F11AE1 1. C = ρν r r. r u z r


 Rosaline Mitchell
 2 years ago
 Views:
Transcription
1 F11AE1 1 Question 1 20 Marks) Consider an infinite horizontal pipe with circular crosssection of radius a, whose centre line is aligned along the zaxis; see Figure 1. Assume noslip boundary conditions at r = a, for all z, i.e. on the inside surface of the cylinder. Using the incompressible Navier Stokes equations in cylindrical polar coordinates see the formulae sheet at the back of the exam paper), look for a stationary solution to the fluid flow in the pipe of the following form. Assume there is no radial flow, u r = 0, and no swirl, u θ = 0. Further assume there is a constant pressure gradient down the pipe, i.e. that p = Cz for some constant C, and there is no body force. Lastly, suppose that the flow down the pipe, i.e. the velocity component u z, has the form u z = u z r) it is a function of r only). a) Using the Navier Stokes equations, show that 1 C = ρν r r b) Integrate the equation above to show that r u z r )). u z = C 4ρν r2 + A log r + B, where A and B are constants. We naturally require that the solution be bounded. Explain why this implies A = 0. Now use the noslip boundary condition to determine B. Hence show that u z = C 4ρν a2 r 2 ). c) Explain why the massflow rate across any cross section of the pipe is given by known as the fourth power law) ρu z ds and then evaluate this quantity for the case above.
2 F11AE1 2 a z Figure 1: Poiseuille flow: a viscous fluid flows along an infinite horizontal pipe with circular crosssection of radius a, whose centre line is aligned along the zaxis. A constant pressure gradient is assumed, as well as axisymmetry, no radial flow and no swirl.
3 F11AE1 3 Question 2 20 Marks) Consider the problem of an ideal, steady, incompressible flow in a channel over a gently undulating bed see Figure 2. Assume that the flow is shallow and uniform in crosssection. Upstream the flow is characterized by flow velocity U and depth H. The flow then impinges on a gently undulating bed of height y = yx) as shown in Figure 2, where x measures distance downstream. The depth of the flow is given by h = hx). The fluid velocity at that point is u = ux) and independent of the depth throughout. Also assume that the fluid depth h = hx) varies slowly as a function of position x. a) Explain why for all x we must have uh = UH. b) By applying Bernoulli s theorem to the surface streamline, for which the pressure is constant and equal to atmospheric pressure P 0, show that for all x: 1 2 U2 + gh = 1 2 u2 + gy + h). c) Using part a), show that the relation in part b) is equivalent to y = U2 2g + H h UH)2 2gh 2. For this relation, sketch y as a function of h, and show that this function has a unique global maximum at where Here F is known as the Froude number. y 0 := H F2 3 2 F2/3), h 0 = HF 2/3 F := U/ gh. d) Assume that the actual given undulation y = yx) attains an actual maximum value y max < y 0. Using part a) show that ) dy 1 dh = u2, gh and hence explain why the sign of 1 u 2 /gh is the same throughout the flow and determined by whether the Froude number F < 1 or F > 1. In each case, assuming the undulation y = yx) is a simple bump with a single maximum y max, sketch how you expect the depth of the fluid h = hx) to vary as a function of x.
4 F11AE1 4 P 0 hx) u H U yx) x Figure 2: Channel flow.
5 F11AE1 5 Question 3 20 Marks) Consider a uniform incompressible ideal flow of velocity U, into which we place a spherical obstacle, radius a. The set up is shown in Figure 3. Assume that the flow around the sphere is steady, incompressible and irrotational. Suppose we use spherical polar coordinates r,θ,ϕ) to represent the flow with the southnorth pole axis passing through the centre of the sphere and aligned with the uniform flow U at infinity. Assume further that the flow is axisymmetric, i.e. independent of the azimuthal angle ϕ, and there is no swirl so that u ϕ = 0. a) Explain why there exists a potential function φ satisfying 2 φ = 0 for this flow. Show that this is equivalent to 1 r 2φ ) + 1 sin θ φ )) = 0. r 2 r r sin θ θ θ b) The general solution to Laplace s equation in part a) is well known and it is given by φr,θ) = A n r n + B ) n P r n+1 n cos θ), n=0 where P n are the Legendre polynomials, with P 1 x) = x. The coefficients A n and B n are constants. State the uniform flow conditions as r and the boundary conditions at r = a. Hence show that the potential for this flow around the sphere is φ = Ur + a 3 /2r 2 ) cos θ. c) Show that the velocity field u = φ has components u r,u θ ) = U1 a 3 /r 3 ) cos θ, U1 + a 3 /2r 3 ) sin θ ). d) Now use Bernoulli s theorem to explain why 1 2 u 2 + P/ρ = 1 2 U2 + P /ρ, where P and u are the pressure and velocity fields, respectively, at any point r,θ,ϕ) with r a. Then using part c) show that on the sphere r = a the relation P P ρ = 1 2 U sin2 θ ), holds. Hence explain why the fluid exerts no net force on the sphere known as D Alembert s paradox).
