MAE210C: Fluid Mechanics III Spring Quarter sgls/mae210c 2013/ Solution II
|
|
- Nancy Kelly
- 5 years ago
- Views:
Transcription
1 MAE210C: Fluid Mechanics III Spring Quarter sgls/mae210c 2013/ Solution II D 4.1 The equations are exactly the same as before, with the difference that the pressure in the fluid is now p and the dynamic boundary condition becomes p η = γ a ) a 2 η θθ + η zz, since the normal which goes out of the fluid) has changed direction. The solution to Laplace s equation bounded at infinity is ˆp = A n αr) it should really be α so use α > 0 to avoid that problem) and the only difference in the boundary conditions is the minus sign. Hence s 2 = γ α nα) a 3 ρ n α) 1 n2 α 2 ). Now α nα)/ n α) < 0 for all α so modes are unstable if and only if n = 0 and 0 < α < 1. The maximum dimensionless growth rate can be found e.g. using MATLAB) as sρa 3 /γ) 1/2 = at α = see Figure 1). alpha=0.01:0.01:1; % note that f is minus s^2 to use matlab s fminbnd command; plot -f f=inline -sqrt-alpha.*-besselk1,alpha))./besselk0,alpha).*1-alpha.^2)) ); plotalpha,-falpha)) xlabel \alpha ) a=fminbndf,0,1) -fa) D 6.10 i) Substituting into the continuity equation gives u = 2 1/2 k 2 + 1)k 1 XkC x C z 2 1/2 k 2 + 1)XC x C z = 0. ii) S z 0) = S z π) = S 2z 0) = S 2z π) = 0 so the boundary conditions are satisfied. iii) The nonlinear terms are uu x + wu z = k 1 k 2 + 1) 2 X 2 sin 2kx, uw x + ww z = k 2 + 1) 2 X 2 S 2z. The vorticity is q = 2 1/2 k 2 + 1) 2 k 1 XS x S z. The curl of the momentum equation takes the form q θ + u )q = σ t x + σ q. The nonlinear term u )q is in fact zero. We read off 2 1/2 k 2 + 1) 2 k 1 X t = σk 2 + 1) 3 k 2 2 1/2 Y k) 2 1/2 k 2 + 1) 2 k 1 X k 2 1). 1
2 α Figure 1: Nondimensional growth rate sρa 3 /γ) 1/2 for n = 0 and 0 < α < 1. Simplifying gives iv) The convective terms are dx dτ = σ Y + X). u θ = k 2 + 1) 4 k 2 XS x C z [ Y kc x )S z + YC x C z + 2 1/2 Z2C 2z )] = k 2 + 1) 4 k 2 XYS 2z + 2 3/2 ZXC x S z C 2z ). Note that S z C 2z = 1 2 S 3z S z ), the first term of which will not appear in the truncations used here. Identifying the C x S z and S 2z terms in turn gives and Simplifying gives k 2 + 1) 3 k 2 2 1/2 Y t + k 2 + 1) 4 k 2 2 3/2 ZX 1 2 ) = R2 1/2 k 2 + 1)X k 2 + 1) 3 k 2 2 1/2 Y k 2 1) k 2 + 1) 3 k 2 Z t + k 2 + 1) 4 k 2 XY = k 2 + 1) 3 k 2 Z 4). dy dτ ZX = Rk2 k 2 + 1) 3 X + Y, dz dτ + XY = 4 k 2 + 1) Z. 2
3 C Since and µ j are constants, U j can be written as the gradient of a velocity potential Φ j. The continuity equation then gives 2 Φ j = Uniform ascension means U j = 0, V). This corresponds to Φ j = Vy + C, where C is a function of time. For convenience, use the new variable z = y Vt so that the unperturbed boundary between the fluids is at z = 0. Then Φ j = Vz and equating potential and pressure from Darcy s law gives the second relation. 3. From Darcy s law and continuity, the perturbation potential and pressure are both harmonic functions with 2 φ j = 2 p j = 0 no linearization required). The conditions at infinity are decay for φ j and p j. The conditions at the front become kinematic and Darcy s law) φ 1 z = φ 2 z = Dh Dt, µ1 V 1g h + p 1 = + ρ 2g h + p 2 at z = hx, t). Linearizing about z = 0 leads to φ 1 z = φ 2 z = h t, µ1 V 1g h + p 1 = + ρ 2g h + p 2 at z = This form is appropriate since the coefficients of the governing equations do not depend on z or t; k is the wavenumber and σ is the growth rate. Writing all quantities as ˆf z) exp ikx + σt) gives d 2 ˆφ j dz 2 k2 ˆφ j = 0, d 2 ˆp j dz 2 k2 ˆp j = 0 as field equations. Solve to get ˆφ 1 = A 1 e k z, ˆφ 2 = A 2 e k z, ˆp 1 = B 1 e k z, ˆφ 2 = B 2 e k z. The conditions at the interface give k A 1 = k A 2 = σĥ, µ1 V 1g ĥ + B 1 = + ρ 2g ĥ + B 2, while Darcy s law corresponds to A j = µ j B j. Hence µ2 µ 1 ) V + ρ 2 ρ 1 )g ĥ = B 2 B 1 = µ 1 A 1 µ 2 A 2 = µ 1 + µ 2 ) σ k ĥ. Hence σ is as required, although with k to be precise. 