Review of the Fundamentals of Groundwater Flow
|
|
- Penelope Higgins
- 5 years ago
- Views:
Transcription
1 Review of te Fundamentals of Groundwater Flow Darc s Law 1 A 1 L A or L 1 datum L : volumetric flow rate [L 3 T -1 ] 1 : draulic ead upstream [L] A : draulic ead downstream [L] : draulic conductivit [L T -1 ] A: cross sectional area normal to flow direction [L ] L: distance between locations were 1 and are measured [L]
2 Darc s Law: 1 A L Alternativel: d A dl A d dl is called specific discarge also Darc flu or Darc velocit. However is not true flow velocit! Te true flow velocit v also referred to as seepage velocit or pore water velocit is defined as v na n d dl Bulk area is A Porosit is n Area for flow is (na)
3 Transmissivit (T) for confined auifer T b Confined auifer Confining bed b Potentiometric surface Confining bed Water table or free surface Unconfined auifer Transmissivit for unconfined auifer T : saturated tickness of auifer Confining bed
4 Darc s Law and Hdraulic Head 1. Hdraulic Head 1 L A 1 and are draulic eads associated wit points 1 and. Te draulic ead or total ead is a measure of te potential of te water fluid at te measurement point. datum Potential of a fluid at a specific point is te work reuired to transform a unit of mass of fluid from an arbitraril cosen state to te state under consideration. p1 1 p 1
5 Tree Tpes of Potentials A. Pressure potential work reuired to raise te water pressure W m m P P VdP m dp ρ w P ρ w ρ w : densit of water assumed to be independent of pressure V: volume P v Reference state P P v v Current state
6 B. Elevation potential work reuired to raise te elevation 1 Z W mgd g m C. inetic potential work reuired to raise te velocit (d vdt) 1 Z 1 Z W mad m m m dv dt d v vdv v 3 Total potential: P Φ g ρ w v Unit [L T -1 ] Total [draulic] ead: Φ P g ρ g Unit [L] w v g
7 Total ead or draulic ead: Pieometer pressure ead [L] P ρ g w v g elevation [L] inetic term P1 ρg P ρg 1 1 datum A fluid moves from were te total ead is iger to were it is lower. For an ideal fluid (frictionless and incompressible) te total ead would sta constant.
8 For Groundwater Flow P v inetic term ρ g g w negligible draulic ead [L] P ρ w g pressure ead [L] elevation ead [L] Important: is relative to datum (reference state) pieometers 1 A B datum flow direction?
9 Hdraulic gradient 1 A L 1 A L ia d dl A L 1 datum Gradient of wit respect to at point A is: 1 d d A 1 1 lim A 1
10 θ dl d Magnitude and direction of gradient vector in D dl d arctan θ
11 Etension of Darc s Law Darc s Law in 1D: dl d ( ) k j i k j i k j i grad dl d ; ; i j k: unit vectors NOTE: above euations are valid onl if is scalar i.e. auifer is isotropic. In oter words does not var wit direction at an location. Darc s Law in 3D:
12 In an anisotropic auifer is a second-order tensor. It as 9 components in te Cartesian coordinate sstem: Darc s Law becomes grad ( ) or in matri form
13 Darc s Law wit epanded terms: In tensor notation : i i ij j 13 j ( ) NOTE are principal components of te tensor; parallel or normal to te direction of te maimum or minimum. ij (were i j) are cross terms of te tensor representing contributions to te flow rate in direction i from te gradient in direction j.
