Chapter 8 Laminar Flows with Dependence on One Dimension

Size: px
Start display at page:

Download "Chapter 8 Laminar Flows with Dependence on One Dimension"

Transcription

1 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 Non-Newtonian luids Steady ilm low down inclined plane Unsteady iscous low Suddenly accelerated plate Deeloping Couette low Reading Assignment: Chapter o BSL, Transport Phenomena One dimensional (1-D) low ields are low ields that ary in only one spatial dimension in Cartesian coordinates. This ecludes turbulent lows because it cannot be one-dimensional. Acoustic waes are an eample o 1-D compressible low. We will concern ourseles here with incompressible 1-D low ields that result rom aial or planar symmetry. Cartesian, 1-D incompressible lows do not hae a elocity component (other than possibly a uniorm translation) in the direction o the spatial dependence because o the condition o ero diergence. Thus the nonlinear conectie deriatie disappears rom the equations o motion in Cartesian coordinates. They may not disappear with curilinear coordinates. ( ) 0 0 ( 0) 0 0 i, i 0 ρ p+ ρ + ρ + µ, 1, t p We can demonstrate that this relation may not apply in curilinear coordinates by considering an eample with cylindrical polar coordinates. Suppose that the only nonero component o elocity is in the direction and the only spatial dependence is on the r coordinate. The radial component o the conectie deriatie is non-ero due to centriugal orces. 8-1

2 [0, ( r), 0] r r [ ] The lows can be classiied as either orced low resulting rom the gradient o the pressure or the potential o the body orce or induced low resulting rom motion o one o the bounding suraces. Some low ields that result in 1-D low are listed below and illustrated in the ollowing igure (Churchill, 1988) 1. Forced low through a round tube. Forced low between parallel plates. Forced low through the annulus between concentric round tubes o dierent diameters 4. Graitational low o a liquid ilm down an inclined or ertical plane 5. Graitational low o a liquid ilm down the inner or outer surace o a round ertical tube 6. Graitational low o a liquid through an inclined hal-ull round tube 7. Flow induced by the moement o one o a pair o parallel planes 8. Flow induced in a concentric annulus between round tubes by the aial moement o either the outer or the inner tube 9. Flow induced in a concentric annulus between round tubes by the aial rotation o either the outer or the inner tube 10. Flow induced in the cylindrical layer o luid between a rotating circular disk and a parallel plane 11. Flow induced by the rotation o a central circular cylinder whose ais is perpendicular to parallel circular disks enclosing a thin cylindrical layer o luid 1. Combined orced and induced low between parallel plates 1. Combined orced and induced longitudinal low in the annulus between concentric round tubes 14. Combined orced and rotationally induced low in the annulus between concentric round tubes 8-

3 8-

4 Geometry and conditions that produce one-dimensional elocity ields (Churchill, 1988) Couette Flow The lows when the luid between two parallel suraces are induced to low by the motion o one surace relatie to the other is called Couette low. This is the generic shear low that is used to illustrate Newton's law o iscosity. Pressure and body orces balance each other and at steady state the equation o motion simpliy to the diergence o the iscous stress tensor or the Laplacian o elocity in the case o a Newtonian luid. Planar Couette low. (case 7). d µ d d 0, 1, d The coordinates system can be deined so that 0 at 0 and the component o elocity is non-ero at L. 0, 0 U, L, 1, The elocity ield is 8-4

5 U. L The shear stress can be determined rom Newton's law o iscosity. d U µ µ, 1, d L Cylindrical Couette low. The aboe eample was the translational moement o two planes relatie to each other. Couette low is also possible in the annular gap between two concentric cylindrical suraces (cases 8 and 9) i secondary lows do not occur due to centriugal orces. We use cylindrical polar coordinates rather than Cartesian and assume anishing Reynolds number. The only independent ariable is the radius. 1 ( rr ) 0 r r 1 [ ] ( rrr ), may not anish i Reynolds number is high r r r r 1 r r [ ] ( r r ) + 0, may not anish i Reynolds number is high r r r 1 [ ] ( rr ) 0 r r 1 ( r ) 0 r r r Newtonian luid r 0 r r The stress proile can be calculated by integration. r r r r r r r r 1 r r r r r r r r r r 1 r r The boundary conditions on elocity depend on whether the cylindrical suraces moe relatie to each other as a result o rotation, aial translation, or both. 0, r r (0, U, U ), r r 1 The elocity ield or cylindrical Couette low o a Newtonian luid is. 8-5

6 U r r 1 r r 1 r1 r r r 1 U log( r/ r1 ) log( r / r) 1 Planer rotational Couette low. The parallel plate iscometer has the coniguration shown in case 10. The system is not strictly 1-D because the elocity o one o the suraces is a unction o radius. Also, there is a centriugal orce present near the rotating surace but is absent at the stationary surace. Howeer, i the Reynolds number is small enough that secondary lows do not occur, then the elocity at a gien alue o the radius may be approimated as a unction o only the distance in the gap. The dierential equations at ero Reynolds number are as ollows. 0 0, Newtonian luid [ ] Suppose the bottom surace is stationary and the top surace is rotating. Then the boundary conditions are as ollows. 0, 0 π rω, L The stress and elocity proiles are as ollows. () r π rωµ ()/ r L π rω/ L, Newtonian luid The stress is a unction o the radius and i the luid is non-newtonian, the iscosity may be changing with radial position. Plane-Poiseuille and Hele-Shaw low Forced low between two stationary, parallel plates, case, is called plane-poiseuille low or i the low depends on two spatial ariables in the plane, it is called Hele-Shaw low. The low is orced by a speciied low rate or a speciied pressure or graity potential gradient. The pressure and graitational potential can be combined into a single ariable, P. 8-6