5. These curves are straight lines. For ρ 1 = ρ 2, the slope is positive if µ 2 µ 1 )V > 0, i.e. there is instability if the front moves into the more viscous fluid. For ρ 1 = ρ 2, the curves are still straight lines but the condition of instability is more complicated. The divergence at large k is not physical. The model is no longer valid beyond wavenumbers of the order of the inverse of the viscous or surface tension length scales, depending which is smaller. 3
4 6. Instability when V > ρ 1 ρ 2 )g µ 2 µ 1 = 10 7 m s 1, which is small. For a speed of 1 cm/s, σ = 1 the gravitional term is negligible), corresponding to a time of t = σ 1 log 10 = 2.3 s. C Conservation of mass; conservation of momentum; continuity of tangential velocity; continuity of normal velocity and definition of front velocity. 2. The Euler equations are trivially satisfied since the variables are constant. The normal velocity is zero and the fourth jump condition shows that U 1 = U L. The third is trivially satisfied since the normal is in the same direction as the base flow. The two other jump conditions reduce to and are clearly satisfied. 3. Small perturbations satisfy Linearize 2.30) and 2.31): at z = 0. The jump conditions become at z = 0. These simplify to ρ 1 U 1 = ρ 2 U 2, P 1 + ρ 1 U 2 1 = P 2 + ρ 2 U 2 2, u j = 0, ρ j t u j + U j x u j ) = p j. n = 1, η y ) w n = t η. ρ 1 u 1 t η) = ρ 2 u 2 t η), p 1 + 2ρ 1 U 1 u 1 t η) = p 2 + 2ρ 2 U 2 u 2 t η), U 1 η y v 1 = U 2 η y v 2, u 1 t η = 0 u 1 = u 2 = t η, p 1 = p 2, U 1 η y + v 1 = U 2 η y + v 2, at z = This form is appropriate since the coefficients of the governing equations do not depend on Y or t; k is the wavenumber and σ is the growth rate. The amplitudes of the normal modes satisfy dû j dx + ik ˆv j = 0, ρ j σû j + U j x û j ) = d ˆp j dx, ρ jσ ˆv j + U j x ˆv j ) = ik ˆp j. 5. Now take k > 0 and σ > 0. The divergence of the momentum equation shows that pressure is harmonic, so we have p 1 = B 1 e kx and p 2 = B 2 e kx. The momentum equations in the x-direction give û 1 = A 1 e kx, û 2 = A 2 e kx + C 2 e σx/u 2, 4
5 with ρ 1 σ + ku 1 )A 1 = kb 1, ρ 2 σ ku 2 )A 2 = kb 2, The homogeneous solution for û 1 vanishes because of the boundary condition as x. The continuity equation gives ˆv 1 = ia 1 e kx, ˆv 2 = ia 2 e kx iσ ku 2 C 2 e σx/u The jump conditions become the four algebraic equations û 1 = û 2 = σ ˆη, ˆp 1 = ˆp 2, iku 1 ˆη + ˆv 1 = iku 2 ˆη + ˆv 2, at z = 0. The last equation can be replaced by iku 1 û 1 + σ ˆv 1 = iku 2 û 2 + σ ˆv 2, at z = 0. The quantities B 1 and B 2 can also be eliminated. The final result is three equations in three unknowns: A 1 = A 2 + C 2, ρ 1 σ + ku 1 )A 1 = ρ 2 σ ku 2 )A 2, iku 1 A 1 + iσa 1 = iku 2 A 2 + C 2 ) iσ This can be rewritten as the matrix system rσ + ku L ) σ rku L 0 σ + ku L σ rku L σ 2 r 1 k 1 UL 1 rku L A 2 + σ ) C 2. ku 2 A 1 A 2 C 2 = 0. In order for a non-trivial solution to exist, the determinant of this system must vanish. This leads to the cubic The first factor is uninteresting. 7. The solution to the quadratic is σ rku L )[1 + r)σ 2 + 2rkU L σ + r1 r)k 2 U 2 L ] = 0. r ± rr σ = ku 2 + r 1) L. 1 + r For r > 1, the square root is real. Hence the system is unstable since it has at least) one positive root. The divergence at large k is not physical: this analysis neglects surface tension and viscosity and below the larger of these two length scales the model is no longer valid. 5