14 General tensor: Auifer formation If te Cartesian coordinate aes are aligned wit te principal directions of te tensor te cross terms are all eual to ero:
15 Use onl one inde for te tensor: ; ; We ave components of specific discarge: or in tensor notation: ( ) i i i i 13
16 In isotropic media is scalar specific discarge is parallel to draulic gradient grad() In anisotropic media is tensor or vector specific discarge is NOT parallel to draulic gradient grad()
17 GROUNDWATER TORAGE a) torage Concept tead tate V w A V A w Non-tead tate if: 1 V w if: 1 V w torage tank A 1 torage Coefficient: Volume of water tat an auifer releases from or takes into storage per unit surface area of te auifer per unit cange in ead. Vw A 3 L LL
18 b) Unconfined Auifer (pecific Yield) n: Porosit; assume 1% drainable V w V w na na auifer torage from unconfined auifer involves psical dewatering V A w : specific ield n A: area Tpicall not 1% drainable V w V fna (f < 1) w fna V A w fn n r specific retention r
19 c) Confined Auifer V A V A w w Vw A water level pieometer pump V w : storage coefficient dimensionless s b [L -1 ] s : specific storage No psical dewatering involved. torage coefficient for confined auifer b water input usuall << porosit A
20 I. Derivation of te Groundwater Flow Euation A. Continuit Principle (conservation of mass) inflow - outflow storage B. Darc s Law pecif ow inflow and outflow are calculated top back Control Volume left side () rigt side front bottom
21 t - Z Z Z Z Y Y Y Y X X X X top bottom front back rigt left ) ( storage sink top front rigt outflow source bottom back left inflow storage outflow inflow Z Y X Z Y X
22 t X X Y Y Z Z Divide bot sides b t t Z Y X Z Z Y Y X X were
23 t Heterogeneous/anisotropic wit sink/source transient 3D t General Flow Euation Tree dimensional transient eterogeneous anisotropic Laplace s Euation tead-state condition in omogeneous isotropic medium witout sink/source ( ) ( ) ( ) () Laplacian Operator
24 Two-dimensional t b b b b t T T ss * Homogeneous & Anisotropic (in terms of transmissivit) t T T ss * Homogeneous & Isotropic t T T ss * b: saturated tickness
25 II. Matematical model of groundwater flow A. Governing Euation (assumptions?) t (assumptions?) impler Eamples: t Assumptions: 3-D transient eterogeneous anisotropic wit sinks/sources (principal components aligned wit coordinate aes)
26 B. Initial Conditions For an transient problem te initial ead distribution must be known in order to solve for ead canges wit time i.e. ( ) f ( ) were f() is known For tis eample ( ) t t t
27 C. Boundar Conditions Heads flues or some combination of te two must be known at boundaries in order to solve for ead canges wit space in te interior of te flow field. (Wang and Anderson 198)
28 1. pecified ead (Diriclet condition) ( t) ( t) Γ1 were (t) is known For eample 1 on te left boundar on te rigt boundar 1
29 . pecified flu (Neumann condition) η o ( t) Γ were o (t) is a known flu across te boundar. Γ η A special case is ero flu boundar condition i.e. η Γ UETION: A river could be treated as eiter a specified-ead or specified flu boundar. Wat s te difference between te two treatments?
30 3. Combination of 1 & (Cauc condition) also referred to as ead-dependent boundar condition te flu into te auifer troug te auitard is dependent on te ead in te auifer i.e. η / / b b Γ 3 b / / b / UETION: Could a river also be treated as a ead-dependent boundar condition? If so wat s te difference wit te previous two treatments?
31 III. olutions of Matematical Model olutions to a matematical model of groundwater flow can be obtained analticall or numericall. Analtical olution: Head is epressed eplicitl as matematical formula (function) of t. For eample stead-state flow in a 1-D confined auifer: ( ) 1 L 1 1 L ceck d 1 d L constant slope!
