unknowns # (conservation of mass) (conservation of momentum)

- Marine Hydrodynamics. Lecture 4. Knowns Equations # Unknowns # (conservation of mass) (conservation of momentum) 2.20 - Marine Hydrodynamics, Spring 2005 Lecture 4 2.20 - Marine Hydrodynamics Lecture 4 Introduction Governing Equations so far: Knowns Equations # Unknowns # density ρ( x, t) Continuity 1 velocities

More information

Lecture 12! Center of mass! Uniform circular motion!

Lecture 12! Center of mass! Uniform circular motion! Lecture 1 Center of mass Uniform circular motion Today s Topics: Center of mass Uniform circular motion Centripetal acceleration and force Banked cures Define the center of mass The center of mass is a

More information

Page 1. Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.)

Page 1. Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Page 1 Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Circle your lecture section (-1 point if not circled, or circled incorrectly): Prof. Vlachos Prof. Ardekani

More information

FLUID FLOW AND HEAT TRANSFER IN A COLLAPSIBLE TUBE

FLUID FLOW AND HEAT TRANSFER IN A COLLAPSIBLE TUBE FLUID DYNAMICS FLUID FLOW AND HEAT TRANSFER IN A COLLAPSIBLE TUBE S. A. ODEJIDE 1, Y. A. S. AREGBESOLA 1, O. D. MAKINDE 1 Obafemi Awolowo Uniersity, Department of Mathematics, Faculty of Science, Ile-Ife,

More information

Numerical Heat and Mass Transfer

Numerical Heat and Mass Transfer Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis

More information

6.1 Steady, One-Dimensional Rectilinear Flows Steady, Spherically Symmetric Radial Flows 42

6.1 Steady, One-Dimensional Rectilinear Flows Steady, Spherically Symmetric Radial Flows 42 Contents 6 UNIDIRECTIONAL FLOWS 1 6.1 Steady, One-Dimensional Rectilinear Flows 6. Steady, Axisymmetric Rectilinear Flows 19 6.3 Steady, Axisymmetric Torsional Flows 8 6.4 Steady, Axisymmetric Radial Flows

More information

Exercise 5: Exact Solutions to the Navier-Stokes Equations I

Exercise 5: Exact Solutions to the Navier-Stokes Equations I Fluid Mechanics, SG4, HT009 September 5, 009 Exercise 5: Exact Solutions to the Navier-Stokes Equations I Example : Plane Couette Flow Consider the flow of a viscous Newtonian fluid between two parallel

More information

Prediction of Coating Thickness

Prediction of Coating Thickness Prediction of Coating Tickness Jon D. Wind Surface Penomena CE 385M 4 May 1 Introduction Tis project involves te modeling of te coating of metal plates wit a viscous liquid by pulling te plate vertically

More information

In this section, mathematical description of the motion of fluid elements moving in a flow field is

In this section, mathematical description of the motion of fluid elements moving in a flow field is Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small

More information

4 Fundamentals of Continuum Thermomechanics

4 Fundamentals of Continuum Thermomechanics 4 Fundamentals of Continuum Thermomechanics In this Chapter, the laws of thermodynamics are reiewed and formulated for a continuum. The classical theory of thermodynamics, which is concerned with simple

More information

Note: the net distance along the path is a scalar quantity its direction is not important so the average speed is also a scalar.

Note: the net distance along the path is a scalar quantity its direction is not important so the average speed is also a scalar. PHY 309 K. Solutions for the first mid-term test /13/014). Problem #1: By definition, aerage speed net distance along the path of motion time. 1) ote: the net distance along the path is a scalar quantity

More information

Entropy generation and transport

Entropy generation and transport Chapter 7 Entropy generation and transport 7.1 Convective form of the Gibbs equation In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?

More information

FLUID MECHANICS EQUATIONS

FLUID MECHANICS EQUATIONS FLUID MECHANIC EQUATION M. Ragheb 11/2/2017 INTRODUCTION The early part of the 18 th -century saw the burgeoning of the field of theoretical fluid mechanics pioneered by Leonhard Euler and the father and

More information

2 Law of conservation of energy

2 Law of conservation of energy 1 Newtonian viscous Fluid 1 Newtonian fluid For a Newtonian we already have shown that σ ij = pδ ij + λd k,k δ ij + 2µD ij where λ and µ are called viscosity coefficient. For a fluid under rigid body motion

More information

Detailed Outline, M E 521: Foundations of Fluid Mechanics I

Detailed Outline, M E 521: Foundations of Fluid Mechanics I Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Second-order tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic

More information

CIRCULAR MOTION EXERCISE 1 1. d = rate of change of angle

CIRCULAR MOTION EXERCISE 1 1. d = rate of change of angle CICULA MOTION EXECISE. d = rate of change of angle as they both complete angle in same time.. c m mg N r m N mg r Since r A r B N A N B. a Force is always perpendicular to displacement work done = 0 4.

