(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 11.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. For additional information concerning CDF *) (* licensing and redistribution see: *) (* *) (* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 1064, 20] NotebookDataLength[ 246608, 5336] NotebookOptionsPosition[ 234467, 4953] NotebookOutlinePosition[ 240109, 5123] CellTagsIndexPosition[ 238807, 5084] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Potential Flow Over an Airfoil", "Text", CellChangeTimes->{{3.6276464268948593`*^9, 3.627646468243655*^9}, { 3.686747298164953*^9, 3.686747310185066*^9}}, TextAlignment->Right], Cell[CellGroupData[{ Cell["3. Review of 2-D Potential Flow", "Title", CellChangeTimes->{ 3.564231525149586*^9, {3.564231631998974*^9, 3.5642316770050535`*^9}, { 3.5642384301531*^9, 3.5642384306367006`*^9}, 3.5644048861893845`*^9, { 3.5653699589212346`*^9, 3.565369972274858*^9}, {3.56664569960443*^9, 3.5666457199000654`*^9}, 3.5916275614505596`*^9, 3.5916276024479046`*^9, { 3.5973991436248155`*^9, 3.597399145256909*^9}, {3.6005194702312784`*^9, 3.6005194762716236`*^9}, {3.613843721776073*^9, 3.6138437273596697`*^9}, { 3.625225374963622*^9, 3.625225381282794*^9}, {3.6276471915110607`*^9, 3.627647193269577*^9}}, CellTags->"potential flow"], Cell["\<\ Richard L. Fearn (rlf@ufl.edu) Professor Emeritus Department of Mechanical and Aerospace Engineering University of Florida January 15, 2017\ \>", "Text", CellChangeTimes->{{3.596196010741573*^9, 3.5961960186030226`*^9}, { 3.596281573065505*^9, 3.5962815739135537`*^9}, {3.596447151691945*^9, 3.596447155900186*^9}, {3.5980023721244545`*^9, 3.598002392541622*^9}, { 3.598197345221053*^9, 3.5981973459130926`*^9}, {3.5981974064065523`*^9, 3.5981974339781294`*^9}, {3.5981974853800693`*^9, 3.598197541846299*^9}, { 3.5982624337007723`*^9, 3.5982624545329638`*^9}, {3.59860761717274*^9, 3.598607618310805*^9}, {3.598786538526937*^9, 3.598786539474991*^9}, { 3.600425038513008*^9, 3.6004250504946938`*^9}, {3.60051098969722*^9, 3.600510990319255*^9}, {3.600596003146227*^9, 3.6005960038762684`*^9}, { 3.604159770481059*^9, 3.604159774733302*^9}, {3.607941005336136*^9, 3.6079410082153006`*^9}, {3.6082827640284376`*^9, 3.608282764511465*^9}, { 3.619427919154167*^9, 3.6194279395799923`*^9}, 3.6240324570753107`*^9, 3.624100246291356*^9, {3.6284345369040747`*^9, 3.628434550215487*^9}, { 3.6587463985708*^9, 3.658746403655876*^9}, {3.66955571272053*^9, 3.669555728192572*^9}, {3.677764220925273*^9, 3.677764242294568*^9}, { 3.680273823936883*^9, 3.680273829429666*^9}, {3.6863092849018908`*^9, 3.686309290258443*^9}, {3.6866701516022253`*^9, 3.6866701527915783`*^9}, { 3.686752999294807*^9, 3.686753000715919*^9}, {3.687613922487953*^9, 3.687613928988291*^9}, {3.688655134070484*^9, 3.688655137298662*^9}, { 3.692108667186254*^9, 3.6921086891116867`*^9}, 3.6925456423084707`*^9}], Cell[TextData[ButtonBox["Table of Contents", BaseStyle->"Hyperlink", ButtonData->{"TableOfContents.cdf", "table of contents"}]], "Text", CellChangeTimes->{{3.686670190486175*^9, 3.6866701964617043`*^9}, { 3.686670237316935*^9, 3.6866702373201323`*^9}}, TextAlignment->Right], Cell[CellGroupData[{ Cell["3.1 Introduction", "Section", CellChangeTimes->{{3.5506747539911995`*^9, 3.550674756767999*^9}, { 3.550866826402*^9, 3.550866831267*^9}, 3.553937946881362*^9, { 3.553938004662362*^9, 3.553938005467362*^9}, {3.553938090500362*^9, 3.553938106714362*^9}, 3.553941429658362*^9, {3.559904582514*^9, 3.559904586651*^9}, {3.559904909689*^9, 3.559904912105*^9}, { 3.59162770394471*^9, 3.591627710208068*^9}, {3.5973991484340906`*^9, 3.597399149247137*^9}, {3.629714598919586*^9, 3.629714607330158*^9}}, CellTags->"introduction"], Cell[TextData[{ StyleBox["This document is a brief review of steady incompressible \ two-dimensional potential flow over an object focusing on the use of complex \ analysis.", FontWeight->"Bold"], " I assume that you have an elementary understanding of both fluid mechanics \ and ", ButtonBox["complex analysis", BaseStyle->"Hyperlink", ButtonData->{"IntroductionToComplexAnalysis.cdf", "complex analysis"}], " Potential flow is introduced in most undergraduate textbooks on fluid \ mechanics or aerodynamics. For example, ", StyleBox["Fluid Flow", FontSlant->"Italic"], " by ", ButtonBox["Sabersky et al.", BaseStyle->"Hyperlink", ButtonData->"sabersky"], ", is written for a first course on fluid mechanics, and ", StyleBox["Fundamentals of Aerodynamics", FontSlant->"Italic"], " by ", ButtonBox["Anderson,", BaseStyle->"Hyperlink", ButtonData->"anderson"], " for a first course on aerodynamics. ", StyleBox["Fluid Flow", FontSlant->"Italic"], " by ", ButtonBox["Fitzpatrick", BaseStyle->"Hyperlink", ButtonData->"fitzpatrick"], " is an electronic book at the advanced undergraduate level for physics \ majors. ", StyleBox["Principles of", FontSlant->"Italic"], " ", StyleBox["Ideal-Fluid Aerodynamics", FontSlant->"Italic"], " by ", ButtonBox["Karamcheti", BaseStyle->"Hyperlink", ButtonData->"karamcheti"], " contains a chapter on two-dimensional motion and the complex variable." }], "Text", CellChangeTimes->{{3.564404831542488*^9, 3.5644048754097652`*^9}, { 3.564404909807826*^9, 3.5644049765291433`*^9}, {3.564405010194002*^9, 3.5644050184308167`*^9}, {3.56440507184531*^9, 3.5644050808933263`*^9}, { 3.5644051346666207`*^9, 3.5644052046171436`*^9}, {3.5644053426305857`*^9, 3.564405729776466*^9}, {3.564412109621065*^9, 3.5644121378259144`*^9}, { 3.5644121806479897`*^9, 3.5644126330799847`*^9}, {3.5644126719240527`*^9, 3.56441274993979*^9}, {3.564414955097263*^9, 3.5644152573477936`*^9}, { 3.5644152890314493`*^9, 3.564415345316348*^9}, {3.565014827489868*^9, 3.5650149593880997`*^9}, {3.565014996765765*^9, 3.5650150311326256`*^9}, { 3.5650154426613483`*^9, 3.5650154426613483`*^9}, {3.565369935723994*^9, 3.565369947158814*^9}, {3.565369985300881*^9, 3.5653700056589165`*^9}, { 3.5653700519909983`*^9, 3.5653701424867573`*^9}, {3.565370205776068*^9, 3.565370250516947*^9}, {3.5653703084866486`*^9, 3.5653703767367687`*^9}, { 3.565370703354542*^9, 3.565370729874589*^9}, {3.5662121088082705`*^9, 3.56621214822954*^9}, {3.5662121967768254`*^9, 3.5662122160740595`*^9}, { 3.5662122605809374`*^9, 3.566212262406141*^9}, {3.566213189677374*^9, 3.566213318362*^9}, {3.5662133501392555`*^9, 3.566213390746127*^9}, { 3.566213427203391*^9, 3.5662135339075785`*^9}, {3.566213613779719*^9, 3.566213613779719*^9}, {3.566213657818596*^9, 3.566213657818596*^9}, { 3.5662137139474945`*^9, 3.5662137215447083`*^9}, {3.566645861719915*^9, 3.5666458679287252`*^9}, {3.5666459185508146`*^9, 3.566645920329218*^9}, { 3.566645989530939*^9, 