7 p + ρ P where P p+ ρ gh The product gh is the graitational potential, where g is the acceleration o graity and h is distance upward relatie to some datum. The pressure, p, is also relatie to a datum, which may be the datum or h. The primary spatial dependence is in the direction normal to the plane o the plates. I there is no dependence on one spatial direction, then the low is truly one-dimensional. Howeer, i the elocity and pressure gradients hae components in two directions in the plane o the plates, the low is not strictly 1-D and nonlinear, inertial terms will be present in the equations o motion. The signiicance o these terms is quantiied by the Reynolds number. I the low is steady, and the Reynolds number negligible, the equations o motion are as ollows. P 0, 1, P 0 0 P 0 + µ, 1,, Newtonian luid Let h be the spacing between the plates and the elocity is ero at each surace. 0, 0, h 1, The elocity proile or a Newtonian luid in plane-poiseuille low is, 1,, 0 µ h h h P h The aerage elocity oer the thickness o the plate can be determined by integrating the proile. h P, 1, 1µ This equation or the aerage elocity can be written as a ector equation i it is recognied that the ectors hae components only in the (1,) directions. 8-7

8 h P, ( 1, ), P P( 1, ) 1µ I the low is incompressible, the diergence o elocity is ero and the potential, P, is a solution o the Laplace equation ecept where sources are present. I the strength o the sources or the lu at boundaries are known, the potential, P, can be determined rom the methods or the solution o the Laplace equation. We now hae the result that the aerage elocity ector is proportional to a potential gradient. Thus the aerage elocity ield in a Hele-Shaw low is irrotational. I the luid is incompressible, the aerage elocity ield is also solenoidal can can be epressed as the curl o a ector potential or the stream unction. The aerage elocity ield o Hele-Shaw low is an physical analog or the irrotational, solenoidal, -D low described by the comple potential. It is also a physical analog or -D low o incompressible luids through porous media by Darcy s law and was used or that purpose beore numerical reseroir simulators were deeloped. Poiseuille Flow Poiseuille law describes laminar low o a Newtonian luid in a round tube (case 1). We will derie Poiseuille law or a Newtonian luid and leae the low o a power-law luid as an assignment. The equation o motion or the steady, deeloped (rom end eects) low o a luid in a round tube o uniorm radius is as ollows. P 0 r P 1 0 ( r r ), 0< r < R r r P µ r, Newtonian luid r r r The boundary conditions are symmetry at r 0 and no slip at r R. r µ 0 r 0 r 0, r R r 0 From the radial component o the equations o motion, P does not depend on radial position. Since the low is steady and ully deeloped, the gradient o P is a constant. The component o the equations o motion can be integrated once to derie the stress proile and wall shear stress. 8-8

9 P r r, 0< r < R P R w I the luid is Newtonian, the equation o motion can be integrated once more to obtain the elocity proile and maimum elocity. R P r 1, 0< r < R 4µ R, Newtonian luid R P,ma 4µ The olumetric rate o low through the pipe can be determined by integration o the elocity proile across the cross-section o the pipe, i.e., 0 < r < R and 0 < < π. 4 π R P Q, Newtonian luid 8µ This relation is the Hagen-Poiseuille law. I the low rate is speciied, then the potential gradient can be epressed as a unction o the low rate and substituted into the aboe epressions. The aerage elocity or olumetric lu can be determined by diiding the olumetric rate by the cross-sectional area. R P, Newtonian luid 8µ Beore one begins to beliee that the Hagen-Poiseuille law is a law that applies under all conditions, the ollowing is a list o assumptions are implicit in this relation (BSL, 1960). a. The low is laminar N Re less than about 100. b. The density ρ is constant ("incompressible low"). c. The low is independent o time ("steady state"). d. The luid is Newtonian. e. End eects are neglected actually an "entrance length" (beyond the tube entrance) on the order o L e 0.05D N Re is required or build-up to the parabolic proiles; i the section o pipe o interest includes the entrance region, a correction must be applied. The ractional correction introduced in either P or Q neer eceeds L e /L i L > L e. 8-9

10 . The luid behaes as a continuum this assumption is alid ecept or ery dilute gases or ery narrow capillary tubes, in which the molecular mean ree path is comparable to the tube diameter ("slip low" regime) or much greater than the tube diameter ("Knudsen low" or "ree molecule low" regime). g. There is no slip at the wall this is an ecellent assumption or pure luids under the conditions assumed in ( ). Friction actor and Reynolds number. Because pressure drop in pipes is commonly used in process design, correlation epressed as riction actor ersus Reynolds number are aailable or laminar and turbulent low. The Hagen- Poiseuille law describes the laminar low portion o the correlation. The correlations in the literature dier when they use dierent deinitions or the riction actor. Correlations are usually are usually epressed in terms o the Fanning riction actor and the Darcy-Weisbach riction actor. u SP F DW m ρ ρ u w um w m 8 ρ u w m, Stanton Pannell riction actor, Fanning riction actor ( ), Darcy Weisbach riction actor Moody, mean elocity The Reynolds number is epressed as a ratio o inertial stress and shear stress. N Re ρum ρum D ρum R µ u / D µ m µ Both the riction actor and the Reynolds number hae as a common actor, the kinetic energy per unit olume ρ u m. This actor may be eliminated between the two equations to epress the riction actor as a unction o the Reynolds number. SP F DW 1 w N µ u / D RE w N µ u / D RE 8 N RE m m w µ u / D m 8-10