6 F11AE1 6 U U r θ U U Figure 3: D Alembert s paradox: there is no net force on a solid sphere obstacle placed in a uniform flow for an ideal fluid.
7 F11AE1 7 Question 4 20 Marks) An incompressible homogeneous fluid occupies the region between two horizontal rigid parallel planes, which are a distance h apart, and outside a rigid cylinder of diameter a) which intersects the planes normally; see Figure 4 for the set up. Suppose that for this question x and y are horizontal Cartesian coordinates and z is the vertical coordinate. Let u,v,w) be the fluid velocity components in the three coordinate directions x, y and z, respectively. Assume throughout that a typical horizontal velocity scale for u,v) is U and a typical vertical velocity scale for w is W. a) Explain very briefly why a is a typical horizontal scale for x,y) and h a typical vertical scale for z. b) Hereafter further assume that h a and further that ρuh 2 aµ where µ = ρν is the first coefficient of viscosity. Using these assumptions, show that the Navier Stokes equations for an incompressible homogeneous fluid are well approximated by µ 2 u z 2 = p x, v µ2 z = p 2 y and p z = 0 for a steady flow, where p = px,y,z) is the pressure. c) We define the vertically averaged velocity components u,v) = ux,y),vx,y) ) for the flow in part b) by ux,y),vx,y) ) := 1 h h 0 ux,y),vx,y) ) dz. Use the incompressibility of u, v, w) to show that the vertically averaged velocity field u,v) is incompressible. Use that p/z = 0 to show that u,v) is irrotational.
8 F11AE1 8 fluid a cylinder h cylinder Side view Above view fluid Figure 4: Hele Shaw cell: an incompressible homogeneous fluid occupies the region between two parallel planes a distance h apart) and outside the cylinder of diameter a normal to the planes).
9 F11AE1 9 Question 5 20 Marks) Hill s spherical vortex is an exact solution of Euler s equation of motion for an incompressible fluid. The vorticity is confined to the interior of a uniformly translating sphere, of radius a, translating with velocity U. Outside the sphere the flow is irrotational. If we use spherical polar coordinates r, θ, ϕ) centred at the middle of the sphere, then without loss of generality, we can assume that the sphere is translating along the south to north pole axis. Further, we can assume the flow is axisymmetric with no swirl. a) Use the properties of axisymmetry and no swirl to show that in spherical polar coordinates the only nonzero component of vorticity ω = ω r,ω θ,ω ϕ ) inside the sphere is ω ϕ. b) Explain why there is a stream function ψ = ψr,θ,t) Stokes stream function) which is a solution of the differential equations u r = 1 ψ r 2 sin θ θ and u θ = 1 ψ r sin θ r in this case, where u r and u θ are the radial and latitudinal components of velocity. Show that the azimuthal vorticity ω ϕ is given in terms of Stokes stream function by ω ϕ = 1 2 ψ r sin θ r + sin θ 2 r 2 1 θ sin θ )) ψ. θ c) Use the two identities given in part IV) of the formulae sheet at the end of the exam paper, to prove that the incompressible Euler equations of motion imply that the vorticity field ω = u satisfies ω t + u ω = ω u. d) Now assume that the flow is steady so that ω/t = 0. Using the assumption of axisymmetry, no swirl, part a) above, and that in these circumstances ω ϕ ω u = 0, 0, ur sin θ + u θ cos θ )), r sin θ which you may assume without proof), show that the equations for the vorticity in part c) above reduce to the equation ω ϕ u r r + u θ ω ϕ r θ = ω ϕ ur sin θ + u θ cos θ ). r sin θ in spherical polar coordinates. Use this last result to prove that ) D ωϕ = 0. Dt r sin θ Question 5 continues overleaf...)