More information

COMPARISON OF ANALYTICAL SOLUTIONS FOR CMSMPR CRYSTALLIZER WITH QMOM POPULATION BALANCE MODELING IN FLUENT

COMPARISON OF ANALYTICAL SOLUTIONS FOR CMSMPR CRYSTALLIZER WITH QMOM POPULATION BALANCE MODELING IN FLUENT CHINA PARTICUOLOGY Vol. 3, No. 4, 13-18, 5 COMPARISON OF ANALYTICAL SOLUTIONS FOR CMSMPR CRYSTALLIZER WITH QMOM POPULATION BALANCE MODELING IN FLUENT Bin Wan 1, Terry A. Ring 1, *, Kumar M. Dhanasekharan

More information

Kinetic plasma description

Kinetic plasma description Kinetic plasma description Distribution function Boltzmann and Vlaso equations Soling the Vlaso equation Examples of distribution functions plasma element t 1 r t 2 r Different leels of plasma description

More information

Physics 11 HW #7 Solutions

Physics 11 HW #7 Solutions hysics HW #7 Solutions Chapter 7: Focus On Concepts: 2, 6, 0, 3 robles: 8, 7, 2, 22, 32, 53, 56, 57 Focus On Concepts 7-2 (d) Moentu is a ector quantity that has a agnitude and a direction. The agnitudes

More information

Conservation of Mass. Computational Fluid Dynamics. The Equations Governing Fluid Motion

Conservation of Mass. Computational Fluid Dynamics. The Equations Governing Fluid Motion http://www.nd.edu/~gtryggva/cfd-course/ http://www.nd.edu/~gtryggva/cfd-course/ Computational Fluid Dynamics Lecture 4 January 30, 2017 The Equations Governing Fluid Motion Grétar Tryggvason Outline Derivation

More information

a by a factor of = 294 requires 1/T, so to increase 1.4 h 294 = h

a by a factor of = 294 requires 1/T, so to increase 1.4 h 294 = h IDENTIFY: If the centripetal acceleration matches g, no contact force is required to support an object on the spinning earth s surface. Calculate the centripetal (radial) acceleration /R using = πr/t to

More information

Fluid Dynamics for Ocean and Environmental Engineering Homework #2 Viscous Flow

Fluid Dynamics for Ocean and Environmental Engineering Homework #2 Viscous Flow OCEN 678-600 Fluid Dynamics for Ocean and Environmental Engineering Homework #2 Viscous Flow Date distributed : 9.18.2005 Date due : 9.29.2005 at 5:00 pm Return your solution either in class or in my mail

More information

Homework #4 Solution. μ 1. μ 2

Homework #4 Solution. μ 1. μ 2 Homework #4 Solution 4.20 in Middleman We have two viscous liquids that are immiscible (e.g. water and oil), layered between two solid surfaces, where the top boundary is translating: y = B y = kb y =

More information

Math 212-Lecture 20. P dx + Qdy = (Q x P y )da. C

Math 212-Lecture 20. P dx + Qdy = (Q x P y )da. C 15. Green s theorem Math 212-Lecture 2 A simple closed curve in plane is one curve, r(t) : t [a, b] such that r(a) = r(b), and there are no other intersections. The positive orientation is counterclockwise.

More information

P = 1 3 (σ xx + σ yy + σ zz ) = F A. It is created by the bombardment of the surface by molecules of fluid.

P = 1 3 (σ xx + σ yy + σ zz ) = F A. It is created by the bombardment of the surface by molecules of fluid. CEE 3310 Thermodynamic Properties, Aug. 27, 2010 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

More information

On my honor, I have neither given nor received unauthorized aid on this examination.