11 Recall the epressions deried earlier or the wall shear stress and the aerage elocity or a Newtonian luid and substitute into the aboe epressions. SP F DW 8 N Re 16 N Re 64 N Re Correlation o riction actor ersus Reynolds number appear in the literature with all three deinitions o the riction actor and usually without a subscript to denote which deinition is being used. Non-Newtonian luids. The elocity proiles aboe were deried or a Newtonian luid. A constitutie relation is necessary to determine the elocity proile and mean elocity or non-newtonian luids. We will consider the cases o a Bingham model luid and a power-law or Ostwald-de Waele model luid. The constitutie relations or these luids are as ollows. The Bingham Model d y µ o ± o, i y > o dy d dy 0, i < The Ostwald de Waele (power law) Model y d m dy n 1 d dy y o The power-law model is an empirical model that is oten alid oer an intermediate range o shear rates. At ery low and ery high shear rates limiting alues o iscosity are approached. Assignment 8.1 Flow in annular space between concentric cylinders as unction o relatie translation, rotation, potential gradient, low or no-net low. Assume incompressible, Newtonian luid with small Reynolds number. The outer radius has ero elocity. Parameters: R outer radius R 1 inner radius, may be ero P potential gradient, may be ero 1 translation elocity o inner radius, may be ero rotational elocity o inner radius

12 q net low rate, may be ero a) Epress dimensionless elocity as a unction o the dimensionless radius and dimensionless groups. Plot the ollowing cases: Table o cases to plot Case R 1 /R P/ 1 1 q b) What is the net low i the inner cylinder is translating and the pressure gradient is ero? c) What is the pressure gradient i the net low is ero? Plot the elocity proile or this case. Assignment 8. Capillary low o power-law model luid. Calculate the ollowing or a power-law model luid (see hint in BSL, 1960). a) Calculate and plot the elocity proile, normalied with respect to the mean elocity or n 1, 0.67, 0.5, and 0.. b) Derie an epression corresponding to Poiseuille law. c) Derie the same relation between riction actor and Reynolds number as or Newtonian low by deining a modiied Reynolds number or power-law luids. Steady ilm low down inclined plane Steady ilm low down an inclined plane corresponds to case 4 (Churchill, 1988) or Section.; Flow o a Falling Film (BSL, 1960). These lows occur in chemical processing with alling ilm sulonation reactors, eaporation and gas adsorption, and ilm-condensation heat transer. It is assumed that the low is steady and there is no dependence on distance in the plane o the surace due to entrance eects, side walls, or ripples. The Reynolds number must be small enough or ripples to be aoided. The coniguration will be similar to that o BSL ecept 0 corresponds to the wall and the thickness is denoted by h rather than δ. It is assumed that the gas has negligible density compared to the liquid such that the pressure at the gas-liquid interace can be assumed to be constant. The potential gradient in the plane o the ilm is constant and can be epressed either in term o the angle rom the ertical, β, or the angle rom the horiontal, α. P ρ gcos β ρ gsin α, α π / β The equations o motion are as ollows. 8-1

13 dp 0 d d 0 ρ g cosβ d d 0 ρ g cos β + µ, Newtonian luid d The boundary conditions are ero stress at the gas-liquid interace and no slip at the wall. 0, 0 0, h The shear stress proile can be determined by integration and application o the ero stress boundary condition. ρ ghcos β 1, 0 h h ρ ghcos β w The elocity proile or a Newtonian luid can be determined by a second integration and application o the no slip boundary condition. ρ gh cos β µ h h,ma ρ gh cos β µ, 0 h The aerage elocity and olumetric low rate can be determined by integration o the elocity proile oer the ilm thickness. ρ gh cos β µ ρ gw h cos β Q µ The ilm thickness, h, can be gien in terms o the aerage elocity, the olume rate o low, or the mass rate o low per unit width o wall ( Γ ρ h ): 8-1

14 µ µ Q µ Γ h ρ gcos β ρ gw cos β ρ gcos β Unsteady iscous low Suddenly accelerated plate. (BSL, 1960) A semi-ininite body o liquid with constant density and iscosity is bounded on one side by a lat surace ( the plane). Initially the luid and solid surace is at rest; but at time t 0 the solid surace is set in motion in the positie -direction with a elocity U. It is desired to know the elocity as a unction o y and t. The pressure is hydrostatic and the low is assumed to be laminar. The only nonero component o elocity is (y,t). Thus the only nonero equation o motion is as ollows. ρ µ, y > 0, t >0 t y The initial condition and boundary conditions are as ollows. 0, t 0, y > 0 U, y 0, t > 0 0, y, t > 0 I we normalie the elocity with respect to the boundary condition, we see that this is the same parabolic PDE and boundary condition as we soled with a similarity transormation. Thus the solution is y Uerc 4 µ t / ρ The presence o the ratio o iscosity and density, the kinematic iscosity, in the epression or the elocity implies that both iscous and inertial orces are operatie. 8-14

15 The elocity proiles or the wall at y 0 suddenly set in motion is illustrated below. 1 Wall set into motion; t Velocity y Deeloping Couette low. The transient deelopment to the steady-state Couette low discussed earlier can now be easily deried. We will let the plane y 0 be the surace with ero elocity and let the elocity be speciied at y L. The initial and boundary conditions are as ollows. 0, t 0, 0 < y< L 0, y 0, t > 0 U, y L, t > 0 or 0, t 0, L< y< L U, y L, t > 0 U, y L, t > 0 It should be apparent that the two ormulations o the boundary conditions gie the same solution. Howeer, the latter gies a clue how one should obtain a solution. The solution is antisymmetric about y 0 and the ero elocity 8-15

16 condition is satisied. A series o additional terms are needed to satisy the boundary conditions at y ± L. The solution is (n+ 1) L y (n+ 1) L+ y U erc erc n 0 4νt 4νt µ ν ρ 1 Deeloping Couette Flow; t Velocity y 8-16

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

OE4625 Dredge Pumps and Slurry Transport. Vaclav Matousek October 13, 2004

OE4625 Dredge Pumps and Slurry Transport. Vaclav Matousek October 13, 2004 OE465 Vaclav Matousek October 13, 004 1 Dredge Vermelding Pumps onderdeel and Slurry organisatie Transport OE465 Vaclav Matousek October 13, 004 Dredge Vermelding Pumps onderdeel and Slurry organisatie

More information

Lesson 6: Apparent weight, Radial acceleration (sections 4:9-5.2)