10 F11AE1 10 Question 5 continued e) You may take as given that the flow outside the sphere is given by an irrotational flow around a solid sphere refer to Question 3 earlier in this exam paper) and described by the stream function ψ = ψ out where ψ = 1 2 U r 2 a 3 /r ) sin 2 θ for r > a. If ψ = ψ in is the stream function in the interior of the sphere, explain why at the boundary the sphere surface r = a) we require that ψ in ψ in a,θ) = ψ out a,θ) and r = ψ out r=a r. r=a Hence deduce that the boundary conditions for ψ = ψ in required are ψ in ψ in a,θ) = 0 and r = 3Ua 2 sin2 θ. r=a f) To finish, let us combine parts b), d) and e) above. From part d) we know that ω ϕ = Ar sin θ for some constant A. Thus inside the sphere Stokes stream function ψ = ψ in must satisfy the boundary value problem 2 ψ r + sin θ 2 r 2 θ 1 sin θ ψ θ ) = Ar 2 sin 2 θ, with the boundary conditions given at the end of part e) above. Show that ψ = A 10 r2 C 2 r 2 ) sin 2 θ is a general solution to the partial differential equation for ψ inside the sphere, where C is some constant. Use the boundary conditions to deduce specific values for C and A in terms of the given physical parameters.
11 F11AE1 11 Formulae I) The Navier Stokes equations for an incompressible homogeneous fluid are u t + u u = ν u 1 p + f, ρ u = 0, where u = ux,t) is the fluid velocity at position x and time t, ρ is the uniform constant density, p = px,t) is the pressure, ν is the constant kinematic viscosity and f is the body force per unit mass. II) The incompressible Navier Stokes equations in cylindrical polar coordinates r,θ,z) with the velocity field u = u r,u θ,u z ) are u r t + u )u r u2 θ r = 1 p ρ r + ν u r u r r 2 ) u θ + f 2 r 2 r, θ u θ t + u )u θ + u ru θ = 1 p r ρr θ + ν u θ + 2 u r r 2 θ u ) θ + f r 2 θ, u z t + u )u z = 1 p ρ z + ν u z + f z, where p = pr,θ,z,t) is the pressure, ρ is the mass density and f = f r,f θ,f z ) is the body force per unit mass. Here we also have and u = u r r + u θ r = 1 r θ + u z z r ) + 1r 2 r r 2 θ z 2 Further the gradient of a scalar function Φ and the divergence of a vector field u are given in cylindrical coordinates, respectively, by Φ Φ = r, 1 Φ r θ, Φ ) z and u = 1 r r ru r) + 1 u θ r θ + u z z. Lastly in cylindrical coordinates u is given by ω r u = ω θ = ω z 1 r 1 u z u θ r θ z u r uz z r r ru θ) 1 r III) The incompressible Navier Stokes equations in spherical polar coordinates r, θ, ϕ) with the velocity field u = u r,u θ,u ϕ ) are note θ is the angle to the southnorth u r θ.