UNIVERSITY 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 informationKelvin Helmholtz Instability
Kelvin Helmholtz Instability A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram November 00 One of the most well known instabilities in fluid
More informationChE 385M Surface Phenomena University of Texas at Austin. Marangoni-Driven Finger Formation at a Two Fluid Interface. James Stiehl
ChE 385M Surface Phenomena University of Texas at Austin Marangoni-Driven Finger Formation at a Two Fluid Interface James Stiehl Introduction Marangoni phenomena are driven by gradients in surface tension
More informationCHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY
1 Lecture Notes on Fluid Dynamics (1.63J/.1J) by Chiang C. Mei, 00 CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY References: Drazin: Introduction to Hydrodynamic Stability Chandrasekar: Hydrodynamic
More informationStability of Shear Flow
Stability of Shear Flow notes by Zhan Wang and Sam Potter Revised by FW WHOI GFD Lecture 3 June, 011 A look at energy stability, valid for all amplitudes, and linear stability for shear flows. 1 Nonlinear
More information1. Comparison of stability analysis to previous work
. Comparison of stability analysis to previous work The stability problem (6.4) can be understood in the context of previous work. Benjamin (957) and Yih (963) have studied the stability of fluid flowing
More informationα 2 )(k x ũ(t, η) + k z W (t, η)
1 3D Poiseuille Flow Over the next two lectures we will be going over the stabilization of the 3-D Poiseuille flow. For those of you who havent had fluids, that means we have a channel of fluid that is
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 informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
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 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 informationBounds on Surface Stress Driven Flows
Bounds on Surface Stress Driven Flows George Hagstrom Charlie Doering January 1, 9 1 Introduction In the past fifteen years the background method of Constantin, Doering, and Hopf has been used to derive
More informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
More informationMAE 210C FLUID MECHANICS III SPRING 2015 Rayleigh s equation for axisymmetric flows Forthestabilityanalysisofparallelaxisymmetricflowswithbasicvelocity profileū = (U x,u r,u θ ) = (U(r),0,0) weintroducemodalperturbationsoftheform(v
More informationPrototype Instabilities
Prototype Instabilities David Randall Introduction Broadly speaking, a growing atmospheric disturbance can draw its kinetic energy from two possible sources: the kinetic and available potential energies
More informationThe Orchestra of Partial Differential Equations. Adam Larios
The Orchestra of Partial Differential Equations Adam Larios 19 January 2017 Landscape Seminar Outline 1 Fourier Series 2 Some Easy Differential Equations 3 Some Not-So-Easy Differential Equations Outline