Lesson 6: Apparent weight, Radial acceleration (sections 4:9-5.2) Beore we start the new material we will do another Newton s second law problem. A bloc is being pulled by a rope as shown in the picture. The coeicient o static riction is 0.7 and the coeicient o inetic

More information

Chapter 11 Collision Theory

Chapter 11 Collision Theory Chapter Collision Theory Introduction. Center o Mass Reerence Frame Consider two particles o masses m and m interacting ia some orce. Figure. Center o Mass o a system o two interacting particles Choose

More information

Chapter 3 Water Flow in Pipes

Chapter 3 Water Flow in Pipes The Islamic University o Gaza Faculty o Engineering Civil Engineering Department Hydraulics - ECI 33 Chapter 3 Water Flow in Pipes 3. Description o A Pipe Flow Water pipes in our homes and the distribution

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

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

FLUID MECHANICS. Lecture 7 Exact solutions

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 information

Department of Physics PHY 111 GENERAL PHYSICS I

Department of Physics PHY 111 GENERAL PHYSICS I EDO UNIVERSITY IYAMHO Department o Physics PHY 111 GENERAL PHYSICS I Instructors: 1. Olayinka, S. A. (Ph.D.) Email: akinola.olayinka@edouniersity.edu.ng Phone: (+234) 8062447411 2. Adekoya, M. A Email:

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

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

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

CRITICAL MASS FLOW RATE THROUGH CAPILLARY TUBES

CRITICAL MASS FLOW RATE THROUGH CAPILLARY TUBES Proceedings o the ASME 010 rd Joint US-European Fluids Engineering Summer Meeting and 8th International Conerence FESM-ICNMM010 August 1-5, 010, Montreal, Canada Proceedings o ASME 010 rd Joint US-European

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

CONVECTIVE HEAT TRANSFER CHARACTERISTICS OF NANOFLUIDS. Convective heat transfer analysis of nanofluid flowing inside a

CONVECTIVE HEAT TRANSFER CHARACTERISTICS OF NANOFLUIDS. Convective heat transfer analysis of nanofluid flowing inside a Chapter 4 CONVECTIVE HEAT TRANSFER CHARACTERISTICS OF NANOFLUIDS Convective heat transer analysis o nanoluid lowing inside a straight tube o circular cross-section under laminar and turbulent conditions

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-III CONVECTION IN A POROUS MEDIUM WITH EFFECT OF MAGNETIC FIELD, VARIABLE FLUID PROPERTIES AND VARYING WALL TEMPERATURE

CHAPTER-III CONVECTION IN A POROUS MEDIUM WITH EFFECT OF MAGNETIC FIELD, VARIABLE FLUID PROPERTIES AND VARYING WALL TEMPERATURE CHAPER-III CONVECION IN A POROUS MEDIUM WIH EFFEC OF MAGNEIC FIELD, VARIABLE FLUID PROPERIES AND VARYING WALL EMPERAURE 3.1. INRODUCION Heat transer studies in porous media ind applications in several

More information

Phase Changes Heat must be added or removed to change a substance from one phase to another. Phases and Phase Changes. Evaporation

Phase Changes Heat must be added or removed to change a substance from one phase to another. Phases and Phase Changes. Evaporation Applied Heat Transer Part One (Heat Phase Changes Heat must be added or remoed to change a substance rom one phase to another. Ahmad RAMAZANI S.A. Associate Proessor Shari Uniersity o Technology انتقال

More information

ROAD MAP... D-1: Aerodynamics of 3-D Wings D-2: Boundary Layer and Viscous Effects D-3: XFLR (Aerodynamics Analysis Tool)

ROAD MAP... D-1: Aerodynamics of 3-D Wings D-2: Boundary Layer and Viscous Effects D-3: XFLR (Aerodynamics Analysis Tool) AE301 Aerodynamics I UNIT D: Applied Aerodynamics ROAD MAP... D-1: Aerodynamics o 3-D Wings D-2: Boundary Layer and Viscous Eects D-3: XFLR (Aerodynamics Analysis Tool) AE301 Aerodynamics I : List o Subjects

More information

39.1 Gradually Varied Unsteady Flow

39.1 Gradually Varied Unsteady Flow 39.1 Gradually Varied Unsteady Flow Gradually varied unsteady low occurs when the low variables such as the low depth and velocity do not change rapidly in time and space. Such lows are very common in

More information

Differential Equations

Differential Equations LOCUS Dierential Equations CONCEPT NOTES 0. Motiation 0. Soling Dierential Equations LOCUS Dierential Equations Section - MOTIVATION A dierential equation can simpl be said to be an equation inoling deriaties

More information

Comments on Magnetohydrodynamic Unsteady Flow of A Non- Newtonian Fluid Through A Porous Medium

Comments on Magnetohydrodynamic Unsteady Flow of A Non- Newtonian Fluid Through A Porous Medium Comments on Magnetohydrodynamic Unsteady Flow o A Non- Newtonian Fluid Through A Porous Medium Mostaa A.A.Mahmoud Department o Mathematics, Faculty o Science, Benha University (358), Egypt Abstract The

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

Microscopic Momentum Balance Equation (Navier-Stokes)

Microscopic Momentum Balance Equation (Navier-Stokes) 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

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

MOTION OF FALLING OBJECTS WITH RESISTANCE

MOTION OF FALLING OBJECTS WITH RESISTANCE DOING PHYSICS WIH MALAB MECHANICS MOION OF FALLING OBJECS WIH RESISANCE Ian Cooper School of Physics, Uniersity of Sydney ian.cooper@sydney.edu.au DOWNLOAD DIRECORY FOR MALAB SCRIPS mec_fr_mg_b.m Computation