12 F11AE1 12 pole axis and ϕ is the azimuthal angle) u r t + u )u r u2 θ r u2 ϕ r = 1 p ρ r + ν u r 2 u r r 2 2 r 2 sin θ u θ t + u )u θ + u ru θ u2 ϕ cos θ r r sin θ = 1 ρr + ν u θ + 2 u r r 2 θ u ϕ t + u )u ϕ + u ru ϕ r + ν u ϕ + 2 r 2 sin θ θ u θ sin θ) 2 r 2 sin θ p θ u θ r 2 sin 2 θ 2 cos θ r 2 sin 2 θ + u θu ϕ cosθ = 1 r sin θ u r ϕ + 2 cos θ r 2 sin 2 θ u θ ϕ ) u ϕ + f r, ϕ ) u ϕ + f θ, ϕ p ρr sin θ ϕ u ) ϕ r 2 sin 2 + f z, θ where p = pr,θ,ϕ,t) is the pressure, ρ is the mass density and f = f r,f θ,f ϕ ) is the body force per unit mass. Here we also have and u = u r r + u θ r = 1 r 2 ) + 1 r 2 r r r 2 sin θ θ + u ϕ r sin θ ϕ sin θ ) + θ θ 1 r 2 sin 2 θ 2 ϕ 2. Further the gradient of a scalar function Φ and the divergence of a vector field u are given in spherical coordinates, respectively, by Φ Φ = r, 1 ) Φ r θ, 1 Φ r sin θ ϕ and u = 1 r 2 r r2 u r ) + 1 r sin θ θ sin θu θ) + 1 u ϕ r sin θ ϕ. Lastly in spherical coordinates u is given by ω r u = ω θ = ω ϕ 1 r sin θ 1 r sin θ 1 r sin θu θ ϕ) u θ ϕ u r 1 ru ϕ r r ϕ). r ru θ) 1 r IV) The following two identities hold for any two vectors u and v: and 1 2 u 2) = u )u + u u) u v) = u v) v u) + v )u u )v. u r θ
F1.9AB2 1. r 2 θ2 + sin 2 α. and. p θ = mr 2 θ. p2 θ. (d) In light of the information in part (c) above, we can express the Hamiltonian in the form
F1.9AB2 1 Question 1 (20 Marks) A cone of semiangle α has its axis vertical and vertex downwards, as in Figure 1 (overleaf). A point mass m slides without friction on the inside of the cone under the
More informationUNIVERSITY OF EAST ANGLIA
UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2007 2008 FLUIDS DYNAMICS WITH ADVANCED TOPICS Time allowed: 3 hours Attempt question ONE and FOUR other questions. Candidates must
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 201213 MODULE TITLE: Introduction to Fluids DURATION OF
More information0 = p. 2 x + 2 w. z +ν w
Solution (Elliptical pipe flow (a Using the Navier Stokes equations in three dimensional cartesian coordinates, given that u =, v = and w = w(x,y only, and assuming no body force, we are left with = p
More informationCandidates must show on each answer book the type of calculator used. Only calculators permitted under UEA Regulations may be used.
UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2011 2012 FLUID DYNAMICS MTH3D41 Time allowed: 3 hours Attempt FIVE questions. Candidates must show on each answer book the type
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the NavierStokes equation for incompressible constant density, i.e. homogeneous flows:
More informationChapter 6: Incompressible Inviscid Flow
Chapter 6: Incompressible Inviscid Flow 61 Introduction 62 Nondimensionalization of the NSE 63 Creeping Flow 64 Inviscid Regions of Flow 65 Irrotational Flow Approximation 66 Elementary Planar Irrotational
More informationV (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 informationFluid 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 information5.1 Fluid momentum equation Hydrostatics Archimedes theorem The vorticity equation... 42
Chapter 5 Euler s equation Contents 5.1 Fluid momentum equation........................ 39 5. Hydrostatics................................ 40 5.3 Archimedes theorem........................... 41 5.4 The
More informationFluid Dynamics Problems M.Sc MathematicsSecond Semester Dr. Dinesh KhattarK.M.College
Fluid Dynamics Problems M.Sc MathematicsSecond Semester Dr. Dinesh KhattarK.M.College 1. (Example, p.74, Chorlton) At the point in an incompressible fluid having spherical polar coordinates,,, the velocity
More informationCandidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request.