More information2 The incompressible Kelvin-Helmholtz instability
Hydrodynamic Instabilities References Chandrasekhar: Hydrodynamic and Hydromagnetic Instabilities Landau & Lifshitz: Fluid Mechanics Shu: Gas Dynamics 1 Introduction Instabilities are an important aspect
More informationFundamentals of Transport Processes Prof. Kumaran Indian Institute of Science, Bangalore Chemical Engineering
Fundamentals of Transport Processes Prof. Kumaran Indian Institute of Science, Bangalore Chemical Engineering Module No # 05 Lecture No # 25 Mass and Energy Conservation Cartesian Co-ordinates Welcome
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 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 MTH-3D41 Time allowed: 3 hours Attempt FIVE questions. Candidates must show on each answer book the type
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 informationOn the weakly nonlinear Kelvin-Helmholtz instability of tangential discontinuities in MHD
On the weakly nonlinear Kelvin-Helmholtz instability of tangential discontinuities in MHD John K. Hunter Department of Mathematics University of California at Davis Davis, California 9566, U.S.A. and J.
More informationUNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS. BSc and MPhys Undergraduate Programmes in Physics LEVEL HE2
Phys/Level /1/9/Semester, 009-10 (1 handout) UNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS BSc and MPhys Undergraduate Programmes in Physics LEVEL HE PAPER 1 MATHEMATICAL,
More informationThe Euler Equation of Gas-Dynamics
The Euler Equation of Gas-Dynamics A. Mignone October 24, 217 In this lecture we study some properties of the Euler equations of gasdynamics, + (u) = ( ) u + u u + p = a p + u p + γp u = where, p and u
More informationThin airfoil theory. Chapter Compressible potential flow The full potential equation
hapter 4 Thin airfoil theory 4. ompressible potential flow 4.. The full potential equation In compressible flow, both the lift and drag of a thin airfoil can be determined to a reasonable level of accuracy
More informationThe first order quasi-linear PDEs
Chapter 2 The first order quasi-linear PDEs The first order quasi-linear PDEs have the following general form: F (x, u, Du) = 0, (2.1) where x = (x 1, x 2,, x 3 ) R n, u = u(x), Du is the gradient of u.
More informationProf. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit
Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit Rough breakdown of MHD shocks Jump conditions: flux in = flux out mass flux: ρv n magnetic flux: B n Normal momentum flux: ρv n
More informationVorticity and Dynamics
Vorticity and Dynamics In Navier-Stokes equation Nonlinear term ω u the Lamb vector is related to the nonlinear term u 2 (u ) u = + ω u 2 Sort of Coriolis force in a rotation frame Viscous term ν u = ν
More informationWeek 6 Notes, Math 865, Tanveer
Week 6 Notes, Math 865, Tanveer. Energy Methods for Euler and Navier-Stokes Equation We will consider this week basic energy estimates. These are estimates on the L 2 spatial norms of the solution u(x,
More information10 Shallow Water Models
10 Shallow Water Models So far, we have studied the effects due to rotation and stratification in isolation. We then looked at the effects of rotation in a barotropic model, but what about if we add stratification
More informationA NOTE ON APPLICATION OF FINITE-DIF TitleMETHOD TO STABILITY ANALYSIS OF BOU LAYER FLOWS.