3.566645989530939*^9}, {3.5692371548617277`*^9, 3.569237180944973*^9}, {3.5740793824142246`*^9, 3.5740793923514423`*^9}, { 3.574079424830699*^9, 3.5740796033886127`*^9}, {3.574079643246683*^9, 3.5740798041921654`*^9}, {3.574079873269087*^9, 3.5740799917980947`*^9}, { 3.5740800256657543`*^9, 3.5740800291601605`*^9}, {3.574080060750216*^9, 3.5740800657422247`*^9}, {3.574083937966282*^9, 3.5740839786667533`*^9}, { 3.5740840302248435`*^9, 3.574084082360135*^9}, {3.574084126726613*^9, 3.574084265738457*^9}, {3.5740843079677315`*^9, 3.5740844527047853`*^9}, { 3.5740844857144437`*^9, 3.574084538192936*^9}, {3.5740845699233913`*^9, 3.5740845790650077`*^9}, {3.5740846140714693`*^9, 3.5740846580323467`*^9}, 3.574084691946806*^9, {3.5740847674665384`*^9, 3.5740848045946035`*^9}, { 3.5740848361534595`*^9, 3.574084934464832*^9}, {3.574084964635285*^9, 3.5740849984717445`*^9}, {3.57408505297824*^9, 3.5740850822282915`*^9}, { 3.5740851418359957`*^9, 3.5740851877312765`*^9}, {3.5740853695807962`*^9, 3.5740853695807962`*^9}, {3.576511236937685*^9, 3.576511285301755*^9}, { 3.5962100061730657`*^9, 3.596210073963943*^9}, {3.596210104347681*^9, 3.596210283984956*^9}, {3.5962103360119314`*^9, 3.5962103360169315`*^9}, { 3.596210435938647*^9, 3.596210445978221*^9}, {3.5962105760776625`*^9, 3.5962105760896626`*^9}, {3.5962107909979553`*^9, 3.5962108522154565`*^9}, {3.596210941320553*^9, 3.596210973667403*^9}, { 3.596274547144182*^9, 3.596274564362167*^9}, {3.596274659663618*^9, 3.5962747038791466`*^9}, {3.5962757353851457`*^9, 3.596275736932234*^9}, { 3.5974003231222787`*^9, 3.597400323945326*^9}, {3.5987993401587486`*^9, 3.5987993575097413`*^9}, {3.604159911738139*^9, 3.604159911742139*^9}, { 3.6083076000839787`*^9, 3.6083076002909904`*^9}, {3.6083076378311377`*^9, 3.6083077747099667`*^9}, {3.60830787931795*^9, 3.608307885829322*^9}, { 3.6083930150061035`*^9, 3.6083930177932634`*^9}, {3.6139233448682833`*^9, 3.613923358051935*^9}, {3.6240325806190653`*^9, 3.6240325867768583`*^9}, { 3.624035103088064*^9, 3.6240351058536253`*^9}, {3.624035147301402*^9, 3.6240352478703337`*^9}, {3.624100298610491*^9, 3.624100360916051*^9}, { 3.62410039129972*^9, 3.624100426969397*^9}, {3.624100473309621*^9, 3.624100501412653*^9}, {3.624100722744689*^9, 3.624100722746335*^9}, 3.629464716419121*^9, {3.6297146236715803`*^9, 3.629714629749522*^9}, { 3.681213334591352*^9, 3.6812133600449333`*^9}, {3.6867530863804092`*^9, 3.686753088402628*^9}, {3.687613939591856*^9, 3.687613948877705*^9}, 3.687614013683889*^9, {3.6930627356750937`*^9, 3.693062744631935*^9}, { 3.693062848996421*^9, 3.693062848999296*^9}}, CellTags->{ "introduction: sabersky", "introduction: karamcheti", "introduction: anderson", "introduction: fitzpatrick"}], Cell[TextData[{ "Incompressible potential flow is a model based on simplifying assumptions \ about the properties of the fluid and the flow. If the curl of the velocity \ field is zero (no ", ButtonBox["vorticity", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Vorticity"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Vorticity"], " in the flow field), the velocity can be represented as the ", ButtonBox["gradient", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/Gradient.html"], None}, ButtonNote->"http://mathworld.wolfram.com/Gradient.html"], " of a scalar ", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["V", "\[RightVector]"], "=", RowBox[{"\[Del]", "\[Phi]"}]}], TraditionalForm]]], ", where ", Cell[BoxData[ FormBox["\[Phi]", TraditionalForm]]], " is called the velocity potential. If the flow is steady, inviscid and \ incompressible, the equation for conservation of mass can be written as ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Del]", RowBox[{"\[CenterDot]", OverscriptBox["V", "\[RightVector]"]}]}], "=", "0"}], TraditionalForm]]], ". This leads to ", ButtonBox["Laplace\[CloseCurlyQuote]s equation", BaseStyle->"Hyperlink", ButtonData->{ URL["http://mathworld.wolfram.com/LaplacesEquation.html"], None}, ButtonNote->"http://mathworld.wolfram.com/LaplacesEquation.html"], " as the governing equation for the velocity potential, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["\[Del]", "2"], "\[Phi]"}], "=", "0"}], TraditionalForm]]], ". This is a linear partial differential equation and has the ", ButtonBox["superposition property", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Superposition_principle"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Superposition_principle"], "." }], "Text", CellChangeTimes->{{3.564404831542488*^9, 3.5644048754097652`*^9}, { 3.564404909807826*^9, 3.5644049765291433`*^9}, {3.564405010194002*^9, 3.5644050184308167`*^9}, {3.56440507184531*^9, 3.5644050808933263`*^9}, { 3.5644051346666207`*^9, 3.5644052046171436`*^9}, {3.5644053426305857`*^9, 3.564405729776466*^9}, {3.564412109621065*^9, 3.5644121378259144`*^9}, { 3.5644121806479897`*^9, 3.5644126330799847`*^9}, {3.5644126719240527`*^9, 3.56441274993979*^9}, {3.564414955097263*^9, 3.5644152573477936`*^9}, { 3.5644152890314493`*^9, 3.564415345316348*^9}, {3.565014827489868*^9, 3.5650149593880997`*^9}, {3.565014996765765*^9, 3.5650150311326256`*^9}, { 3.5650154426613483`*^9, 3.5650154426613483`*^9}, {3.565369935723994*^9, 3.565369947158814*^9}, {3.565369985300881*^9, 3.5653700056589165`*^9}, { 3.5653700519909983`*^9, 3.5653701424867573`*^9}, {3.565370205776068*^9, 3.565370250516947*^9}, {3.5653703084866486`*^9, 3.5653703767367687`*^9}, { 3.565370703354542*^9, 3.565370729874589*^9}, {3.5662121088082705`*^9, 3.56621214822954*^9}, {3.5662121967768254`*^9, 3.5662122160740595`*^9}, { 3.5662122605809374`*^9, 3.566212262406141*^9}, {3.566213189677374*^9, 3.566213318362*^9}, {3.5662133501392555`*^9, 3.566213390746127*^9}, { 3.566213427203391*^9, 3.5662135339075785`*^9}, {3.566213613779719*^9, 3.566213613779719*^9}, {3.566213657818596*^9, 3.5662136592693987`*^9}, { 3.566213806299657*^9, 3.566213808577261*^9}, {3.5692372460750875`*^9, 3.569237279911547*^9}, {3.574085251550989*^9, 3.5740852558565965`*^9}, { 3.574085659210905*^9, 3.5740857118297973`*^9}, {3.574085744652255*^9, 3.574085744667855*^9}, {3.5740857848847256`*^9, 3.574085788129531*^9}, { 3.5766614005004644`*^9, 3.5766614017824674`*^9}, {3.596275957768865*^9, 3.596275958569911*^9}, {3.608393129371645*^9, 3.6083931303507013`*^9}, { 3.629714632338389*^9, 3.62971463522509*^9}, {3.6297149489674664`*^9, 3.629714952909954*^9}, {3.681212586577937*^9, 3.681212619902528*^9}, { 3.681212660517885*^9, 3.681212719060424*^9}, 3.687614250294033*^9}], Cell[TextData[{ "Two-dimensional potential flow is usually described in introductory \ textbooks without using the powerful computational tool of complex