UNIVERSITY OF EAST ANGLIA School of Mathematics Spring Semester Examination 2004 FLUID DYNAMICS Time allowed: 3 hours Attempt Question 1 and FOUR other questions. Candidates must show on each answer book
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Thursday 27 May, 2004 1.30 to 3.30 PAPER 64 ASTROPHYSICAL FLUID DYNAMICS Attempt THREE questions. There are four questions in total. The questions carry equal weight. Candidates
More information3.5 Vorticity Equation
.0  Marine Hydrodynamics, Spring 005 Lecture 9.0  Marine Hydrodynamics Lecture 9 Lecture 9 is structured as follows: In paragraph 3.5 we return to the full NavierStokes equations (unsteady, viscous
More informationMAE 101A. Homework 7  Solutions 3/12/2018
MAE 101A Homework 7  Solutions 3/12/2018 Munson 6.31: The stream function for a twodimensional, nonviscous, incompressible flow field is given by the expression ψ = 2(x y) where the stream function has
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More informationIntroductory fluid mechanics: solutions
Detailed solutions Introductory fluid mechanics: solutions Simon J.A. Malham Simon J.A. Malham 22nd October 214 Maxwell Institute for Mathematical Sciences and School of Mathematical and Computer Sciences
More informationWater is sloshing back and forth between two infinite vertical walls separated by a distance L: h(x,t) Water L
ME9a. SOLUTIONS. Nov., 29. Due Nov. 7 PROBLEM 2 Water is sloshing back and forth between two infinite vertical walls separated by a distance L: y Surface Water L h(x,t x Tank The flow is assumed to be
More informationOffshore Hydromechanics Module 1
Offshore Hydromechanics Module 1 Dr. ir. Pepijn de Jong 4. Potential Flows part 2 Introduction Topics of Module 1 Problems of interest Chapter 1 Hydrostatics Chapter 2 Floating stability Chapter 2 Constant
More information18.325: Vortex Dynamics
8.35: Vortex Dynamics Problem Sheet. Fluid is barotropic which means p = p(. The Euler equation, in presence of a conservative body force, is Du Dt = p χ. This can be written, on use of a vector identity,
More informationMA3D1 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 informationExercise 5: Exact Solutions to the NavierStokes Equations I
Fluid Mechanics, SG4, HT009 September 5, 009 Exercise 5: Exact Solutions to the NavierStokes Equations I Example : Plane Couette Flow Consider the flow of a viscous Newtonian fluid between two parallel
More information1. Introduction  Tutorials
1. Introduction  Tutorials 1.1 Physical properties of fluids Give the following fluid and physical properties(at 20 Celsius and standard pressure) with a 4digit accuracy. Air density : Water density
More information7 EQUATIONS OF MOTION FOR AN INVISCID FLUID
7 EQUATIONS OF MOTION FOR AN INISCID FLUID iscosity is a measure of the thickness of a fluid, and its resistance to shearing motions. Honey is difficult to stir because of its high viscosity, whereas water
More informationSome Basic Plane Potential Flows
Some Basic Plane Potential Flows Uniform Stream in the x Direction A uniform stream V = iu, as in the Fig. (Solid lines are streamlines and dashed lines are potential lines), possesses both a stream function
More informationAE301 Aerodynamics I UNIT B: Theory of Aerodynamics
AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B1: Mathematics for Aerodynamics B: Flow Field Representations B3: Potential Flow Analysis B4: Applications of Potential Flow Analysis
More informationFluid 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 informationOPEN QUIZ WHEN TOLD AT 7:00 PM
2.25 ADVANCED FLUID MECHANICS Fall 2013 QUIZ 1 THURSDAY, October 10th, 7:009:00 P.M. OPEN QUIZ WHEN TOLD AT 7:00 PM THERE ARE TWO PROBLEMS OF EQUAL WEIGHT Please answer each question in DIFFERENT books
More informationVorticity Equation Marine Hydrodynamics Lecture 9. Return to viscous incompressible flow. NS equation: v. Now: v = v + = 0 incompressible
13.01 Marine Hydrodynamics, Fall 004 Lecture 9 Copyright c 004 MIT  Department of Ocean Engineering, All rights reserved. Vorticity Equation 13.01  Marine Hydrodynamics Lecture 9 Return to viscous incompressible
More informationPart IB. Fluid Dynamics. Year
Part IB Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2018 14 Paper 1, Section I 5D Show that the flow with velocity potential φ = q 2π ln r in twodimensional,
More information3 Generation and diffusion of vorticity
Version date: March 22, 21 1 3 Generation and diffusion of vorticity 3.1 The vorticity equation We start from Navier Stokes: u t + u u = 1 ρ p + ν 2 u 1) where we have not included a term describing a
More informationPart II. Fluid Dynamics II. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 2, Section II 38C 41 An initially unperturbed twodimensional inviscid jet in h < y < h has uniform speed U
More informationMath 575Lecture 19. In this lecture, we continue to investigate the solutions of the Stokes equations.