A NOTE ON APPLICATION OF FINITE-DIF TitleMETHOD TO STABILITY ANALYSIS OF BOU LAYER FLOWS Author(s) ASAI, Tomio; NAKASUJI, Isao Citation Contributions of the Geophysical In (1971), 11: 25-33 Issue Date
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 informationQuick Recapitulation of Fluid Mechanics
Quick Recapitulation of Fluid Mechanics Amey Joshi 07-Feb-018 1 Equations of ideal fluids onsider a volume element of a fluid of density ρ. If there are no sources or sinks in, the mass in it will change
More informationBoiling Heat Transfer and Two-Phase Flow Fall 2012 Rayleigh Bubble Dynamics. Derivation of Rayleigh and Rayleigh-Plesset Equations:
Boiling Heat Transfer and Two-Phase Flow Fall 2012 Rayleigh Bubble Dynamics Derivation of Rayleigh and Rayleigh-Plesset Equations: Let us derive the Rayleigh-Plesset Equation as the Rayleigh equation can
More informationOPAC102. The Acoustic Wave Equation
OPAC102 The Acoustic Wave Equation Acoustic waves in fluid Acoustic waves constitute one kind of pressure fluctuation that can exist in a compressible fluid. The restoring forces responsible for propagating
More informationEquations of linear stellar oscillations
Chapter 4 Equations of linear stellar oscillations In the present chapter the equations governing small oscillations around a spherical equilibrium state are derived. The general equations were presented
More informationChapter 5. Shallow Water Equations. 5.1 Derivation of shallow water equations
Chapter 5 Shallow Water Equations So far we have concentrated on the dynamics of small-scale disturbances in the atmosphere and ocean with relatively simple background flows. In these analyses we have
More informationReview of Fundamental Equations Supplementary notes on Section 1.2 and 1.3
Review of Fundamental Equations Supplementary notes on Section. and.3 Introduction of the velocity potential: irrotational motion: ω = u = identity in the vector analysis: ϕ u = ϕ Basic conservation principles:
More informationOrdinary Differential Equations (ODEs)
Ordinary Differential Equations (ODEs) 1 Computer Simulations Why is computation becoming so important in physics? One reason is that most of our analytical tools such as differential calculus are best
More informationDynamics of Strongly Nonlinear Internal Solitary Waves in Shallow Water
Dynamics of Strongly Nonlinear Internal Solitary Waves in Shallow Water By Tae-Chang Jo and Wooyoung Choi We study the dynamics of large amplitude internal solitary waves in shallow water by using a strongly
More information1 Assignment 1: Nonlinear dynamics (due September
Assignment : Nonlinear dynamics (due September 4, 28). Consider the ordinary differential equation du/dt = cos(u). Sketch the equilibria and indicate by arrows the increase or decrease of the solutions.
More informationNumerical Heat and Mass Transfer
Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis
More informationStability of two-layer viscoelastic plane Couette flow past a deformable solid layer
J. Non-Newtonian Fluid Mech. 117 (2004) 163 182 Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer V. Shankar Department of Chemical Engineering, Indian Institute of Technology,
More information9. Boundary layers. Flow around an arbitrarily-shaped bluff body. Inner flow (strong viscous effects produce vorticity) BL separates
9. Boundary layers Flow around an arbitrarily-shaped bluff body Inner flow (strong viscous effects produce vorticity) BL separates Wake region (vorticity, small viscosity) Boundary layer (BL) Outer flow
More informationTheory of linear gravity waves April 1987
April 1987 By Tim Palmer European Centre for Medium-Range Weather Forecasts Table of contents 1. Simple properties of internal gravity waves 2. Gravity-wave drag REFERENCES 1. SIMPLE PROPERTIES OF INTERNAL
More information0.2. CONSERVATION LAW FOR FLUID 9
0.2. CONSERVATION LAW FOR FLUID 9 Consider x-component of Eq. (26), we have D(ρu) + ρu( v) dv t = ρg x dv t S pi ds, (27) where ρg x is the x-component of the bodily force, and the surface integral is
More informationGoverning Equations of Fluid Dynamics
Chapter 3 Governing Equations of Fluid Dynamics The starting point of any numerical simulation are the governing equations of the physics of the problem to be solved. In this chapter, we first present
More informationASTR 320: Solutions to Problem Set 2
ASTR 320: Solutions to Problem Set 2 Problem 1: Streamlines A streamline is defined as a curve that is instantaneously tangent to the velocity vector of a flow. Streamlines show the direction a massless
More informationSeveral forms of the equations of motion
Chapter 6 Several forms of the equations of motion 6.1 The Navier-Stokes equations Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Seies UG Examination 2015 16 FLUID DYNAMICS WITH ADVANCED TOPICS MTH-MD59 Time allowed: 3 Hous Attempt QUESTIONS 1 and 2, and THREE othe questions.