analysis. \ This document develops the topic in terms of analytic functions of a complex \ variable by introducing the complex potential, complex velocity and mapping \ by an analytic function. ", StyleBox["Some simple examples are given that serve as building blocks for \ constructing a solution for potential flow over a circle with circulation.", FontWeight->"Bold"], " Although potential flow over a circle is of limited physical interest, it \ serves as the starting point for computing some properties of interest for \ airfoils." }], "Text", CellChangeTimes->{{3.564404831542488*^9, 3.5644048754097652`*^9}, { 3.564404909807826*^9, 3.5644049765291433`*^9}, {3.564405010194002*^9, 3.5644050184308167`*^9}, {3.56440507184531*^9, 3.5644050808933263`*^9}, { 3.5644051346666207`*^9, 3.5644052046171436`*^9}, {3.5644053426305857`*^9, 3.564405729776466*^9}, {3.564412109621065*^9, 3.5644121378259144`*^9}, { 3.5644121806479897`*^9, 3.5644126330799847`*^9}, {3.5644126719240527`*^9, 3.56441274993979*^9}, {3.564414955097263*^9, 3.5644152573477936`*^9}, { 3.5644152890314493`*^9, 3.564415345316348*^9}, {3.565014827489868*^9, 3.5650149593880997`*^9}, {3.565014996765765*^9, 3.5650150311326256`*^9}, { 3.5650154426613483`*^9, 3.5650154426613483`*^9}, {3.565369935723994*^9, 3.565369947158814*^9}, {3.565369985300881*^9, 3.5653700056589165`*^9}, { 3.5653700519909983`*^9, 3.5653701424867573`*^9}, {3.565370205776068*^9, 3.565370250516947*^9}, {3.5653703084866486`*^9, 3.5653703767367687`*^9}, { 3.565370703354542*^9, 3.565370729874589*^9}, {3.5662121088082705`*^9, 3.56621214822954*^9}, {3.5662121967768254`*^9, 3.5662122160740595`*^9}, { 3.5662122605809374`*^9, 3.566212262406141*^9}, {3.566213189677374*^9, 3.566213318362*^9}, {3.5662133501392555`*^9, 3.566213390746127*^9}, { 3.566213427203391*^9, 3.5662135339075785`*^9}, {3.566213613779719*^9, 3.5662136160729227`*^9}, {3.569237333528841*^9, 3.569237335354045*^9}, { 3.574085819719587*^9, 3.574085849656039*^9}, {3.574085897501323*^9, 3.5740858996853275`*^9}, {3.57666143232251*^9, 3.5766614334925117`*^9}, { 3.576661513862624*^9, 3.5766615211026344`*^9}, {3.596275690585583*^9, 3.596275714315941*^9}, {3.596275764885833*^9, 3.5962758002868576`*^9}, { 3.596276047662007*^9, 3.5962760577175817`*^9}, {3.596276132286847*^9, 3.596276165316736*^9}, {3.596276203409915*^9, 3.5962762830024676`*^9}, { 3.5962763552556*^9, 3.596276371544532*^9}, {3.596276478727662*^9, 3.5962765002698946`*^9}, {3.5962765397781544`*^9, 3.5962765476866064`*^9}, {3.596276618366649*^9, 3.596276695706073*^9}, { 3.5962767327971945`*^9, 3.596276770756366*^9}, {3.597400423235005*^9, 3.5974004262291765`*^9}, {3.59879947165527*^9, 3.5987994755824947`*^9}, { 3.6083931738411884`*^9, 3.6083931744592237`*^9}, {3.624101510371101*^9, 3.624101552065093*^9}, {3.6241017920543213`*^9, 3.624101792056086*^9}, { 3.629465224219514*^9, 3.629465231419858*^9}, {3.6294652888974657`*^9, 3.629465322421523*^9}, {3.6294653596183577`*^9, 3.6294653871003437`*^9}, { 3.6294724218824997`*^9, 3.629472491388397*^9}, {3.62971463645116*^9, 3.629714641054812*^9}, 3.629714977463469*^9, {3.680274721263647*^9, 3.680274749051265*^9}, {3.681212797449603*^9, 3.681212919415185*^9}, { 3.6812129742690268`*^9, 3.681213024427792*^9}, {3.686753247498839*^9, 3.686753252795326*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["3.2 Basic Concepts and Notation", "Section", CellChangeTimes->{{3.564412870387601*^9, 3.564412906642065*^9}, { 3.5668162198626227`*^9, 3.566816221828226*^9}, {3.5740880934859805`*^9, 3.5740881097100086`*^9}, {3.5916277161934104`*^9, 3.5916277173134747`*^9}, {3.5973991551494746`*^9, 3.5973991556305017`*^9}, 3.629714645176374*^9}, CellTags->"concepts"], Cell[TextData[{ "Two-dimensional flow is used to describe flows that take place in a plane. \ Label the Cartesian coordinates as ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], ". The vector velocity describing the flow of the fluid is denoted by ", Cell[BoxData[ FormBox[ OverscriptBox["V", "\[RightVector]"], TraditionalForm]]], " and has components (", Cell[BoxData[ FormBox[ RowBox[{"u", ",", "v"}], TraditionalForm]]], "). Two-dimensional objects are interpreted in three dimensions as objects \ of infinite extent along the axis of the orthogonal third dimension, and have \ constant cross section. For example, two-dimensional flow over a circle is \ interpreted in three dimensions as flow over a circular cylinder of constant \ cross section extending to infinity in both directions. The velocity far from \ the object is assumed to be constant." }], "Text", CellChangeTimes->{{3.5740859712894526`*^9, 3.574086147273362*^9}, 3.5740861932778425`*^9, {3.5740862564735537`*^9, 3.574086285552005*^9}, { 3.5740863244740734`*^9, 3.5740863367200947`*^9}, {3.5740864534238997`*^9, 3.5740865158708096`*^9}, {3.5740865534824753`*^9, 3.574086553700876*^9}, { 3.5740866041201644`*^9, 3.5740866729318852`*^9}, {3.574086708234747*^9, 3.574086881020651*^9}, 3.5740879970778112`*^9, 3.5740881205988283`*^9, 3.574088310560362*^9, {3.5770252384010963`*^9, 3.5770253231512156`*^9}, { 3.5962125136604853`*^9, 3.5962125310684814`*^9}, {3.596212581480365*^9, 3.596212628341045*^9}, 3.596212662016971*^9, {3.5962129455181866`*^9, 3.596212990201742*^9}, {3.5987996053439164`*^9, 3.598799605913949*^9}, 3.598799644345147*^9, {3.608393618030595*^9, 3.608393618188604*^9}, { 3.6083936798981333`*^9, 3.6083937885133457`*^9}, {3.6194286203655357`*^9, 3.619428651863958*^9}, {3.619428709636819*^9, 3.619428734349248*^9}, { 3.6294725716478024`*^9, 3.629472604233553*^9}, {3.629714646558028*^9, 3.6297146502781143`*^9}, {3.68027495497762*^9, 3.680275010199079*^9}, { 3.680275040886292*^9, 3.6802750610533743`*^9}, 3.6813148426249027`*^9, { 3.6876143740079412`*^9, 3.687614401775765*^9}, {3.687614445734365*^9, 3.6876145059252443`*^9}, {3.692628777266501*^9, 3.692628851461941*^9}, { 3.692628902530719*^9, 3.692628908450116*^9}}], Cell[TextData[{ "The dimensionless measure for pressure in the flow field for flow is called \ the pressure coefficient:\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["C", "p"], "=", FractionBox[ RowBox[{"p", "-", SubscriptBox["p", "\[Infinity]"]}], RowBox[{ FractionBox["1", "2"], "\[Rho]", " ", SubsuperscriptBox["V", "\[Infinity]", "2"]}]]}], TraditionalForm]]], "\nwhere:\n\t\t", Cell[BoxData[ FormBox["p", TraditionalForm]]], "\tpressure at a point in the flow field (", Cell[BoxData[ FormBox[ RowBox[{"force", "/", SuperscriptBox["length", "2"]}], TraditionalForm]]], ")\n\t\t", Cell[BoxData[ FormBox["\[Rho]", TraditionalForm]]], "\tdensity of fluid (", Cell[BoxData[ FormBox[ RowBox[{"mass", "/", SuperscriptBox["length", "3"]}], TraditionalForm]]], ")\n\t\t", Cell[BoxData[ FormBox["V", TraditionalForm]]], "\tspeed of the flow (", Cell[BoxData[ FormBox[ RowBox[{"length", "/", "time"}], TraditionalForm]]], ")\nThe following subscripts are used to denote:\n\t\t", Cell[BoxData[ FormBox["\[Infinity]", TraditionalForm]]], "\tvalue of variable far from object\n\t\t", Cell[BoxData[ FormBox["0", TraditionalForm]], FormatType->"TraditionalForm"], "\tvalue of variable at stagnation point " }], "Text", CellChangeTimes->{ 3.565266954385577*^9, {3.565370486217761*^9, 3.565370606805972*^9}, { 3.565370797079507*^9, 3.5653708030387173`*^9}, {3.565370837561578*^9, 3.5653708457983923`*^9}, 3.565371809427685*^9, {3.565371860408574*^9, 3.565371892763031*^9}, {3.5653719813243866`*^9, 3.565371981339987*^9}, { 3.5653720119160404`*^9, 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ButtonBox["Bernoulli\[CloseCurlyQuote]s equation", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.grc.nasa.gov/WWW/k-12/airplane/bern.html"], None}, ButtonNote->"http://www.grc.nasa.gov/WWW/k-12/airplane/bern.html"], " for potential flow, evaluating the constant in terms of conditions far \ from the object of interest, or at a stagnation point\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "2"], "\[Rho]", " ", SuperscriptBox["V", "2"]}], "+", "p"}], "=", RowBox[{ RowBox[{ FractionBox["1", "2"], "\[Rho]", " ", SubsuperscriptBox["V", "\[Infinity]", "2"]}], "+", SubscriptBox["p", "\[Infinity]"]}]}], TraditionalForm]]], " or ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ FractionBox["1", "2"], "\[Rho]", " ", SuperscriptBox["V", "2"]}], "+", "p"}], "=", SubscriptBox["p", "0"]}], TraditionalForm]], FormatType->"TraditionalForm"], " \nRearrange Bernoulli's equation, and use the definition of pressure \ coefficient to write the pressure coefficient at a point in the flow field in \ terms of the fluid speed.\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["C", "p"], "=", RowBox[{"1", "-", FractionBox[ SuperscriptBox["V", "2"], RowBox[{" ", SubsuperscriptBox["V", "\[Infinity]", "2"]}]]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{ 3.565372626978321*^9, {3.5653727548985453`*^9, 3.5653727549141455`*^9}, 3.5653728440371017`*^9, {3.569237860450967*^9, 3.56923786199537*^9}, { 3.5692379718975625`*^9, 3.569238071300937*^9}, {3.569238132328244*^9, 3.5692381456038675`*^9}, {3.5766655712588854`*^9, 3.5766655712688856`*^9}, {3.576665609136945*^9, 3.5766656647470226`*^9}, { 3.692109193079461*^9, 3.692109197893203*^9}, {3.692628360686288*^9, 3.692628385187182*^9}, {3.6926285028125772`*^9, 3.6926285376558857`*^9}, { 3.692628992332847*^9, 3.692628993047543*^9}}, CellTags->{"cp", "bernoulli"}], Cell["\<\ The pressure coefficient is zero far from an object and one at stagnation \ points.\ \>", "Text", CellChangeTimes->{{3.692629161867269*^9, 3.69262921607288*^9}, { 3.692629266966353*^9, 3.692629285741313*^9}}], Cell[TextData[{ "Circulation is the line integral around a closed contour of the component \ of the velocity tangent to the curve.\n\t\t", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalGamma]", "=", RowBox[{ SubscriptBox["\[ContourIntegral]", "\[ScriptCapitalC]"], RowBox[{ RowBox[{ RowBox[{ OverscriptBox["V", "\[RightVector]"], "(", RowBox[{"x", ",", "y"}], ")"}], "\[CenterDot]", OverscriptBox["t", "^"]}], RowBox[{"\[DifferentialD]", "\[ScriptS]"}]}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.6516777716940193`*^9, 3.651677879612083*^9}, { 3.651677925495083*^9, 3.651678326913927*^9}, {3.6516783923178167`*^9, 3.651678435496451*^9}, {3.651678514518013*^9, 3.651678542963567*^9}, { 3.68027527065833*^9, 3.68027527213015*^9}, {3.681217871905127*^9, 3.681217886638958*^9}, {3.681312459889202*^9, 3.681312508931226*^9}, { 3.6813149268045387`*^9, 3.681314939147928*^9}, {3.6921092211641493`*^9, 3.692109233139455*^9}}], Cell[TextData[{ "The volume flux per unit span across a contour in two-dimensional flow is \ the line integral of the component of velocity normal to the curve.\n\t\t", Cell[BoxData[ FormBox[ RowBox[{"Q", "=", RowBox[{ SubscriptBox["\[Integral]", "\[ScriptCapitalC]"], RowBox[{ RowBox[{ RowBox[{ OverscriptBox["V", "\[RightVector]"], "(", RowBox[{"x", ",", "y"}], ")"}], "\[CenterDot]", OverscriptBox["n", "^"]}], " ", RowBox[{"\[DifferentialD]", "\[ScriptS]"}]}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{{3.651678585482333*^9, 3.651678672192402*^9}, { 3.6516787099224157`*^9, 3.651678814923341*^9}, {3.651678850014072*^9, 3.6516788547341022`*^9}, {3.651678901919436*^9, 3.651678906911214*^9}, { 3.68121790950217*^9, 3.6812179106061707`*^9}, 3.681314968186602*^9, { 3.681315002857484*^9, 3.681315012105386*^9}, {3.692109294609272*^9, 3.692109296097126*^9}}], Cell[TextData[{ "The units of ", Cell[BoxData[ FormBox["\[CapitalGamma]", 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CellChangeTimes->{{3.6082930117535734`*^9, 3.608293025746374*^9}, { 3.608293061570423*^9, 3.6082930633245234`*^9}, 3.608293259111722*^9}, CellTags->"properties: potential"], Cell["\<\ Simple potential models are of historical interest in several disciplines of \ physics and engineering. If you are studying a topic where a two-dimensional \ vector field can be represented by the gradient of a scalar field which \ satisfies Poisson\[CloseCurlyQuote]s equation or Laplace\[CloseCurlyQuote]s \ equation, then the ideas presented in this section are relevant. Although \ each discipline may use different terminology, potential models are \ mathematically equivalent in terms of the properties of the complex potential.\ \>", "Text", CellChangeTimes->{ 3.6082931320854564`*^9, 3.608293170402648*^9, {3.6083940505663347`*^9, 3.608394097215002*^9}, {3.624105019661615*^9, 3.6241050274987993`*^9}, { 3.629472962838681*^9, 3.6294729637010517`*^9}, {3.629714652703792*^9, 3.629714654063589*^9}, {3.629714988226235*^9, 3.6297149950854177`*^9}}], Cell[TextData[{ "For steady incompressible two-dimensional potential flow, ", StyleBox["the components of velocity can be written as follows in terms of a \ scalar ", FontWeight->"Bold"], StyleBox[ButtonBox["velocity potential", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Velocity_potential"], None}, ButtonNote->"https://en.wikipedia.org/wiki/Velocity_potential"], FontWeight->"Bold"], ".