On my honor, I have neither given nor received unauthorized aid on this examination. Instructor(s): Field/Furic PHYSICS DEPARTENT PHY 2053 Exam 1 October 5, 2011 Name (print, last first): Signature: On my honor, I hae neither gien nor receied unauthorized aid on this examination. YOUR

More information

Review for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa

Review for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Review for Exam2 11. 13. 2015 Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Assistant Research Scientist IIHR-Hydroscience & Engineering, University

More information

FOCUS ON CONCEPTS Section 7.1 The Impulse Momentum Theorem

FOCUS ON CONCEPTS Section 7.1 The Impulse Momentum Theorem WEEK-6 Recitation PHYS 3 FOCUS ON CONCEPTS Section 7. The Impulse Momentum Theorem Mar, 08. Two identical cars are traeling at the same speed. One is heading due east and the other due north, as the drawing

More information

Note on Posted Slides. Motion Is Relative

Note on Posted Slides. Motion Is Relative Note on Posted Slides These are the slides that I intended to show in class on Tue. Jan. 9, 2014. They contain important ideas and questions from your reading. Due to time constraints, I was probably not

More information

Prediction of anode arc root position in a DC arc plasma torch

Prediction of anode arc root position in a DC arc plasma torch Prediction of anode arc root position in a DC arc plasma torch He-Ping Li 1, E. Pfender 1, Xi Chen 1 Department of Mechanical Engineering, Uniersity of Minnesota, Minneapolis, MN 55455, USA Department

More information

FLUID MECHANICS. Chapter 9 Flow over Immersed Bodies

FLUID 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 information

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2013

University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2013 Lecture 1 3/13/13 University of Washington Department of Chemistry Chemistry 53 Winter Quarter 013 A. Definition of Viscosity Viscosity refers to the resistance of fluids to flow. Consider a flowing liquid

More information

Chapter 4 Continuity Equation and Reynolds Transport Theorem

Chapter 4 Continuity Equation and Reynolds Transport Theorem Chapter 4 Continuity Equation and Reynolds Transport Theorem 4.1 Control Volume 4. The Continuity Equation for One-Dimensional Steady Flow 4.3 The Continuity Equation for Two-Dimensional Steady Flow 4.4

More information

PIPE FLOWS: LECTURE /04/2017. Yesterday, for the example problem Δp = f(v, ρ, μ, L, D) We came up with the non dimensional relation

PIPE FLOWS: LECTURE /04/2017. Yesterday, for the example problem Δp = f(v, ρ, μ, L, D) We came up with the non dimensional relation /04/07 ECTURE 4 PIPE FOWS: Yesterday, for the example problem Δp = f(v, ρ, μ,, ) We came up with the non dimensional relation f (, ) 3 V or, p f(, ) You can plot π versus π with π 3 as a parameter. Or,

More information

SPACE-TIME HOLOMORPHIC TIME-PERIODIC SOLUTIONS OF NAVIER-STOKES EQUATIONS. 1. Introduction We study Navier-Stokes equations in Lagrangean coordinates

SPACE-TIME HOLOMORPHIC TIME-PERIODIC SOLUTIONS OF NAVIER-STOKES EQUATIONS. 1. Introduction We study Navier-Stokes equations in Lagrangean coordinates Electronic Journal of Differential Equations, Vol. 2013 2013, No. 218, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SPACE-TIME HOLOMORPHIC

More information

MCAT Physics - Problem Drill 06: Translational Motion

MCAT Physics - Problem Drill 06: Translational Motion MCAT Physics - Problem Drill 06: Translational Motion Question No. 1 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as 1. An object falls from rest

More information

Basic Fluid Mechanics

Basic Fluid Mechanics Basic Fluid Mechanics Chapter 6A: Internal Incompressible Viscous Flow 4/16/2018 C6A: Internal Incompressible Viscous Flow 1 6.1 Introduction For the present chapter we will limit our study to incompressible

More information

Chapter 2: Basic Governing Equations

Chapter 2: Basic Governing Equations -1 Reynolds Transport Theorem (RTT) - Continuity Equation -3 The Linear Momentum Equation -4 The First Law of Thermodynamics -5 General Equation in Conservative Form -6 General Equation in Non-Conservative

More information

AMME2261: Fluid Mechanics 1 Course Notes

AMME2261: Fluid Mechanics 1 Course Notes Module 1 Introduction and Fluid Properties Introduction Matter can be one of two states: solid or fluid. A fluid is a substance that deforms continuously under the application of a shear stress, no matter

More information

FLOW-FORCE COMPENSATION IN A HYDRAULIC VALVE

FLOW-FORCE COMPENSATION IN A HYDRAULIC VALVE FLOW-FORCE COMPENSATION IN A HYDRAULIC VALVE Jan Lugowski Purdue Uniersity West Lafayette, IN, USA lugowskj@purdue.edu ABSTRACT Flow-reaction forces acting in hydraulic ales hae been studied for many decades.

More information