Math 575Lecture 9 In this lecture, we continue to investigate the solutions of the Stokes equations. Energy balance Rewrite the equation to σ = f. We begin the energy estimate by dotting u in the Stokes
More informationIran University of Science & Technology School of Mechanical Engineering Advance Fluid Mechanics
1. Consider a sphere of radius R immersed in a uniform stream U0, as shown in 3 R Fig.1. The fluid velocity along streamline AB is given by V ui U i x 1. 0 3 Find (a) the position of maximum fluid acceleration
More informationGeneral Solution of the Incompressible, Potential Flow Equations
CHAPTER 3 General Solution of the Incompressible, Potential Flow Equations Developing the basic methodology for obtaining the elementary solutions to potential flow problem. Linear nature of the potential
More informationFluid Dynamics for Ocean and Environmental Engineering Homework #2 Viscous Flow
OCEN 678600 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 information2.25 Advanced Fluid Mechanics Fall 2013
.5 Advanced Fluid Mechanics Fall 013 Solution to Problem 1Final Exam Fall 013 r j g u j ρ, µ,σ,u j u r 1 h(r) r = R Figure 1: Viscous Savart Sheet. Image courtesy: Villermaux et. al. [1]. This kind of
More informationSTEADY VISCOUS FLOW THROUGH A VENTURI TUBE
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 12, Number 2, Summer 2004 STEADY VISCOUS FLOW THROUGH A VENTURI TUBE K. B. RANGER ABSTRACT. Steady viscous flow through an axisymmetric convergentdivergent
More informationNumber of pages in the question paper : 05 Number of questions in the question paper : 48 Modeling Transport Phenomena of Microparticles Note: Follow the notations used in the lectures. Symbols have their
More informationTopics in Fluid Dynamics: Classical physics and recent mathematics
Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:
More informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATHGA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More information2. FLUIDFLOW EQUATIONS SPRING 2019
2. FLUIDFLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Nonconservative differential equations 2.4 Nondimensionalisation Summary Examples 2.1 Introduction Fluid
More informationME 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 lettersized (8.5 11
More informationCE Final Exam. December 12, Name. Student I.D.
CE 100  December 12, 2009 Name Student I.D. This exam is closed book. You are allowed three sheets of paper (8.5 x 11, both sides) of your own notes. You will be given three hours to complete four problems.
More informationLecture 1: Introduction to Linear and NonLinear Waves
Lecture 1: Introduction to Linear and NonLinear Waves Lecturer: Harvey Segur. Writeup: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More information2 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 informationFLUID MECHANICS. Chapter 9 Flow over Immersed Bodies
FLUID MECHANICS Chapter 9 Flow over Immersed Bodies CHAP 9. FLOW OVER IMMERSED BODIES CONTENTS 9.1 General External Flow Characteristics 9.3 Drag 9.4 Lift 9.1 General External Flow Characteristics 9.1.1
More informationVortex motion. Wasilij Barsukow, July 1, 2016
The concept of vorticity We call Vortex motion Wasilij Barsukow, mail@sturzhang.de July, 206 ω = v vorticity. It is a measure of the swirlyness of the flow, but is also present in shear flows where the
More information21 Laplace s Equation and Harmonic Functions
2 Laplace s Equation and Harmonic Functions 2. Introductory Remarks on the Laplacian operator Given a domain Ω R d, then 2 u = div(grad u) = in Ω () is Laplace s equation defined in Ω. If d = 2, in cartesian
More informationThe Bernoulli Equation
The Bernoulli Equation The most used and the most abused equation in fluid mechanics. Newton s Second Law: F = ma In general, most real flows are 3D, unsteady (x, y, z, t; r,θ, z, t; etc) Let consider
More informationBasic concepts in viscous flow
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic Adapted from Chapter 1 of Cambridge Texts in Applied Mathematics 1 The fluid dynamic equations NavierStokes equations Dimensionless
More informationMultiple Integrals and Vector Calculus: Synopsis
Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors  recap of basic principles. Things which are (and are not) vectors. Differentiation and integration
More informationShell 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 information1. Introduction, tensors, kinematics
1. Introduction, tensors, kinematics Content: Introduction to fluids, Cartesian tensors, vector algebra using tensor notation, operators in tensor form, Eulerian and Lagrangian description of scalar and
More informationFLUID DYNAMICS, THEORY AND COMPUTATION MTHA5002Y
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2017 18 FLUID DYNAMICS, THEORY AND COMPUTATION MTHA5002Y Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
More informationMathematical Tripos Part IA Lent Term Example Sheet 1. Calculate its tangent vector dr/du at each point and hence find its total length.