\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"\[Phi]", "(", RowBox[{"x", ",", "y"}], ")"}]}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"\[Phi]", "(", RowBox[{"x", ",", "y"}], ")"}]}]}], TraditionalForm]]], "\n", StyleBox["The ", FontWeight->"Bold"], StyleBox[ButtonBox["stream function", BaseStyle->"Hyperlink", ButtonData->{ URL["https://en.wikipedia.org/wiki/Stream_function#Two-dimensional_stream_\ function"], None}, ButtonNote-> "https://en.wikipedia.org/wiki/Stream_function#Two-dimensional_stream_\ function"], FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ FormBox["\[Psi]", TraditionalForm]], FontWeight->"Bold"], StyleBox[" is defined to satisfy identically the continuity equation", FontWeight->"Bold"], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Del]", "\[CenterDot]", OverscriptBox["V", "\[RightVector]"]}], "=", "0"}], TraditionalForm]]], ".\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{ SubscriptBox["\[PartialD]", "y"], RowBox[{"\[Psi]", "(", RowBox[{"x", ",", "y"}], ")"}]}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"v", "(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{"-", RowBox[{ SubscriptBox["\[PartialD]", "x"], RowBox[{"\[Psi]", "(", RowBox[{"x", ",", "y"}], ")"}]}]}]}], TraditionalForm]]], "\nLines of constant velocity potential are called equipotential lines, and \ lines of constant stream function are called stream lines and can form a \ phase portrait of the velocity field. Note that since the velocity field is \ determined by taking spatial derivatives of ", Cell[BoxData[ FormBox[ RowBox[{"\[Phi]", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], " or ", Cell[BoxData[ FormBox[ RowBox[{"\[Psi]", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]]], ", changing these functions by an additive constant does not change the \ velocity field." }], "Text", CellChangeTimes->{ 3.564241144491063*^9, {3.564242337670355*^9, 3.564242398245261*^9}, 3.5650192758380747`*^9, {3.5650193878930717`*^9, 3.5650195089336843`*^9}, { 3.565019610614663*^9, 3.5650196125334663`*^9}, {3.56501966604156*^9, 3.565019812260617*^9}, {3.5650198549422917`*^9, 3.5650198726951227`*^9}, 3.5650199389484396`*^9, {3.565020021628585*^9, 3.5650200267453938`*^9}, { 3.565020132061178*^9, 3.5650202254897423`*^9}, {3.565020273927828*^9, 3.565020310619092*^9}, {3.5650203513663635`*^9, 3.5650203546579695`*^9}, 3.5650206861897516`*^9, {3.5666466489284973`*^9, 3.5666466515649023`*^9}, { 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"=", RowBox[{ RowBox[{"\[Phi]", "(", RowBox[{"x", ",", "y"}], ")"}], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{"\[Psi]", "(", RowBox[{"x", ",", "y"}], ")"}]}]}]}], TraditionalForm]]], "\nFrom the defining equations for velocity potential and stream function in \ terms of velocity components, we see that the real and imaginary parts of the \ complex potential satisfy the ", ButtonBox["Cauchy-Riemann equations", BaseStyle->"Hyperlink", ButtonData->{ "IntroductionToComplexAnalysis.cdf", "functions: differentiable"}], " so ", StyleBox["the complex potential is an analytic function of the complex \ variable ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{"z", "=", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}]}], TraditionalForm]], FontWeight->"Bold"], "." }], "Text", CellChangeTimes->{ 3.59740093262514*^9, {3.5974010228473005`*^9, 3.597401186322651*^9}, 3.5988005389073133`*^9, {3.604160162066457*^9, 3.604160169850902*^9}, 3.6082930976394863`*^9, {3.624105122068906*^9, 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This \ integral can be interpreted in terms of the circulation and flux integrals of \ the P\[OAcute]lya vector field associated with the complex velocity. Work and \ flux integrals were initially discussed in ", ButtonBox["subsection 1.3.1 ", BaseStyle->"Hyperlink", ButtonData->{"ReviewOfVectorFields.cdf", "calculus: integrals"}], "for a general vector field, circulation as a alternative to work was \ discussed in ", ButtonBox["subsection 2.3.5", BaseStyle->"Hyperlink", ButtonData->{ "IntroductionToComplexAnalysis.cdf", "functions: integration: circulation"}], " and illustrated in ", ButtonBox["Figure 2.3", BaseStyle->"Hyperlink", ButtonData->{ "IntroductionToComplexAnalysis.cdf", "visualizing: polya: figure vortex"}], " for a source/vortex combination.\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[ContourIntegral]", "\[ScriptCapitalC]"], RowBox[{ RowBox[{"w", "(", "z", ")"}], "dz"}]}], "=", RowBox[{ RowBox[{ SubscriptBox["\[ContourIntegral]", "\[ScriptCapitalC]"], RowBox[{ RowBox[{ OverscriptBox["V", "\[RightVector]"], "\[CenterDot]", OverscriptBox["t", "^"]}], RowBox[{"\[DifferentialD]", "\[ScriptS]"}]}]}], "+", 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", StyleBox["If we are interested in the flow field external to an object and \ require finite velocity everywhere in the region of flow, then singularities \ will be interior to the object, and the closed contour will be on or outside \ the object. For potential flow, the flux integral and the circulation \ integral are independent of the path of integration. If the object is \ impervious, the flux integral is zero.", FontWeight->"Bold"] }], "Text", CellChangeTimes->{{3.6301501465888853`*^9, 3.630150150603161*^9}, { 3.630150182589137*^9, 3.630150285938634*^9}, {3.681393622995296*^9, 3.6813936450169497`*^9}, {3.681393697135377*^9, 3.681393698079*^9}, { 3.681393738077235*^9, 3.681393778291424*^9}, {3.68675361267873*^9, 3.6867536350127068`*^9}, {3.687101557405616*^9, 3.6871015816601553`*^9}, { 3.687101635304174*^9, 3.687101657766982*^9}, 3.687615579744928*^9, { 3.692874712371686*^9, 3.692874717203443*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["3.4 Some Simple Flows", "Section", CellChangeTimes->{{3.5650206580161023`*^9, 3.5650206660813165`*^9}, { 3.566816237007053*^9, 3.5668162387542562`*^9}, {3.5752763677966986`*^9, 3.575276373356706*^9}, {3.5916277460421176`*^9, 3.5916277470021725`*^9}, { 3.597399166891146*^9, 3.5973991680852146`*^9}, 3.629714755994298*^9}, CellTags->"simple"], Cell[CellGroupData[{ Cell["3.4.1 Uniform Flow", "Subsection", CellChangeTimes->{{3.5650208123783736`*^9, 3.5650208158571796`*^9}, { 3.5701921523162613`*^9, 3.5701921553894663`*^9}, {3.575276477860874*^9, 3.5752764787608757`*^9}, {3.5973991825610423`*^9, 3.5973991834570937`*^9}, { 3.600518657245778*^9, 3.600518657746807*^9}}, CellTags->"simple: uniform"], Cell[TextData[{ "Any analytic function of a complex variable is a candidate for a complex \ potential; the function may or may not result in a velocity field of \ interest. Tests for determining if a complex function is analytic are \ discussed in ", ButtonBox["Section 2.3", BaseStyle->"Hyperlink", ButtonData->{"IntroductionToComplexAnalysis.cdf", "functions"}], " and include representing the complex potential as a ", ButtonBox["function of z alone", BaseStyle->"Hyperlink", ButtonData->{ "IntroductionToComplexAnalysis.cdf", "functions: representation: gradient"}], ", satisfying the ", ButtonBox["Cauchy-Riemann equations", BaseStyle->"Hyperlink", ButtonData->{ "IntroductionToComplexAnalysis.cdf", "functions: derivative: cr"}], " or satisfying the condition that the ", ButtonBox["complex gradient is zero.", BaseStyle->"Hyperlink", ButtonData->{ "IntroductionToComplexAnalysis.cdf", "functions: representation: del"}] }], "Text", CellChangeTimes->{{3.681308680089004*^9, 3.681308696189436*^9}, { 3.681309204881957*^9, 3.681309289990951*^9}, {3.681393925400721*^9, 3.68139393830219*^9}, {3.681393972917864*^9, 3.6813940172353354`*^9}, { 3.681394050858471*^9, 3.681394094648284*^9}, {3.687101707787573*^9, 3.687101726106731*^9}, {3.687615643920467*^9, 3.68761568251828*^9}}], Cell[TextData[{ "Consider the simple function ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " as a complex potential. This function is analytic because it is written as \ a function of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " alone. The complex velocity is the derivative of the complex potential, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "z", ")"}]}], "=", "1"}], TraditionalForm]]], "; thus, ", Cell[BoxData[ FormBox[ RowBox[{"u", "=", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{"v", "=", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], ". 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circulation and flux integrals can be calculated by integrating the \ complex velocity on a contour enclosing the origin. The complex velocity has \ a pole-type singularity at ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", "0"}], TraditionalForm]]], ", and the integral can be evaluated using the method of residues. After \ evaluating the integral, list the real and imaginary parts using ", StyleBox[ButtonBox["ReIm", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/ReIm.html?q=ReIm"], None}, ButtonNote->"http://reference.wolfram.com/language/ref/ReIm.html?q=ReIm"], "Input"], ", and use ", StyleBox[ButtonBox["ComplexExpand", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/language/ref/ComplexExpand.html?q=\ ComplexExpand"], None}, ButtonNote-> "http://reference.wolfram.com/language/ref/ComplexExpand.html?q=\ ComplexExpand"], "Input"], " to expand the expression assuming that all variables are real." }], "Text", CellChangeTimes->{ 3.6246165771027927`*^9, {3.624617409834898*^9, 3.6246174225846987`*^9}, { 3.6246177034973288`*^9, 3.624617717937092*^9}, {3.624617755793839*^9, 3.624617886914297*^9}, {3.624617948565755*^9, 3.624617958541492*^9}, { 3.624618011289988*^9, 3.624618035022236*^9}, {3.6246184982215357`*^9, 3.624618555609243*^9}, 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An elegant procedure has replaced numerical integration." }], "Text", CellChangeTimes->{{3.681397442211474*^9, 3.681397539430242*^9}, { 3.6813977265382643`*^9, 3.68139777488548*^9}, {3.6871785154497766`*^9, 3.6871786128838377`*^9}, {3.687615977139303*^9, 3.6876159771422157`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["3.4.3 Point Doublet", "Subsection", CellChangeTimes->{{3.5650219372915497`*^9, 3.565021940037154*^9}, { 3.570192172175096*^9, 3.5701921744995003`*^9}, {3.59739919283563*^9, 3.5973991933416595`*^9}, {3.6005186693934727`*^9, 3.6005186698865013`*^9}, { 3.65669095486473*^9, 3.656690956520505*^9}}, CellTags->"simple: doublet"], Cell[TextData[{ "We have seen that taking the derivative of a complex potential leads to a \ complex velocity. For the case of the source-vortex combination, successive \ derivatives also generate a collection of complex potentials for the family \ of multipoles. Not only is the function ", Cell[BoxData[ FormBox[ FractionBox["c", "z"], TraditionalForm]]], " the complex velocity for the combined source and vortex, but it can be \ considered a new complex potential describing a doublet." }], "Text", CellChangeTimes->{{3.5466874797450323`*^9, 3.546687541419032*^9}, { 3.546687641721032*^9, 3.546687668027032*^9}, {3.547131461939*^9, 3.547131463238*^9}, {3.547131512472*^9, 3.54713156249*^9}, { 3.5471316334379997`*^9, 3.5471316388129997`*^9}, {3.547131704403*^9, 3.5471317293970003`*^9}, {3.5471317647460003`*^9, 3.547131779369*^9}, { 3.547131819145*^9, 3.547131835028*^9}, 3.547649733054578*^9, 3.5476497980565777`*^9, {3.5507672192451987`*^9, 3.5507672242839985`*^9}, { 3.550767554741799*^9, 3.550767556083399*^9}, 3.553941429684362*^9, { 3.565022059283764*^9, 3.5650221267226825`*^9}, {3.5666473024136453`*^9, 3.5666473034276466`*^9}, 3.5692407700836773`*^9, {3.5766776251745625`*^9, 3.576677628054567*^9}, {3.598803346021871*^9, 3.598803346021871*^9}, { 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This is because of two long-established conventions. \ First, the flow is shown as being from left to right in illustrations of flow \ over an object. Second, positive lift is associated with positive \ circulation. Potential flow over a circle is a building block for the \ simplest model for potential flow over an airfoil. ", StyleBox["For the remainder of this document, I will use the aerodynamics \ convention of clockwise circulation being positive, but will not change \ notation.", FontWeight->"Bold"], " " }], "Text", CellChangeTimes->{{3.681401083657894*^9, 3.681401404030141*^9}, { 3.681401434173347*^9, 3.681401633398241*^9}, {3.6814714713772783`*^9, 3.681471544143393*^9}, {3.687183119728101*^9, 3.68718312611096*^9}, { 3.687183156974819*^9, 3.687183203869172*^9}, {3.6876162935184927`*^9, 3.68761629528953*^9}, {3.692111716880001*^9, 3.692111725917983*^9}, { 3.69306380668894*^9, 3.693063809177115*^9}}], Cell[TextData[{ "Consider uniform potential flow over an impervious circle with circulation. \ ", StyleBox["The complex potential for this flow is a result of the", FontWeight->"Bold"], StyleBox[ButtonBox[" superposition", BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Superposition_principle"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Superposition_principle"], FontWeight->"Bold"], StyleBox[" of the elementary solutions for uniform flow, a point doublet and \ a point vortex.", FontWeight->"Bold"], " For convenience, place the center the circle of radius ", Cell[BoxData[ FormBox["a", TraditionalForm]]], " at the origin, and orient the positive real axis in the direction of the \ onset flow. The strength of the doublet is determined by requiring that the \ flow be tangent to the circle. \n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"f", "(", "z", ")"}], "=", RowBox[{ RowBox[{ SubscriptBox["V", "\[Infinity]"], " ", RowBox[{"z", "(", RowBox[{"1", "+", FractionBox[ SuperscriptBox["a", "2"], SuperscriptBox["z", "2"]]}], ")"}]}], "+", RowBox[{"\[ImaginaryI]", " ", FractionBox["\[CapitalGamma]", RowBox[{"2", "\[Pi]"}]], RowBox[{"ln", "(", FractionBox["z", "a"], ")"}]}]}]}], TraditionalForm]]], "\nYou should recognize the form of the additive terms from the simple flows \ described above. The complex potential is written in dimensional form and has \ units of ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["length", "2"], "/", "time"}], TraditionalForm]]], ". 