32 Eample of Analtical olutions 1 b datum L 1. Governing euation:. Boundar conditions: T t ( ) ( ) X 1 X L
33 d d d T d d d d d d d a ( ) d d d ad ( ) a b wit boundar condition (1): () 1 we found: b 1 wit boundar condition (): L () 1 we found: a L 1 ( ) L 1
34 Numerical olution: Head is solved approimatel at predefined nodal points as illustrated below. Numerical solution is tpicall obtained troug a computer code. (Wang and Anderson 198)
Brief Review of Vector Calculus
Darc s Law in 3D Toda Vector Calculus Darc s Law in 3D q " A scalar as onl a magnitude A vector is caracteried b bot direction and magnitude. e.g, g, q, v,"," Vectors are represented b : boldface in boos,
More informationDarcy s law in 3-D. K * xx K * yy K * zz
PART 7 Equations of flow Darcy s law in 3-D Specific discarge (vector) is calculated by multiplying te ydraulic conductivity (second-order tensor) by te ydraulic gradient (vector). We obtain a general
More information1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet #9: Numerical Modeling of Groundwater Flow
1.7, Groundwater Hydrology Prof. Carles Harvey Lecture Packet #9: Numerical Modeling of Groundwater Flow Simulation: Te prediction of quantities of interest (dependent variables) based upon an equation
More informationModule 1 : The equation of continuity. Lecture 4: Fourier s Law of Heat Conduction
1 Module 1 : The equation of continuit Lecture 4: Fourier s Law of Heat Conduction NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering Fourier s Law of Heat Conduction According
More informationGG655/CEE623 Groundwater Modeling. Aly I. El-Kadi
GG655/CEE63 Groundwater Modeling Model Theory Water Flow Aly I. El-Kadi Hydrogeology 1 Saline water in oceans = 97.% Ice caps and glaciers =.14% Groundwater = 0.61% Surface water = 0.009% Soil moisture
More informationpancakes. A typical pancake also appears in the sketch above. The pancake at height x (which is the fraction x of the total height of the cone) has
Volumes One can epress volumes of regions in tree dimensions as integrals using te same strateg as we used to epress areas of regions in two dimensions as integrals approimate te region b a union of small,
More informationPERSPECTIVE OF COASTAL VEGETATION PATCHES WITH TOPOGRAPHY VARIATIONS FOR TSUNAMI PROTECTION IN 2D - NUMERICAL MODELING
PERSPECTIVE OF COASTAL VEGETATION PATCHES WITH TOPOGRAPHY VARIATIONS FOR TSUNAMI PROTECTION IN D - NUMERICAL MODELING N. A. K. NANDASENA ), Norio TANAKA ) and Katsutosi TANIMOTO ) Member of JSCE, B.Sc.Eng
More informationFilm thickness Hydrodynamic pressure Liquid saturation pressure or dissolved gases saturation pressure. dy. Mass flow rates per unit length
NOTES DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Te lecture presents te derivation of te Renolds equation of classical lubrication teor. Consider a liquid flowing troug a tin film region
More informationCONSERVATION OF ANGULAR MOMENTUM FOR A CONTINUUM
Chapter 4 CONSERVATION OF ANGULAR MOMENTUM FOR A CONTINUUM Figure 4.1: 4.1 Conservation of Angular Momentum Angular momentum is defined as the moment of the linear momentum about some spatial reference
More informationIntegral Calculus, dealing with areas and volumes, and approximate areas under and between curves.
Calculus can be divided into two ke areas: Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and minima problems Integral
More informationMATH CALCULUS I 2.1: Derivatives and Rates of Change
MATH 12002 - CALCULUS I 2.1: Derivatives and Rates of Cange Professor Donald L. Wite Department of Matematical Sciences Kent State University D.L. Wite (Kent State University) 1 / 1 Introduction Our main
More informationy = 3 2 x 3. The slope of this line is 3 and its y-intercept is (0, 3). For every two units to the right, the line rises three units vertically.
Mat 2 - Calculus for Management and Social Science. Understanding te basics of lines in te -plane is crucial to te stud of calculus. Notes Recall tat te and -intercepts of a line are were te line meets
More informationDifferentiation. Area of study Unit 2 Calculus
Differentiation 8VCE VCEco Area of stud Unit Calculus coverage In tis ca 8A 8B 8C 8D 8E 8F capter Introduction to limits Limits of discontinuous, rational and brid functions Differentiation using first
More informationINTRODUCTION TO CALCULUS LIMITS
Calculus can be divided into two ke areas: INTRODUCTION TO CALCULUS Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and
More informationIn all of the following equations, is the coefficient of permeability in the x direction, and is the hydraulic head.