More information

Controlling the Heat Flux Distribution by Changing the Thickness of Heated Wall

Controlling the Heat Flux Distribution by Changing the Thickness of Heated Wall J. Basic. Appl. Sci. Res., 2(7)7270-7275, 2012 2012, TextRoad Publication ISSN 2090-4304 Journal o Basic and Applied Scientiic Research www.textroad.com Controlling the Heat Flux Distribution by Changing

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

2015 American Journal of Engineering Research (AJER)

2015 American Journal of Engineering Research (AJER) American Journal o Engineering Research (AJER) 2015 American Journal o Engineering Research (AJER) e-issn: 2320-0847 p-issn : 2320-0936 Volume-4, Issue-7, pp-33-40.ajer.org Research Paper Open Access The

More information

Exercise: concepts from chapter 10

Exercise: concepts from chapter 10 Reading:, Ch 10 1) The flow of magma with a viscosity as great as 10 10 Pa s, let alone that of rock with a viscosity of 10 20 Pa s, is difficult to comprehend because our common eperience is with s like

More information

Physics 107 HOMEWORK ASSIGNMENT #9b

Physics 107 HOMEWORK ASSIGNMENT #9b Physics 07 HOMEWORK SSIGNMENT #9b Cutnell & Johnson, 7 th edition Chapter : Problems 5, 58, 66, 67, 00 5 Concept Simulation. reiews the concept that plays the central role in this problem. (a) The olume

More information

NON-SIMILAR SOLUTIONS FOR NATURAL CONVECTION FROM A MOVING VERTICAL PLATE WITH A CONVECTIVE THERMAL BOUNDARY CONDITION

NON-SIMILAR SOLUTIONS FOR NATURAL CONVECTION FROM A MOVING VERTICAL PLATE WITH A CONVECTIVE THERMAL BOUNDARY CONDITION NON-SIMILAR SOLUTIONS FOR NATURAL CONVECTION FROM A MOVING VERTICAL PLATE WITH A CONVECTIVE THERMAL BOUNDARY CONDITION by Asterios Pantokratoras School o Engineering, Democritus University o Thrace, 67100

More information

Analysis of Non-Thermal Equilibrium in Porous Media

Analysis of Non-Thermal Equilibrium in Porous Media Analysis o Non-Thermal Equilibrium in Porous Media A. Nouri-Borujerdi, M. Nazari 1 School o Mechanical Engineering, Shari University o Technology P.O Box 11365-9567, Tehran, Iran E-mail: anouri@shari.edu

More information

Resistance in Open Channel Hydraulics

Resistance in Open Channel Hydraulics 16.3.1 Resistance in Open Channel Hdraulics I Manning and Chez equations are compared e1 2 3 1 1 1 2 0 = 2 2 0 1 R S CR S n 21 1-3 2 6 R R C= = n n 1 6 R C= n For laminar low: K R R e1 VR = VR K= 2 2 8gSR

More information

Section 6: PRISMATIC BEAMS. Beam Theory

Section 6: PRISMATIC BEAMS. Beam Theory Beam Theory There are two types of beam theory aailable to craft beam element formulations from. They are Bernoulli-Euler beam theory Timoshenko beam theory One learns the details of Bernoulli-Euler beam

More information

Non-newtonian Rabinowitsch Fluid Effects on the Lubrication Performances of Sine Film Thrust Bearings

Non-newtonian Rabinowitsch Fluid Effects on the Lubrication Performances of Sine Film Thrust Bearings International Journal o Mechanical Engineering and Applications 7; 5(): 6-67 http://www.sciencepublishinggroup.com/j/ijmea doi:.648/j.ijmea.75.4 ISSN: -X (Print); ISSN: -48 (Online) Non-newtonian Rabinowitsch

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

COMBINED EFFECTS OF RADIATION AND HEAT GENERATION ON MHD NATURAL CONVECTION FLOW ALONG A VERTICAL FLAT PLATE IN PRESENCE OF HEAT CONDUCTION

COMBINED EFFECTS OF RADIATION AND HEAT GENERATION ON MHD NATURAL CONVECTION FLOW ALONG A VERTICAL FLAT PLATE IN PRESENCE OF HEAT CONDUCTION BRAC University Journal, vol.vi, no., 9, pp 11- COMBINED EFFECTS OF RADIATION AND HEAT GENERATION ON MHD NATURAL CONVECTION FLOW ALONG A VERTICAL FLAT PLATE IN PRESENCE OF HEAT CONDUCTION Mohammad Mokaddes

More information

Two Phase Pressure Drop of CO2, Ammonia, and R245fa in Multiport Aluminum Microchannel Tubes

Two Phase Pressure Drop of CO2, Ammonia, and R245fa in Multiport Aluminum Microchannel Tubes Purdue Uniersity Purdue e-pubs International Refrigeration and Air Conditioning Conference School of Mechanical Engineering 6 Two Phase Pressure Drop of CO, Ammonia, and R45fa in Multiport Aluminum Microchannel

More information

Chapter (3) Motion. in One. Dimension

Chapter (3) Motion. in One. Dimension Chapter (3) Motion in One Dimension Pro. Mohammad Abu Abdeen Dr. Galal Ramzy Chapter (3) Motion in one Dimension We begin our study o mechanics by studying the motion o an object (which is assumed to be

More information

HYDROMAGNETIC DIVERGENT CHANNEL FLOW OF A VISCO- ELASTIC ELECTRICALLY CONDUCTING FLUID

HYDROMAGNETIC DIVERGENT CHANNEL FLOW OF A VISCO- ELASTIC ELECTRICALLY CONDUCTING FLUID Rita Choudhury et al. / International Journal o Engineering Science and Technology (IJEST) HYDROAGNETIC DIVERGENT CHANNEL FLOW OF A VISCO- ELASTIC ELECTRICALLY CONDUCTING FLUID RITA CHOUDHURY Department