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Notation used \ in Mathematica makes this straightforward because arguments are explicitly \ stated in function definitions. Here are Mathematica definitions for the \ dimensionless and dimensional complex potentials for flow over a circle. You \ choose the appropriate function by the arguments specified.\ \>", "Text", CellChangeTimes->{{3.5477419829154015`*^9, 3.5477420654254017`*^9}, { 3.5478027665963917`*^9, 3.547802780101392*^9}, 3.5478029634963913`*^9, { 3.5490448834431715`*^9, 3.5490448970307713`*^9}, {3.549129171903779*^9, 3.5491292521033792`*^9}, 3.5497357268451767`*^9, {3.549736346150777*^9, 3.549736389596777*^9}, {3.549736425788777*^9, 3.5497365591955767`*^9}, { 3.5497366126723766`*^9, 3.549736658037177*^9}, {3.549736688223177*^9, 3.549736771605177*^9}, 3.549906777448*^9, 3.550768088207399*^9, { 3.550845165934789*^9, 3.5508452059453893`*^9}, 3.553941429689362*^9, 3.565268456917816*^9, {3.5666483137766194`*^9, 3.5666483218574333`*^9}, { 3.566648356255494*^9, 3.5666483991711693`*^9}, {3.569241088979438*^9, 3.569241092645444*^9}, {3.5692411411615295`*^9, 3.5692411606927633`*^9}, { 3.57668500793288*^9, 3.57668500793288*^9}, {3.5770234974126434`*^9, 3.5770235042326527`*^9}, {3.577457947974365*^9, 3.5774579768544064`*^9}, { 3.577458012714456*^9, 3.577458029064479*^9}, {3.577458074374542*^9, 3.577458102564582*^9}, {3.5774581415946364`*^9, 3.577458218264744*^9}, { 3.577458253994794*^9, 3.577458340864916*^9}, 3.597402155812103*^9, { 3.5974022234529715`*^9, 3.5974024188531475`*^9}, {3.597495776750374*^9, 3.5974957792525167`*^9}, {3.6041596793128448`*^9, 3.6041596888783917`*^9}, {3.62963090696964*^9, 3.6296309986847486`*^9}, 3.6297147758453903`*^9, 3.629715136766925*^9, {3.656691159333625*^9, 3.656691165710657*^9}, {3.658746802317606*^9, 3.658746804797423*^9}, { 3.677072840401475*^9, 3.677072886580543*^9}, {3.677072961929468*^9, 3.677072963025359*^9}, {3.68140184607777*^9, 3.681401926043023*^9}, { 3.6814718112182198`*^9, 3.681471851649314*^9}, {3.687616430792631*^9, 3.6876164381923637`*^9}, {3.692111894111248*^9, 3.6921118964392977`*^9}, 3.692632780453248*^9}], Cell[BoxData[{ RowBox[{ RowBox[{"f", "[", RowBox[{"z_", ",", "\[CapitalGamma]_"}], "]"}], ":=", " ", RowBox[{ RowBox[{"z", RowBox[{"(", RowBox[{"1", "+", FractionBox["1", SuperscriptBox["z", "2"]]}], ")"}]}], "+", RowBox[{"\[ImaginaryI]", " ", FractionBox["\[CapitalGamma]", RowBox[{"2", "\[Pi]"}]], RowBox[{"Log", "[", "z", "]"}]}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"f", "[", RowBox[{"z_", ",", "\[CapitalGamma]_", ",", "V\[Infinity]_", ",", "a_"}], "]"}], ":=", RowBox[{ RowBox[{"V\[Infinity]", " ", "z", RowBox[{"(", RowBox[{"1", "+", FractionBox[ SuperscriptBox["a", "2"], SuperscriptBox["z", "2"]]}], ")"}]}], "+", RowBox[{"\[ImaginaryI]", " ", FractionBox["\[CapitalGamma]", RowBox[{"2", "\[Pi]"}]], RowBox[{"Log", "[", FractionBox["z", "a"], "]"}]}]}]}]}], "Input", CellChangeTimes->{{3.547742083811402*^9, 3.5477422817674017`*^9}, { 3.5477424167014017`*^9, 3.547742419584402*^9}, {3.547801228153392*^9, 3.547801232054392*^9}, {3.5486153765142*^9, 3.5486153799774*^9}, { 3.5486164476446*^9, 3.5486164794686003`*^9}, {3.5486165182502003`*^9, 3.5486165256602*^9}, {3.5486165766888*^9, 3.5486165903076*^9}, { 3.549735738591977*^9, 3.549735761789177*^9}, {3.5497370653183765`*^9, 3.5497370672995768`*^9}, 3.5499067853571997`*^9, {3.550768097640399*^9, 3.550768117436799*^9}, {3.550845102146389*^9, 3.550845117652789*^9}, 3.550845148431589*^9, {3.5512823129739714`*^9, 3.5512823274351716`*^9}, 3.553941429690362*^9, 3.565268463937828*^9, {3.565287318063796*^9, 3.565287342743039*^9}, {3.597402465539818*^9, 3.5974024996447687`*^9}, { 3.6566912157788477`*^9, 3.65669122284282*^9}, {3.6566913311780853`*^9, 3.65669135857143*^9}}, CellTags->"complex_potential_circle"], Cell["\<\ The first definition will be used in this section, and the second definition \ will be used in the next section when we use potential flow over a circle as \ an example to compute the force and moment on a body in potential flow.\ \>", "Text", CellChangeTimes->{{3.6252275032113028`*^9, 3.625227505713624*^9}, { 3.629631014989855*^9, 3.629631034063875*^9}, {3.692112049697126*^9, 3.692112126462308*^9}}], Cell[TextData[{ "The complex potential has a pole-type singularity at ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", "0"}], TraditionalForm]]], ". Write the complex potential in terms of the velocity potential and stream \ function ", Cell[BoxData[ FormBox[ RowBox[{"f", "=", RowBox[{"\[Phi]", "+", RowBox[{"\[ImaginaryI]", " ", "\[Psi]"}]}]}], TraditionalForm]]], ". Streamlines are obtained by plotting lines of ", Cell[BoxData[ FormBox[ RowBox[{"\[Psi]", "=", "const"}], TraditionalForm]]], ". The Mathematica function ", StyleBox["ContourPlot", "Input"], " will be used to visualize the flow by plotting streamlines." }], "Text", CellChangeTimes->{{3.597402533738719*^9, 3.597402687646522*^9}, { 3.5975038487110634`*^9, 3.597503909091517*^9}, {3.5975039962385015`*^9, 3.597504234812147*^9}, {3.5975042732083435`*^9, 3.5975044253870473`*^9}, { 3.6083860784553556`*^9, 3.6083862069527054`*^9}, {3.6083862428747597`*^9, 3.6083863562892466`*^9}, 3.6083864376869025`*^9, {3.6083865353114862`*^9, 3.6083865899136095`*^9}, {3.6083871355178165`*^9, 3.608387136034846*^9}, { 3.608387223283836*^9, 3.6083872232958364`*^9}, {3.6083890840292645`*^9, 3.608389100138186*^9}, {3.608389138089357*^9, 3.6083891511221023`*^9}, { 3.6083899069923353`*^9, 3.608389910232521*^9}, {3.625227071424111*^9, 3.625227088958879*^9}, {3.6252271329643593`*^9, 3.625227141523829*^9}, { 3.625227498723962*^9, 3.6252275208723927`*^9}, {3.625228262546628*^9, 3.6252282792504473`*^9}, {3.625228417224098*^9, 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We only need to show \ that the tangent-flow boundary condition is met on the circle to establish \ that we have a solution for potential flow over an impervious circle.", FontWeight->"Bold"], " The tangent-flow boundary condition can be expressed as ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ OverscriptBox["V", "\[RightVector]"], "\[CenterDot]", OverscriptBox["n", "^"]}], "=", "0"}], TraditionalForm]]], " on the circle. 