More informationTwo dimensional oscillator and central forces
Two dimensional oscillator and central forces September 4, 04 Hooke s law in two dimensions Consider a radial Hooke s law force in -dimensions, F = kr where the force is along the radial unit vector and
More informationGoal: Use understanding of physically-relevant scales to reduce the complexity of the governing equations
Scale analysis relevant to the tropics [large-scale synoptic systems]* Goal: Use understanding of physically-relevant scales to reduce the complexity of the governing equations *Reminder: Midlatitude scale
More informationChapter 7. Linear Internal Waves. 7.1 Linearized Equations. 7.2 Linear Vertical Velocity Equation
Chapter 7 Linear Internal Waves Here we now derive linear internal waves in a Boussinesq fluid with a steady (not time varying) stratification represented by N 2 (z) andnorotationf =0. 7.1 Linearized Equations
More informationρ Du i Dt = p x i together with the continuity equation = 0, x i
1 DIMENSIONAL ANALYSIS AND SCALING Observation 1: Consider the flow past a sphere: U a y x ρ, µ Figure 1: Flow past a sphere. Far away from the sphere of radius a, the fluid has a uniform velocity, u =
More informationAE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known
More informationApplication of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate
Physics Letters A 37 007) 33 38 www.elsevier.com/locate/pla Application of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate M. Esmaeilpour, D.D. Ganji
More informationAST242 LECTURE NOTES PART 5
AST242 LECTURE NOTES PART 5 Contents 1. Waves and instabilities 1 1.1. Sound waves compressive waves in 1D 1 2. Jeans Instability 5 3. Stratified Fluid Flows Waves or Instabilities on a Fluid Boundary
More informationChapter 3. Stability theory for zonal flows :formulation
Chapter 3. Stability theory for zonal flows :formulation 3.1 Introduction Although flows in the atmosphere and ocean are never strictly zonal major currents are nearly so and the simplifications springing
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationLecture 1: Introduction to Linear and Non-Linear Waves
Lecture 1: Introduction to Linear and Non-Linear Waves Lecturer: Harvey Segur. Write-up: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More informationChapter 9. Barotropic Instability. 9.1 Linearized governing equations
Chapter 9 Barotropic Instability The ossby wave is the building block of low ossby number geophysical fluid dynamics. In this chapter we learn how ossby waves can interact with each other to produce a
More informationSecond-Order Linear ODEs
Chap. 2 Second-Order Linear ODEs Sec. 2.1 Homogeneous Linear ODEs of Second Order On pp. 45-46 we extend concepts defined in Chap. 1, notably solution and homogeneous and nonhomogeneous, to second-order
More informationThe Whitham Equation. John D. Carter April 2, Based upon work supported by the NSF under grant DMS
April 2, 2015 Based upon work supported by the NSF under grant DMS-1107476. Collaborators Harvey Segur, University of Colorado at Boulder Diane Henderson, Penn State University David George, USGS Vancouver
More informationOn Fluid Maxwell Equations
On Fluid Maxwell Equations Tsutomu Kambe, (Former Professor, Physics), University of Tokyo, Abstract Fluid mechanics is a field theory of Newtonian mechanics of Galilean symmetry, concerned with fluid
More informationResistive MHD, reconnection and resistive tearing modes
DRAFT 1 Resistive MHD, reconnection and resistive tearing modes Felix I. Parra Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK (This version is of 6 May 18 1. Introduction
More informationThe Higgins-Selkov oscillator
The Higgins-Selkov oscillator May 14, 2014 Here I analyse the long-time behaviour of the Higgins-Selkov oscillator. The system is ẋ = k 0 k 1 xy 2, (1 ẏ = k 1 xy 2 k 2 y. (2 The unknowns x and y, being