More information

Contents. Microfluidics - Jens Ducrée Physics: Laminar and Turbulent Flow 1

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

More information

A Semi-Analytical Solution for a Porous Channel Flow of a Non-Newtonian Fluid

A Semi-Analytical Solution for a Porous Channel Flow of a Non-Newtonian Fluid Journal o Applied Fluid Mechanics, Vol. 9, No. 6, pp. 77-76, 6. Available online at www.jamonline.net, ISSN 735-357, EISSN 735-3645. A Semi-Analytical Solution or a Porous Channel Flow o a Non-Newtonian

More information

FLOW CHARACTERISTICS OF HFC-134a IN AN ADIABATIC HELICAL CAPILLARY TUBE

FLOW CHARACTERISTICS OF HFC-134a IN AN ADIABATIC HELICAL CAPILLARY TUBE E HEFAT7 5 th International Conerence on Heat Transer, Fluid Mechanics and Thermodynamics Sun City, South Arica Paper number: KM1 FLOW CHARACTERISTICS OF HFC-1a IN AN ADIABATIC HELICAL CAPILLARY TUBE Khan

More information

BOUNDARY LAYER ANALYSIS ALONG A STRETCHING WEDGE SURFACE WITH MAGNETIC FIELD IN A NANOFLUID

BOUNDARY LAYER ANALYSIS ALONG A STRETCHING WEDGE SURFACE WITH MAGNETIC FIELD IN A NANOFLUID Proceedings o the International Conerence on Mechanical Engineering and Reneable Energy 7 (ICMERE7) 8 December, 7, Chittagong, Bangladesh ICMERE7-PI- BOUNDARY LAYER ANALYSIS ALONG A STRETCHING WEDGE SURFACE

More information

Exam 3 Review. F P av A. m V

Exam 3 Review. F P av A. m V Chapter 9: luids Learn the physics o liquids and gases. States o Matter Solids, liquids, and gases. Exam 3 Reiew ressure a ascal s rinciple change in pressure at any point in a conined luid is transmitted

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

Chapter 4: Properties of Pure Substances. Pure Substance. Phases of a Pure Substance. Phase-Change Processes of Pure Substances

Chapter 4: Properties of Pure Substances. Pure Substance. Phases of a Pure Substance. Phase-Change Processes of Pure Substances Chapter 4: roperties o ure Substances ure Substance A substance that has a ixed chemical composition throughout is called a pure substance such as water, air, and nitrogen A pure substance does not hae

More information

Boundary 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 information

UNIT II CONVECTION HEAT TRANSFER

UNIT II CONVECTION HEAT TRANSFER UNIT II CONVECTION HEAT TRANSFER Convection is the mode of heat transfer between a surface and a fluid moving over it. The energy transfer in convection is predominately due to the bulk motion of the fluid

More information

Get Solution of These Packages & Learn by Video Tutorials on WAVES ON A STRING

Get Solution of These Packages & Learn by Video Tutorials on  WAVES ON A STRING WVES ON STRING WVES Wae motion is the phenomenon that can be obsered almost eerywhere around us, as well it appears in almost eery branch o physics. Surace waes on bodies o mater are commonly obsered.

More information

Demonstrating the Quantization of Electrical Charge Millikan s Experiment

Demonstrating the Quantization of Electrical Charge Millikan s Experiment Demonstrating the Quantization o Electrical Charge Millikan s Experiment Objecties o the experiment To demonstrate that electrical charge is quantized, and to determine the elementary electron charge by

More information

Dynamics ( 동역학 ) Ch.2 Motion of Translating Bodies (2.1 & 2.2)

Dynamics ( 동역학 ) Ch.2 Motion of Translating Bodies (2.1 & 2.2) Dynamics ( 동역학 ) Ch. Motion of Translating Bodies (. &.) Motion of Translating Bodies This chapter is usually referred to as Kinematics of Particles. Particles: In dynamics, a particle is a body without

More information

Buoyancy Driven Heat Transfer of Water-Based CuO Nanofluids in a Tilted Enclosure with a Heat Conducting Solid Cylinder on its Center

Buoyancy Driven Heat Transfer of Water-Based CuO Nanofluids in a Tilted Enclosure with a Heat Conducting Solid Cylinder on its Center July 4-6 2012 London U.K. Buoyancy Driven Heat Transer o Water-Based CuO Nanoluids in a Tilted Enclosure with a Heat Conducting Solid Cylinder on its Center Ahmet Cihan Kamil Kahveci and Çiğdem Susantez

More information

Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition

Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition C. Pozrikidis m Springer Contents Preface v 1 Introduction to Kinematics 1 1.1 Fluids and solids 1 1.2 Fluid parcels and flow

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

Geostrophy & Thermal wind

Geostrophy & Thermal wind Lecture 10 Geostrophy & Thermal wind 10.1 f and β planes These are planes that are tangent to the earth (taken to be spherical) at a point of interest. The z ais is perpendicular to the plane (anti-parallel

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

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

Figure 3: Problem 7. (a) 0.9 m (b) 1.8 m (c) 2.7 m (d) 3.6 m

Figure 3: Problem 7. (a) 0.9 m (b) 1.8 m (c) 2.7 m (d) 3.6 m 1. For the manometer shown in figure 1, if the absolute pressure at point A is 1.013 10 5 Pa, the absolute pressure at point B is (ρ water =10 3 kg/m 3, ρ Hg =13.56 10 3 kg/m 3, ρ oil = 800kg/m 3 ): (a)

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

RESOLUTION MSC.362(92) (Adopted on 14 June 2013) REVISED RECOMMENDATION ON A STANDARD METHOD FOR EVALUATING CROSS-FLOODING ARRANGEMENTS

RESOLUTION MSC.362(92) (Adopted on 14 June 2013) REVISED RECOMMENDATION ON A STANDARD METHOD FOR EVALUATING CROSS-FLOODING ARRANGEMENTS (Adopted on 4 June 203) (Adopted on 4 June 203) ANNEX 8 (Adopted on 4 June 203) MSC 92/26/Add. Annex 8, page THE MARITIME SAFETY COMMITTEE, RECALLING Article 28(b) o the Convention on the International