Groundwater Seepage 1 Groundwater Seepage Simplified Steady State Fluid Flow The finite element method can be used to model both steady state and transient groundwater flow, and it has been used to incorporate
More information1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet #4: Continuity and Flow Nets
1.7, Groundwater Hydrology Prof. Charles Harvey Lecture Packet #4: Continuity and Flow Nets Equation of Continuity Our equations of hydrogeology are a combination of o Conservation of mass o Some empirical
More informationTHE HEATED LAMINAR VERTICAL JET IN A LIQUID WITH POWER-LAW TEMPERATURE DEPENDENCE OF DENSITY. V. A. Sharifulin.
THE HEATED LAMINAR VERTICAL JET IN A LIQUID WITH POWER-LAW TEMPERATURE DEPENDENCE OF DENSITY 1. Introduction V. A. Sharifulin Perm State Technical Universit, Perm, Russia e-mail: sharifulin@perm.ru Water
More information= h. Geometrically this quantity represents the slope of the secant line connecting the points
Section 3.7: Rates of Cange in te Natural and Social Sciences Recall: Average rate of cange: y y y ) ) ), ere Geometrically tis quantity represents te slope of te secant line connecting te points, f (
More informationLines, Conics, Tangents, Limits and the Derivative
Lines, Conics, Tangents, Limits and te Derivative Te Straigt Line An two points on te (,) plane wen joined form a line segment. If te line segment is etended beond te two points ten it is called a straigt
More informationMATH1901 Differential Calculus (Advanced)
MATH1901 Dierential Calculus (Advanced) Capter 3: Functions Deinitions : A B A and B are sets assigns to eac element in A eactl one element in B A is te domain o te unction B is te codomain o te unction
More informationNOTES ON OPEN CHANNEL FLOW
NOTES ON OPEN CANNEL FLOW Prof. Marco Pilotti Facoltà di Ingegneria, Università degli Studi di Brescia Profili di moto permanente in un canale e in una serie di due canali - Boudine, 86 OPEN CANNEL FLOW:
More informationDifferential Calculus (The basics) Prepared by Mr. C. Hull
Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit
More information2.2 Derivative. 1. Definition of Derivative at a Point: The derivative of the function f x at x a is defined as
. Derivative. Definition of Derivative at a Point: Te derivative of te function f at a is defined as f fa fa a lim provided te limit eists. If te limit eists, we sa tat f is differentiable at a, oterwise,
More information1.5 Functions and Their Rates of Change
66_cpp-75.qd /6/8 4:8 PM Page 56 56 CHAPTER Introduction to Functions and Graps.5 Functions and Teir Rates of Cange Identif were a function is increasing or decreasing Use interval notation Use and interpret
More informationChapter 2 Basic Conservation Equations for Laminar Convection
Chapter Basic Conservation Equations for Laminar Convection Abstract In this chapter, the basic conservation equations related to laminar fluid flow conservation equations are introduced. On this basis,
More informationConservation of Linear Momentum for a Differential Control Volume
Conservation of Linear Momentum for a Differential Control Volume When we applied the rate-form of the conservation of mass equation to a differential control volume (open sstem in Cartesian coordinates,
More informationSeepage Analysis through Earth Dam Based on Finite Difference Method
J. Basic. Appl. Sci. Res., (11)111-1, 1 1, TetRoad Publication ISSN -44 Journal of Basic and Applied Scientific Researc www.tetroad.com Seepage Analysis troug Eart Dam Based on Finite Difference Metod
More information11-19 PROGRESSION. A level Mathematics. Pure Mathematics
SSaa m m pplle e UCa ni p t ter DD iff if erfe enren tiatia tiotio nn - 9 RGRSSIN decel Slevel andmatematics level Matematics ure Matematics NW FR 07 Year/S Year decel S and level Matematics Sample material
More informationNumerical Solution of Partial Differential Equations
Numerical Solution of Partial Differential Equations Partial differential equations (PDEs) pla an important role in several areas of engineering, ranging from fluid mecanics, eat transfer, and applied
More informationFluids and Buoyancy. 1. What will happen to the scale reading as the mass is lowered?