More informationMath 575-Lecture 26. KdV equation. Derivation of KdV
Math 575-Lecture 26 KdV equation We look at the KdV equations and the so-called integrable systems. The KdV equation can be written as u t + 3 2 uu x + 1 6 u xxx = 0. The constants 3/2 and 1/6 are not
More informationAnswers to Problem Set # 01, MIT (Winter-Spring 2018)
Answers to Problem Set # 01, 18.306 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Contents 1 Nonlinear solvable ODEs 2 1.1 Statement:
More informationHomework #4 Solution. μ 1. μ 2
Homework #4 Solution 4.20 in Middleman We have two viscous liquids that are immiscible (e.g. water and oil), layered between two solid surfaces, where the top boundary is translating: y = B y = kb y =
More information2, where dp is the constant, R is the radius of
Dynamics of Viscous Flows (Lectures 8 to ) Q. Choose the correct answer (i) The average velocity of a one-dimensional incompressible fully developed viscous flow between two fixed parallel plates is m/s.
More informationOn the displacement of two immiscible Stokes fluids in a 3D Hele-Shaw cell
On the displacement of two immiscible Stokes fluids in a 3D Hele-Shaw cell Gelu Paşa Abstract. In this paper we study the linear stability of the displacement of two incompressible Stokes fluids in a 3D
More information6 Two-layer shallow water theory.
6 Two-layer shallow water theory. Wewillnowgoontolookatashallowwatersystemthathastwolayersofdifferent density. This is the next level of complexity and a simple starting point for understanding the behaviour
More informationF11AE1 1. C = ρν r r. r u z r
F11AE1 1 Question 1 20 Marks) Consider an infinite horizontal pipe with circular cross-section of radius a, whose centre line is aligned along the z-axis; see Figure 1. Assume no-slip boundary conditions
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 informationAn acoustic wave equation for orthorhombic anisotropy
Stanford Exploration Project, Report 98, August 1, 1998, pages 6?? An acoustic wave equation for orthorhombic anisotropy Tariq Alkhalifah 1 keywords: Anisotropy, finite difference, modeling ABSTRACT Using
More informationAnalysis of Least Stable Mode of Buoyancy Assisted Mixed Convective Flow in Vertical Pipe Filled with Porous Medium
Proceedings of the World Congress on Engineering 2 Vol I WCE 2, July 6-8, 2, London, U.K. Analysis of Least Stable Mode of Buoyancy Assisted Mixed Convective Flow in Vertical Pipe Filled with Porous Medium
More information2.25 Advanced Fluid Mechanics Fall 2013
.5 Advanced Fluid Mechanics Fall 013 Solution to Problem 1-Final 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 informationSecond Order ODEs. Second Order ODEs. In general second order ODEs contain terms involving y, dy But here only consider equations of the form
Second Order ODEs Second Order ODEs In general second order ODEs contain terms involving y, dy But here only consider equations of the form A d2 y dx 2 + B dy dx + Cy = 0 dx, d2 y dx 2 and F(x). where
More informationGFD 2013 Lecture 4: Shallow Water Theory
GFD 213 Lecture 4: Shallow Water Theory Paul Linden; notes by Kate Snow and Yuki Yasuda June 2, 214 1 Validity of the hydrostatic approximation In this lecture, we extend the theory of gravity currents
More informationReflection of Plane Electromagnetic Wave from Conducting Plane
Reflection of Plane Electromagnetic Wave from Conducting Plane Zafar Turakulov August 19, 2014 Abstract The phenomenon of reflection from conducting surface is considered in terms of exact solutions of
More informationEKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009)
EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009) Dr. Mohamad Hekarl Uzir-chhekarl@eng.usm.my School of Chemical Engineering Engineering Campus, Universiti
More informationSuppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber
Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber J.C. Ji, N. Zhang Faculty of Engineering, University of Technology, Sydney PO Box, Broadway,
More informationChapter 2. General concepts. 2.1 The Navier-Stokes equations
Chapter 2 General concepts 2.1 The Navier-Stokes equations The Navier-Stokes equations model the fluid mechanics. This set of differential equations describes the motion of a fluid. In the present work
More informationSolar Physics & Space Plasma Research Center (SP 2 RC) MHD Waves
MHD Waves Robertus vfs Robertus@sheffield.ac.uk SP RC, School of Mathematics & Statistics, The (UK) What are MHD waves? How do we communicate in MHD? MHD is kind! MHD waves are propagating perturbations
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF
More informationMagnetostatics and the vector potential
Magnetostatics and the vector potential December 8, 2015 1 The divergence of the magnetic field Starting with the general form of the Biot-Savart law, B (x 0 ) we take the divergence of both sides with
More information12.1 Viscous potential flow (VPF)
1 Energy equation for irrotational theories of gas-liquid flow:: viscous potential flow (VPF), viscous potential flow with pressure correction (VCVPF), dissipation method (DM) 1.1 Viscous potential flow
More informationExercise 5: Exact Solutions to the Navier-Stokes Equations I
Fluid Mechanics, SG4, HT009 September 5, 009 Exercise 5: Exact Solutions to the Navier-Stokes Equations I Example : Plane Couette Flow Consider the flow of a viscous Newtonian fluid between two parallel
More informationPHYS 432 Physics of Fluids: Instabilities
PHYS 432 Physics of Fluids: Instabilities 1. Internal gravity waves Background state being perturbed: A stratified fluid in hydrostatic balance. It can be constant density like the ocean or compressible
More informationθ α W Description of aero.m
Description of aero.m Determination of the aerodynamic forces, moments and power by means of the blade element method; for known mean wind speed, induction factor etc. Simplifications: uniform flow (i.e.
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 informationMATHS 267 Answers to Stokes Practice Dr. Jones
MATH 267 Answers to tokes Practice Dr. Jones 1. Calculate the flux F d where is the hemisphere x2 + y 2 + z 2 1, z > and F (xz + e y2, yz, z 2 + 1). Note: the surface is open (doesn t include any of the
More informationSolutions of Spring 2008 Final Exam
Solutions of Spring 008 Final Exam 1. (a) The isocline for slope 0 is the pair of straight lines y = ±x. The direction field along these lines is flat. The isocline for slope is the hyperbola on the left
More informationIn two-dimensional barotropic flow, there is an exact relationship between mass
19. Baroclinic Instability In two-dimensional barotropic flow, there is an exact relationship between mass streamfunction ψ and the conserved quantity, vorticity (η) given by η = 2 ψ.the evolution of the
More informationPAPER 331 HYDRODYNAMIC STABILITY
MATHEMATICAL TRIPOS Part III Thursday, 6 May, 016 1:30 pm to 4:30 pm PAPER 331 HYDRODYNAMIC STABILITY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal
More informationGeneral introduction to Hydrodynamic Instabilities
KTH ROYAL INSTITUTE OF TECHNOLOGY General introduction to Hydrodynamic Instabilities L. Brandt & J.-Ch. Loiseau KTH Mechanics, November 2015 Luca Brandt Professor at KTH Mechanics Email: luca@mech.kth.se
More informationType III migration in a low viscosity disc
Type III migration in a low viscosity disc Min-Kai Lin John Papaloizou mkl23@cam.ac.uk, minkailin@hotmail.com DAMTP University of Cambridge KIAA, Beijing, December 13, 2009 Outline Introduction: type III
More informationENGI 4430 PDEs - d Alembert Solutions Page 11.01
ENGI 4430 PDEs - d Alembert Solutions Page 11.01 11. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit
More information