Mathematical Tripos Part IA Lent Term 205 ector Calculus Prof B C Allanach Example Sheet Sketch the curve in the plane given parametrically by r(u) = ( x(u), y(u) ) = ( a cos 3 u, a sin 3 u ) with 0 u
More informationContinuum Mechanics Lecture 7 Theory of 2D potential flows
Continuum Mechanics ecture 7 Theory of 2D potential flows Prof. http://www.itp.uzh.ch/~teyssier Outline  velocity potential and stream function  complex potential  elementary solutions  flow past a
More informationFigure 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 informationThe vorticity field. A dust devil
The vorticity field The vector ω = u curl u is twice the local angular velocity in the flow, and is called the vorticity of the flow (from Latin for a whirlpool). Vortex lines are everywhere in the direction
More informationPage 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 informationENGI 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 noncartesian coordinate systems. Sections
More informationφ(r, θ, t) = a 2 U(t) cos θ. (7.1)
BioFluids Lectures 78: Slender Fish Added Mass for Lateral Motion At high Reynolds number, most of the effort required in swimming is pushing water out of the way, that is our energy goes in providing
More informationExercise 9, Ex. 6.3 ( submarine )
Exercise 9, Ex. 6.3 ( submarine The flow around a submarine moving at at velocity V can be described by the flow caused by a source and a sink with strength Q at a distance a from each other. V x Submarine
More informationCHAPTER 2 INVISCID FLOW
CHAPTER 2 INVISCID FLOW Changes due to motion through a field; Newton s second law (f = ma) applied to a fluid: Euler s equation; Euler s equation integrated along a streamline: Bernoulli s equation; Bernoulli
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More informationPEMP ACD2505. M.S. Ramaiah School of Advanced Studies, Bengaluru
TwoDimensional Potential Flow Session delivered by: Prof. M. D. Deshpande 1 Session Objectives  At the end of this session the delegate would have understood PEMP The potential theory and its application
More informationChapter 2 Dynamics of Perfect Fluids
hapter 2 Dynamics of Perfect Fluids As discussed in the previous chapter, the viscosity of fluids induces tangential stresses in relatively moving fluids. A familiar example is water being poured into
More informationPRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.
PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part IB Thursday 7 June 2007 9 to 12 PAPER 3 Before you begin read these instructions carefully. Each question in Section II carries twice the number of marks of each question in Section
More informationStream Tube. When density do not depend explicitly on time then from continuity equation, we have V 2 V 1. δa 2. δa 1 PH6L24 1
Stream Tube A region of the moving fluid bounded on the all sides by streamlines is called a tube of flow or stream tube. As streamline does not intersect each other, no fluid enters or leaves across the
More informationNumber of pages in the question paper : 06 Number of questions in the question paper : 48 Modeling Transport Phenomena of Microparticles Note: Follow the notations used in the lectures. Symbols have their
More informationContinuum Mechanics Lecture 5 Ideal fluids
Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline  Helmholtz decomposition  Divergence and curl theorem  Kelvin s circulation theorem  The vorticity equation
More informationDetailed 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. Secondorder tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic
More information10.52 Mechanics of Fluids Spring 2006 Problem Set 3
10.52 Mechanics of Fluids Spring 2006 Problem Set 3 Problem 1 Mass transfer studies involving the transport of a solute from a gas to a liquid often involve the use of a laminar jet of liquid. The situation
More informationSimplifications to Conservation Equations
Chater 5 Simlifications to Conservation Equations 5.1 Steady Flow If fluid roerties at a oint in a field do not change with time, then they are a function of sace only. They are reresented by: ϕ = ϕq 1,
More informationHW6. 1. Book problems 8.5, 8.6, 8.9, 8.23, 8.31
HW6 1. Book problems 8.5, 8.6, 8.9, 8.3, 8.31. Add an equal strength sink and a source separated by a small distance, dx, and take the limit of dx approaching zero to obtain the following equations for
More informationREE 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 information6.1 Momentum Equation for Frictionless Flow: Euler s Equation The equations of motion for frictionless flow, called Euler s
Chapter 6 INCOMPRESSIBLE INVISCID FLOW All real fluids possess viscosity. However in many flow cases it is reasonable to neglect the effects of viscosity. It is useful to investigate the dynamics of an