More information

Fluid Dynamics Exercises and questions for the course

Fluid Dynamics Exercises and questions for the course Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r

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

V/ t = 0 p/ t = 0 ρ/ t = 0. V/ s = 0 p/ s = 0 ρ/ s = 0

V/ t = 0 p/ t = 0 ρ/ t = 0. V/ s = 0 p/ s = 0 ρ/ s = 0 UNIT III FLOW THROUGH PIPES 1. List the types of fluid flow. Steady and unsteady flow Uniform and non-uniform flow Laminar and Turbulent flow Compressible and incompressible flow Rotational and ir-rotational

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

Displacement, Time, Velocity

Displacement, Time, Velocity Lecture. Chapter : Motion along a Straight Line Displacement, Time, Velocity 3/6/05 One-Dimensional Motion The area of physics that we focus on is called mechanics: the study of the relationships between

More information

Transport Properties: Momentum Transport, Viscosity

Transport Properties: Momentum Transport, Viscosity Transport Properties: Momentum Transport, Viscosity 13th February 2011 1 Introduction Much as mass(material) is transported within luids (gases and liquids), linear momentum is also associated with transport,

More information

Chapter 16. Kinetic Theory of Gases. Summary. molecular interpretation of the pressure and pv = nrt

Chapter 16. Kinetic Theory of Gases. Summary. molecular interpretation of the pressure and pv = nrt Chapter 16. Kinetic Theory of Gases Summary molecular interpretation of the pressure and pv = nrt the importance of molecular motions elocities and speeds of gas molecules distribution functions for molecular

More information

1. Introduction, tensors, kinematics

1. 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 information

FLUID MECHANICS. 1. Division of Fluid Mechanics. Hydrostatics Aerostatics Hydrodynamics Gasdynamics. v velocity p pressure ρ density

FLUID MECHANICS. 1. Division of Fluid Mechanics. Hydrostatics Aerostatics Hydrodynamics Gasdynamics. v velocity p pressure ρ density FLUID MECHANICS. Diision of Fluid Mechanics elocit p pressure densit Hdrostatics Aerostatics Hdrodnamics asdnamics. Properties of fluids Comparison of solid substances and fluids solid fluid τ F A [Pa]

More information

Fluid Physics 8.292J/12.330J

Fluid Physics 8.292J/12.330J Fluid Phsics 8.292J/12.0J Problem Set 4 Solutions 1. Consider the problem of a two-dimensional (infinitel long) airplane wing traeling in the negatie x direction at a speed c through an Euler fluid. In

More information

FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER

FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY OF AGRICULTURE DEPARTMENT OF AGRICULTURAL MACHINERY AND TECHNOLOGIES ENGINEERING 1 5. FLOW IN PIPES 5.1.3. Pressure and Shear Stress

More information

Principles of Convection

Principles of Convection Principles of Convection Point Conduction & convection are similar both require the presence of a material medium. But convection requires the presence of fluid motion. Heat transfer through the: Solid

More information

12d Model. Civil and Surveying Software. Version 7. Drainage Analysis Module Hydraulics. Owen Thornton BE (Mech), 12d Model Programmer

12d Model. Civil and Surveying Software. Version 7. Drainage Analysis Module Hydraulics. Owen Thornton BE (Mech), 12d Model Programmer 1d Model Civil and Surveying Sotware Version 7 Drainage Analysis Module Hydraulics Owen Thornton BE (Mech), 1d Model Programmer owen.thornton@1d.com 9 December 005 Revised: 10 January 006 8 February 007

More information

Chapter 8: MULTIPLE CONTINUOUS RANDOM VARIABLES

Chapter 8: MULTIPLE CONTINUOUS RANDOM VARIABLES Charles Boncelet Probabilit Statistics and Random Signals" Oord Uniersit Press 06. ISBN: 978-0-9-0005-0 Chapter 8: MULTIPLE CONTINUOUS RANDOM VARIABLES Sections 8. Joint Densities and Distribution unctions

More information

CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW

CHAPTER 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 information

Lesson 6 Review of fundamentals: Fluid flow

Lesson 6 Review of fundamentals: Fluid flow Lesson 6 Review of fundamentals: Fluid flow The specific objective of this lesson is to conduct a brief review of the fundamentals of fluid flow and present: A general equation for conservation of mass

More information

CEE 3310 Open Channel Flow,, Nov. 18,

CEE 3310 Open Channel Flow,, Nov. 18, CEE 3310 Open Channel Flow,, Nov. 18, 2016 165 8.1 Review Drag & Lit Laminar vs Turbulent Boundary Layer Turbulent boundary layers stay attached to bodies longer Narrower wake! Lower pressure drag! C D

More information

INFLUENCE OF POROSITY AND RADIATION IN A VISCO ELASTIC FLUID OF SECOND ORDER FLUID WITHIN A CHANNEL WITH PERMEABLE WALLS

INFLUENCE OF POROSITY AND RADIATION IN A VISCO ELASTIC FLUID OF SECOND ORDER FLUID WITHIN A CHANNEL WITH PERMEABLE WALLS International Journal o Physics and Mathematical Sciences ISSN: 77- (Online) Vol. () January-March, pp.8-9/murthy and Rajani INFLUENCE OF POROSITY AND RADIATION IN A VISCO ELASTIC FLUID OF SECOND ORDER

More information

Introduction to Thermodynamic Cycles Part 1 1 st Law of Thermodynamics and Gas Power Cycles

Introduction to Thermodynamic Cycles Part 1 1 st Law of Thermodynamics and Gas Power Cycles Introduction to Thermodynamic Cycles Part 1 1 st Law of Thermodynamics and Gas Power Cycles by James Doane, PhD, PE Contents 1.0 Course Oeriew... 4.0 Basic Concepts of Thermodynamics... 4.1 Temperature

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

Work and Kinetic Energy

Work and Kinetic Energy Work Work an Kinetic Energy Work (W) the prouct of the force eerte on an object an the istance the object moes in the irection of the force (constant force only). W = " = cos" (N " m = J)! is the angle

More information

12d Model. Civil and Surveying Software. Drainage Analysis Module Hydraulics. Owen Thornton BE (Mech), 12d Model Programmer.