Fluids and Buoyancy. Wat will appen to te scale reading as te mass is lowered? M Using rcimedes Principle: any body fully or partially submerged in a fluid is buoyed up by a force equal to te weigt of
More informationLimits and an Introduction to Calculus
Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem In Matematics If a
More informationName: Sept 21, 2017 Page 1 of 1
MATH 111 07 (Kunkle), Eam 1 100 pts, 75 minutes No notes, books, electronic devices, or outside materials of an kind. Read eac problem carefull and simplif our answers. Name: Sept 21, 2017 Page 1 of 1
More informationContinuity and Differentiability Worksheet
Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;
More informationExponentials and Logarithms Review Part 2: Exponentials
Eponentials and Logaritms Review Part : Eponentials Notice te difference etween te functions: g( ) and f ( ) In te function g( ), te variale is te ase and te eponent is a constant. Tis is called a power
More information. Compute the following limits.
Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim
More information3.1 Extreme Values of a Function
.1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find
More informationExam in Fluid Mechanics SG2214
Exam in Fluid Mecanics G2214 Final exam for te course G2214 23/10 2008 Examiner: Anders Dalkild Te point value of eac question is given in parentesis and you need more tan 20 points to pass te course including
More informationNotes: DERIVATIVES. Velocity and Other Rates of Change
Notes: DERIVATIVES Velocity and Oter Rates of Cange I. Average Rate of Cange A.) Def.- Te average rate of cange of f(x) on te interval [a, b] is f( b) f( a) b a secant ( ) ( ) m troug a, f ( a ) and b,
More informationFinding and Using Derivative The shortcuts
Calculus 1 Lia Vas Finding and Using Derivative Te sortcuts We ave seen tat te formula f f(x+) f(x) (x) = lim 0 is manageable for relatively simple functions like a linear or quadratic. For more complex
More informationSOIL MECHANICS
4.330 SOIL MECHANICS BERNOULLI S EQUATION Were: u w g Z = Total Head u = Pressure = Velocity g = Acceleration due to Graity w = Unit Weigt of Water Slide of 37 4.330 SOIL MECHANICS BERNOULLI S EQUATION
More informationSFU UBC UNBC Uvic Calculus Challenge Examination June 5, 2008, 12:00 15:00
SFU UBC UNBC Uvic Calculus Callenge Eamination June 5, 008, :00 5:00 Host: SIMON FRASER UNIVERSITY First Name: Last Name: Scool: Student signature INSTRUCTIONS Sow all your work Full marks are given only
More informationMath 124. Section 2.6: Limits at infinity & Horizontal Asymptotes. 1 x. lim
Mat 4 Section.6: Limits at infinity & Horizontal Asymptotes Tolstoy, Count Lev Nikolgevic (88-90) A man is like a fraction wose numerator is wat e is and wose denominator is wat e tinks of imself. Te larger
More informationMath 31A Discussion Notes Week 4 October 20 and October 22, 2015
Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes
More informationChapter-2: A Generalized Ratio and Product Type Estimator for the Population Mean in Stratified Random Sampling CHAPTER-2
Capter-: A Generalized Ratio and Product Tpe Eimator for te Population Mean in tratified Random ampling CHAPTER- A GEERALIZED RATIO AD PRODUCT TYPE ETIMATOR FOR THE POPULATIO MEA I TRATIFIED RADOM AMPLIG.
More informationlim 1 lim 4 Precalculus Notes: Unit 10 Concepts of Calculus
Syllabus Objectives: 1.1 Te student will understand and apply te concept of te limit of a function at given values of te domain. 1. Te student will find te limit of a function at given values of te domain.
More informationDifferentiation. introduction to limits
9 9A Introduction to limits 9B Limits o discontinuous, rational and brid unctions 9C Dierentiation using i rst principles 9D Finding derivatives b rule 9E Antidierentiation 9F Deriving te original unction
More informationProblem Solving. Problem Solving Process
Problem Solving One of te primary tasks for engineers is often solving problems. It is wat tey are, or sould be, good at. Solving engineering problems requires more tan just learning new terms, ideas and
More informationFundamentals of Applied Electromagnetics. Chapter 2 - Vector Analysis
Fundamentals of pplied Electromagnetics Chapter - Vector nalsis Chapter Objectives Operations of vector algebra Dot product of two vectors Differential functions in vector calculus Divergence of a vector
More informationWYSE Academic Challenge 2004 Sectional Mathematics Solution Set
WYSE Academic Callenge 00 Sectional Matematics Solution Set. Answer: B. Since te equation can be written in te form x + y, we ave a major 5 semi-axis of lengt 5 and minor semi-axis of lengt. Tis means
More informationHigher Derivatives. Differentiable Functions
Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.