More informationFluid 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 informationControl Volume. Dynamics and Kinematics. Basic Conservation Laws. Lecture 1: Introduction and Review 1/24/2017
Lecture 1: Introduction and Review Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study of motion without regard for the cause. Dynamics: On the other hand, dynamics
More informationLecture 1: Introduction and Review
Lecture 1: Introduction and Review Review of fundamental mathematical tools Fundamental and apparent forces Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIERSTOKES EQUATIONS Under the assumption of a Newtonian stressrateofstrain constitutive equation and a linear, thermally conductive medium,
More informationChapter 6 Equations of Continuity and Motion
Chapter 6 Equations of Continuity and Motion Derivation of 3D Eq. conservation of mass Continuity Eq. conservation of momentum Eq. of motion NavierStrokes Eq. 6.1 Continuity Equation Consider differential
More informationFluid 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 (doublesided) formula sheet. There are five questions on
More informationThe dependence of the crosssectional shape on the hydraulic resistance of microchannels
3weeks course report, s0973 The dependence of the crosssectional shape on the hydraulic resistance of microchannels Hatim Azzouz a Supervisor: Niels Asger Mortensen and Henrik Bruus MIC Department of
More informationChapter 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 informationAPPH 4200 Physics of Fluids
APPH 4200 Physics of Fluids Review (Ch. 3) & Fluid Equations of Motion (Ch. 4) September 21, 2010 1.! Chapter 3 (more notes) 2.! Vorticity and Circulation 3.! NavierStokes Equation 1 Summary: CauchyStokes
More information2.5 Stokes flow past a sphere
Lecture Notes on Fluid Dynamics.63J/.J) by Chiang C. Mei, MIT 007 Spring 5Stokes.tex.5 Stokes flow past a sphere Refs] Lamb: Hydrodynamics Acheson : Elementary Fluid Dynamics, p. 3 ff One of the fundamental
More informationConnection to Laplacian in spherical coordinates (Chapter 13)
Connection to Laplacian in spherical coordinates (Chapter 13) We might often encounter the Laplace equation and spherical coordinates might be the most convenient 2 u(r, θ, φ) = 0 We already saw in Chapter
More information10 minutes reading time is allowed for this paper.
EGT1 ENGINEERING TRIPOS PART IB Tuesday 31 May 2016 2 to 4 Paper 4 THERMOFLUID MECHANICS Answer not more than four questions. Answer not more than two questions from each section. All questions carry the
More informationMAE 222 Mechanics of Fluids Final Exam with Answers January 13, Give succinct answers to the following word questions.
MAE 222 Mechanics of Fluids Final Exam with Answers January 13, 1994 Closed Book Only, three hours: 1:30PM to 4:30PM 1. Give succinct answers to the following word questions. (a) Why is dimensional analysis
More informationChapter 5. Sound Waves and Vortices. 5.1 Sound waves
Chapter 5 Sound Waves and Vortices In this chapter we explore a set of characteristic solutions to the uid equations with the goal of familiarizing the reader with typical behaviors in uid dynamics. Sound
More informationComplex Analysis MATH 6300 Fall 2013 Homework 4
Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner,
More informationContents. MATH 32B2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B2 (8W)
More informationOffshore Hydromechanics
Offshore Hydromechanics Module 1 : Hydrostatics Constant Flows Surface Waves OE4620 Offshore Hydromechanics Ir. W.E. de Vries Offshore Engineering Today First hour: Schedule for remainder of hydromechanics
More informationPHYS 281: Midterm Exam
PHYS 28: Midterm Exam October 28, 200, 8:009:20 Last name (print): Initials: No calculator or other aids allowed PHYS 28: Midterm Exam Instructor: B. R. Sutherland Date: October 28, 200 Time: 8:009:20am
More informationIn 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 informationAPPH 4200 Physics of Fluids
APPH 4200 Physics of Fluids Review 23 Lectures > 700 pages of text Final Exam Questions 1. Bernoulli s 2. Dimensional analysis, Potential flow, Scaling 3. Complex velocity potential 4. Motion of a line
More informationFundamentals of Fluid Dynamics: Waves in Fluids
Fundamentals of Fluid Dynamics: Waves in Fluids Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/ tzielins/ Institute
More informationMathematical Concepts & Notation
Mathematical Concepts & Notation Appendix A: Notation x, δx: a small change in x t : the partial derivative with respect to t holding the other variables fixed d : the time derivative of a quantity that
More information