12d Model. Civil and Surveying Software. Drainage Analysis Module Hydraulics. Owen Thornton BE (Mech), 12d Model Programmer. 1d Model Civil and Surveying Sotware Drainage Analysis Module Hydraulics Owen Thornton BE (Mech), 1d Model Programmer owen.thornton@1d.com 04 June 007 Revised: 3 August 007 (V8C1i) 04 February 008 (V8C1p)

More information

Work and Energy Problems

Work and Energy Problems 09//00 Multiple hoice orce o strength 0N acts on an object o ass 3kg as it oes a distance o 4. I is perpendicular to the 4 displaceent, the work done is equal to: Work and Energy Probles a) 0J b) 60J c)

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

(a) Taking the derivative of the position vector with respect to time, we have, in SI units (m/s),

(a) Taking the derivative of the position vector with respect to time, we have, in SI units (m/s), Chapter 4 Student Solutions Manual. We apply Eq. 4- and Eq. 4-6. (a) Taking the deriatie of the position ector with respect to time, we hae, in SI units (m/s), d ˆ = (i + 4t ˆj + tk) ˆ = 8tˆj + k ˆ. dt

More information

Department of Hydro Sciences, Institute for Urban Water Management. Urban Water

Department of Hydro Sciences, Institute for Urban Water Management. Urban Water Department of Hydro Sciences, Institute for Urban Water Management Urban Water 1 Global water aspects Introduction to urban water management 3 Basics for systems description 4 Water transport 5 Matter

More information

A. unchanged increased B. unchanged unchanged C. increased increased D. increased unchanged

A. unchanged increased B. unchanged unchanged C. increased increased D. increased unchanged IB PHYSICS Name: DEVIL PHYSICS Period: Date: BADDEST CLASS ON CAMPUS CHAPTER B TEST REVIEW. A rocket is fired ertically. At its highest point, it explodes. Which one of the following describes what happens

More information

Aerodynamic Admittance Function of Tall Buildings

Aerodynamic Admittance Function of Tall Buildings Aerodynamic Admittance Function o Tall Buildings in hou a Ahsan Kareem b a alou Engineering Int l, Inc., 75 W. Campbell Rd, Richardson, T, USA b Nataz odeling Laboratory, Uniersity o Notre Dame, Notre

More information

Mechanics for Vibratory Manipulation

Mechanics for Vibratory Manipulation Mechanics or Vibratory Manipulation Wesley H. Huang Matthew T. Mason Robotics Institute Carnegie Mellon Uniersity Abstract Vibratory manipulation is any mode o manipulation inoling repeated impacts due

More information

Class #4. Retarding forces. Worked Problems

Class #4. Retarding forces. Worked Problems Class #4 Retarding forces Stokes Law (iscous drag) Newton s Law (inertial drag) Reynolds number Plausibility of Stokes law Projectile motions with iscous drag Plausibility of Newton s Law Projectile motions

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

Second Order Slip Flow of Cu-Water Nanofluid Over a Stretching Sheet With Heat Transfer

Second Order Slip Flow of Cu-Water Nanofluid Over a Stretching Sheet With Heat Transfer Second Order Slip Flow o Cu-Water Nanoluid Over a Stretching Sheet With Heat Transer RAJESH SHARMA AND ANUAR ISHAK School o Mathematical Sciences, Faculty o Science and Technology Universiti Kebangsaan

More information

Lecture #8-6 Waves and Sound 1. Mechanical Waves We have already considered simple harmonic motion, which is an example of periodic motion in time.

Lecture #8-6 Waves and Sound 1. Mechanical Waves We have already considered simple harmonic motion, which is an example of periodic motion in time. Lecture #8-6 Waes and Sound 1. Mechanical Waes We hae already considered simple harmonic motion, which is an example of periodic motion in time. The position of the body is changing with time as a sinusoidal

More information

Chapter 1: Kinematics of Particles

Chapter 1: Kinematics of Particles Chapter 1: Kinematics of Particles 1.1 INTRODUCTION Mechanics the state of rest of motion of bodies subjected to the action of forces Static equilibrium of a body that is either at rest or moes with constant

More information

d dt T R m n p 1. (A) 4. (4) Carnot engine T Refrigerating effect W COPref. = 1 4 kw 5. (A)

d dt T R m n p 1. (A) 4. (4) Carnot engine T Refrigerating effect W COPref. = 1 4 kw 5. (A) . (A). (C) 5. (C) 7. ( to 5) 9. (C) 6. (C). (C). (D) 6. (A) 8. (0.6 to 0.66) 50. (D) 6. (C). (A) 5. (C) 7. (A) 9. (C) 5. (D) 6. (C). () 6. (C) 8. (600) 0. (D) 5. (B) 6. (D) 5. (A) 7. (A) 9. (D). (C) 5.

More information

Wind-Driven Circulation: Stommel s gyre & Sverdrup s balance

Wind-Driven Circulation: Stommel s gyre & Sverdrup s balance Wind-Driven Circulation: Stommel s gyre & Sverdrup s balance We begin by returning to our system o equations or low o a layer o uniorm density on a rotating earth. du dv h + [ u( H + h)] + [ v( H t y d

More information

UNIT I FLUID PROPERTIES AND STATICS

UNIT I FLUID PROPERTIES AND STATICS SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : Fluid Mechanics (16CE106) Year & Sem: II-B.Tech & I-Sem Course & Branch:

More information