More information5.1 We will begin this section with the definition of a rational expression. We
Basic Properties and Reducing to Lowest Terms 5.1 We will begin tis section wit te definition of a rational epression. We will ten state te two basic properties associated wit rational epressions and go
More informationSection 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,
More informationThe derivative function
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative
More informationLIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT
LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as
More informationMMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS PRELIMINARIES TO FEM
B Course Content: A INTRODUCTION AND OVERVIEW Numerical method and Computer-Aided Engineering; Phsical problems; Mathematical models; Finite element method;. B Elements and nodes, natural coordinates,
More informationVector Calculus. Vector Fields. Reading Trim Vector Fields. Assignment web page assignment #9. Chapter 14 will examine a vector field.
Vector alculus Vector Fields Reading Trim 14.1 Vector Fields Assignment web page assignment #9 hapter 14 will eamine a vector field. For eample, if we eamine the temperature conditions in a room, for ever
More informationPractice Problem Solutions: Exam 1
Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical
More informationPolynomial Functions. Linear Functions. Precalculus: Linear and Quadratic Functions
Concepts: definition of polynomial functions, linear functions tree representations), transformation of y = x to get y = mx + b, quadratic functions axis of symmetry, vertex, x-intercepts), transformations
More information1 ode.mcd. Find solution to ODE dy/dx=f(x,y). Instructor: Nam Sun Wang
Fin solution to ODE /=f(). Instructor: Nam Sun Wang oe.mc Backgroun. Wen a sstem canges wit time or wit location, a set of ifferential equations tat contains erivative terms "/" escribe suc a namic sstem.
More informationKINEMATIC RELATIONS IN DEFORMATION OF SOLIDS
Chapter 8 KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS Figure 8.1: 195 196 CHAPTER 8. KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS 8.1 Motivation In Chapter 3, the conservation of linear momentum for a
More informationThe Control-Volume Finite-Difference Approximation to the Diffusion Equation
The Control-Volume Finite-Difference Approimation to the Diffusion Equation ME 448/548 Notes Gerald Recktenwald Portland State Universit Department of Mechanical Engineering gerr@mepdedu ME 448/548: D
More informationPart 2: Introduction to Open-Channel Flow SPRING 2005
Part : Introduction to Open-Cannel Flow SPRING 005. Te Froude number. Total ead and specific energy 3. Hydraulic jump. Te Froude Number Te main caracteristics of flows in open cannels are tat: tere is
More informationCHAPTER-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 informationMath 34A Practice Final Solutions Fall 2007
Mat 34A Practice Final Solutions Fall 007 Problem Find te derivatives of te following functions:. f(x) = 3x + e 3x. f(x) = x + x 3. f(x) = (x + a) 4. Is te function 3t 4t t 3 increasing or decreasing wen
More information11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
SECTION 11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 633 wit speed v o along te same line from te opposite direction toward te source, ten te frequenc of te sound eard b te observer is were c is
More informationPre-Calculus Review Preemptive Strike
Pre-Calculus Review Preemptive Strike Attaced are some notes and one assignment wit tree parts. Tese are due on te day tat we start te pre-calculus review. I strongly suggest reading troug te notes torougly
More informationLogarithmic functions
Roberto s Notes on Differential Calculus Capter 5: Derivatives of transcendental functions Section Derivatives of Logaritmic functions Wat ou need to know alread: Definition of derivative and all basic
More informationDirectional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables.
Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of
More informationINTRODUCTION DEFINITION OF FLUID. U p F FLUID IS A SUBSTANCE THAT CAN NOT SUPPORT SHEAR FORCES OF ANY MAGNITUDE WITHOUT CONTINUOUS DEFORMATION
INTRODUCTION DEFINITION OF FLUID plate solid F at t = 0 t > 0 = F/A plate U p F fluid t 0 t 1 t 2 t 3 FLUID IS A SUBSTANCE THAT CAN NOT SUPPORT SHEAR FORCES OF ANY MAGNITUDE WITHOUT CONTINUOUS DEFORMATION
More informationNUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,
NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing
More informationFunctions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
More information158 Calculus and Structures
58 Calculus and Structures CHAPTER PROPERTIES OF DERIVATIVES AND DIFFERENTIATION BY THE EASY WAY. Calculus and Structures 59 Copyrigt Capter PROPERTIES OF DERIVATIVES. INTRODUCTION In te last capter you
More informationCONSERVATION LAWS AND CONSERVED QUANTITIES FOR LAMINAR RADIAL JETS WITH SWIRL
Mathematical and Computational Applications,Vol. 15, No. 4, pp. 742-761, 21. c Association for Scientific Research CONSERVATION LAWS AND CONSERVED QUANTITIES FOR LAMINAR RADIAL JETS WITH SWIRL R. Naz 1,
More informationREVIEW LAB ANSWER KEY
REVIEW LAB ANSWER KEY. Witout using SN, find te derivative of eac of te following (you do not need to simplify your answers): a. f x 3x 3 5x x 6 f x 3 3x 5 x 0 b. g x 4 x x x notice te trick ere! x x g
More informationCHAPTER 2 MODELING OF THREE-TANK SYSTEM
7 CHAPTER MODELING OF THREE-TANK SYSTEM. INTRODUCTION Te interacting tree-tank system is a typical example of a nonlinear MIMO system. Heiming and Lunze (999) ave regarded treetank system as a bencmark
More information10 Derivatives ( )
Instructor: Micael Medvinsky 0 Derivatives (.6-.8) Te tangent line to te curve yf() at te point (a,f(a)) is te line l m + b troug tis point wit slope Alternatively one can epress te slope as f f a m lim
More informationA STATIC PDE APPROACH FOR MULTI-DIMENSIONAL EXTRAPOLATION USING FAST SWEEPING METHODS
A STATIC PDE APPROACH FOR MULTI-DIMENSIONAL EXTRAPOLATION USING FAST SWEEPING METHODS TARIQ ASLAM, SONGTING LUO, AND HONGKAI ZHAO Abstract. A static Partial Differential Equation (PDE) approac is presented
More informationHOMEWORK HELP 2 FOR MATH 151
HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,
More informationJune : 2016 (CBCS) Body. Load
Engineering Mecanics st Semester : Common to all rances Note : Max. marks : 6 (i) ttempt an five questions (ii) ll questions carr equal marks. (iii) nswer sould be precise and to te point onl (iv) ssume
More informationESCUELA LATINOAMERICANA DE COOPERACIÓN Y DESARROLLO Especialización en Cooperación Internacional para el Desarrollo
ESCUELA LATINOAMERICANA DE COOPERACIÓN Y DESARROLLO Especialización en Cooperación Internacional para el Desarrollo A SURVIVAL KIT IN CASE O.MATHEMATICS b Marco Missaglia * Universit of Pavia September
More informationAE/ME 339. K. M. Isaac Professor of Aerospace Engineering. December 21, 2001 topic13_grid_generation 1
AE/ME 339 Professor of Aerospace Engineering December 21, 2001 topic13_grid_generation 1 The basic idea behind grid generation is the creation of the transformation laws between the phsical space and the
More informationMATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION
MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of
More informationUNIVERSITY OF SASKATCHEWAN Department of Physics and Engineering Physics
UNIVERSITY O SASKATCHEWAN Department of Pysics and Engineering Pysics Pysics 117.3 MIDTERM EXAM Regular Sitting NAME: (Last) Please Print (Given) Time: 90 minutes STUDENT NO.: LECTURE SECTION (please ceck):
More informationChapter 1 Functions and Graphs. Section 1.5 = = = 4. Check Point Exercises The slope of the line y = 3x+ 1 is 3.
Capter Functions and Graps Section. Ceck Point Exercises. Te slope of te line y x+ is. y y m( x x y ( x ( y ( x+ point-slope y x+ 6 y x+ slope-intercept. a. Write te equation in slope-intercept form: x+
More informationGradient Descent etc.
1 Gradient Descent etc EE 13: Networked estimation and control Prof Kan) I DERIVATIVE Consider f : R R x fx) Te derivative is defined as d fx) = lim dx fx + ) fx) Te cain rule states tat if d d f gx) )
More informationIn Leibniz notation, we write this rule as follows. DERIVATIVE OF A CONSTANT FUNCTION. For n 4 we find the derivative of f x x 4 as follows: lim
.1 DERIVATIVES OF POLYNOIALS AND EXPONENTIAL FUNCTIONS c =c slope=0 0 FIGURE 1 Te grap of ƒ=c is te line =c, so fª()=0. In tis section we learn ow to ifferentiate constant functions, power functions, polnomials,
More informationDefinition of the Derivative
Te Limit Definition of te Derivative Tis Handout will: Define te limit grapically and algebraically Discuss, in detail, specific features of te definition of te derivative Provide a general strategy of
More informationMATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart
MATH 121 - Fall 08 Review Problems for te Midterm Eamination Covers [1.1, 4.3] in Stewart 1. (a) Use te definition of te derivative to find f (3) wen f() = π 1 2. (b) Find an equation of te tangent line
More informationNST1A: Mathematics II (Course A) End of Course Summary, Lent 2011
General notes Proofs NT1A: Mathematics II (Course A) End of Course ummar, Lent 011 tuart Dalziel (011) s.dalziel@damtp.cam.ac.uk This course is not about proofs, but rather about using different techniques.
More informationFluid Mechanics II. Newton s second law applied to a control volume
Fluid Mechanics II Stead flow momentum equation Newton s second law applied to a control volume Fluids, either in a static or dnamic motion state, impose forces on immersed bodies and confining boundaries.
More informationChapter. Differentiation: Basic Concepts. 1. The Derivative: Slope and Rates. 2. Techniques of Differentiation. 3. The Product and Quotient Rules
Differentiation: Basic Concepts Capter 1. Te Derivative: Slope and Rates 2. Tecniques of Differentiation 3. Te Product and Quotient Rules 4. Marginal Analsis: Approimation b Increments 5. Te Cain Rule
More informationAverage Rate of Change
Te Derivative Tis can be tougt of as an attempt to draw a parallel (pysically and metaporically) between a line and a curve, applying te concept of slope to someting tat isn't actually straigt. Te slope
More informationPREPARATORY EXAMINATION
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P PREPARATORY EXAMINATION 008 MEMORANDUM MARKS: 50 TIME: ours Tis memorandum consists of pages. Matematics/P DoE/Preparatory Eamination 008 QUESTION. 0. 0
More informationTHE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225
THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:
More informationENGI Gradient, Divergence, Curl Page 5.01
ENGI 940 5.0 - Gradient, Divergence, Curl Page 5.0 5. e Gradient Operator A brief review is provided ere for te gradient operator in bot Cartesian and ortogonal non-cartesian coordinate systems. Sections
More informationLecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.
Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative
More information1. Which one of the following expressions is not equal to all the others? 1 C. 1 D. 25x. 2. Simplify this expression as much as possible.
004 Algebra Pretest answers and scoring Part A. Multiple coice questions. Directions: Circle te letter ( A, B, C, D, or E ) net to te correct answer. points eac, no partial credit. Wic one of te following
More informationMAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points
MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim
More informationFluid Mechanics for International Engineers HW #4: Conservation of Linear Momentum and Conservation of Energy
2141-365 Fluid Mechanics for International Engineers 1 Problem 1 RTT and Time Rate of Change of Linear Momentum and The Corresponding Eternal Force Notation: Here a material volume (MV) is referred to
More informationTime (hours) Morphine sulfate (mg)
Mat Xa Fall 2002 Review Notes Limits and Definition of Derivative Important Information: 1 According to te most recent information from te Registrar, te Xa final exam will be eld from 9:15 am to 12:15
More information