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Use complex variables to \ compute ", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["V", "\[RightVector]"], "\[CenterDot]", OverscriptBox["n", "^"]}], TraditionalForm]], FormatType->"TraditionalForm"], " on the circle." }], "Text", CellChangeTimes->{{3.6921135325299263`*^9, 3.692113742162575*^9}, { 3.692113794808681*^9, 3.692113859740726*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"Re", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"w", "[", RowBox[{"z", ",", "\[CapitalGamma]"}], "]"}], "/.", RowBox[{"z", "\[Rule]", SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]]}]}], ")"}], SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]]}], "]"}], "//", "ComplexExpand"}], "//", "Simplify"}]], "Input", CellChangeTimes->{{3.687618382854681*^9, 3.687618498706852*^9}, { 3.687618597143485*^9, 3.687618641820174*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.687619239599588*^9, 3.6876215929413853`*^9, 3.687621642314987*^9, 3.688300970320224*^9, 3.6883010560779953`*^9, 3.688744991128621*^9, 3.693328203343061*^9}] }, Open ]], Cell["\<\ This complex potential is thus a solution for potential flow over a circle \ for any value of circulation. To obtain a unique solution, the circulation \ \[CapitalGamma] must be specified.\ \>", "Text", CellChangeTimes->{{3.608389296883439*^9, 3.608389441924735*^9}, 3.6296314265525723`*^9, 3.629635243469193*^9, {3.629635346105033*^9, 3.629635354440737*^9}, {3.6296355421917887`*^9, 3.629635617153722*^9}, 3.629715149650516*^9, {3.6814723095841017`*^9, 3.6814723978530273`*^9}}], Cell[TextData[{ "Points in the flow field where the velocity is zero are called stagnation \ points.", " ", "Locate stagnation points as a function of circulation by ", ButtonBox["solving the equation", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/tutorial/SolvingEquations.\ html"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/tutorial/SolvingEquations.html"], " obtained by setting the complex velocity equal to zero." }], "Text", CellChangeTimes->{{3.5459925612571754`*^9, 3.5459925721771755`*^9}, { 3.545993440586375*^9, 3.545993449415975*^9}, 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function ", StyleBox[ButtonBox["DensityPlot", BaseStyle->"Hyperlink", ButtonData->{ URL["http://reference.wolfram.com/mathematica/ref/DensityPlot.html?q=\ DensityPlot&lang=en"], None}, ButtonNote-> "http://reference.wolfram.com/mathematica/ref/DensityPlot.html?q=\ DensityPlot&lang=en"], "Input"], " to visualize of the pressure field throughout the flow." }], "Text", CellChangeTimes->{{3.5975059447949524`*^9, 3.5975060168220725`*^9}, { 3.598804605813927*^9, 3.5988046058249273`*^9}, {3.608390569602235*^9, 3.6083905971548104`*^9}, {3.625228772526321*^9, 3.625228790077457*^9}, { 3.629631587729073*^9, 3.629631592546073*^9}, {3.687621671859453*^9, 3.687621681650028*^9}, {3.6921140668769627`*^9, 3.692114078908486*^9}}], Cell[TextData[{ "Figure 3.1 is an interactive graph showing the flow field in the form \ streamlines and the pressure field. Although there is a mathematical solution \ for flow interior to the circle, only the external flow is shown. Streamlines \ are shown in black and the pressure field is color coded with a legend shown \ to the right of the graph. Regions with ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["C", "p"], "<", RowBox[{"-", "5"}]}], TraditionalForm]]], " are colored the same as ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["C", "p"], "=", RowBox[{"-", "5"}]}], TraditionalForm]]], ". External stagnation points are shown in yellow. Circulation is specified \ by a slider control, and has an initial setting of zero. As the circulation \ increases, stagnation points move downward along the circle until they \ coalesce for a certain value of circulation. With further increase, one \ stagnation point moves from the circle into the external flow; the other \ stagnation point moves to the interior of the circle and out of the region of \ interest. 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These computations are too slow to provide smooth response to the \ slider control so low precision calculations are made while you are moving \ the slider and replaced by higher precision computations when you release the \ control. When you are finished, open the Bookmarks/Autorun menu and select \ Initial Settings to return the graph to the default value of circulation." }], "Text", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{{3.610553639880825*^9, 3.6105536461030273`*^9}, 3.610554445888129*^9, {3.610554550720607*^9, 3.6105546377511673`*^9}, { 3.610554696287266*^9, 3.610554718383009*^9}, {3.610554776969058*^9, 3.610554838951166*^9}, {3.610554886832674*^9, 3.610555159655409*^9}, { 3.6296318506249313`*^9, 3.629632059551652*^9}, {3.629632157516583*^9, 3.629632170661921*^9}, {3.629632246607667*^9, 3.629632250301984*^9}, 3.629714805785987*^9, 3.629715180804804*^9}], Cell["\<\ It\[CloseCurlyQuote]s important to realize that potential flow over a bluff \ body such as a circle has little practical application. The symmetry of the \ streamlines through the imaginary axis indicates zero drag. For real fluids, \ there will be massive separation of the flow and drag will be significant. \ \>", "Text", CellFrame->{{0, 0}, {0, 0.5}}, CellChangeTimes->{ 3.5507681684799986`*^9, {3.5507687435885987`*^9, 3.5507687708261986`*^9}, { 3.550854972637189*^9, 3.550855014425189*^9}, {3.5512899824777718`*^9, 3.5512899925397716`*^9}, 3.5539414297123623`*^9, {3.5690601987783184`*^9, 3.5690603263241425`*^9}, 3.5692421481120977`*^9, {3.5770249786907268`*^9, 3.5770250029807606`*^9}, {3.6018910789194183`*^9, 3.6018913094796057`*^9}, {3.60189135379014*^9, 3.601891460384237*^9}, { 3.6018914950732207`*^9, 3.601891590752693*^9}, {3.60189162705577*^9, 3.601891656923478*^9}, {3.601891687063202*^9, 3.6018917427353864`*^9}, { 3.625239803318137*^9, 3.625239806620986*^9}, 3.62971480701562*^9, 3.6297151847803087`*^9, {3.677770913030588*^9, 3.67777093927602*^9}, { 3.67777103360044*^9, 3.677771082390622*^9}}], Cell[TextData[{ StyleBox["The primary importance of potential flow over a circle is that it \ serves as a known solution for potential flow over an object of simple \ geometry.", FontWeight->"Bold"], " The process of conformal mapping can be used to transform potential flow \ over a circle to potential flow over an object of more complicated geometry \ such as an airfoil, which provides useful approximations for lift and \ pitching moment for limited conditions." }], "Text", CellChangeTimes->{{3.601891697951825*^9, 3.601891708124407*^9}, { 3.601891748117694*^9, 3.601892128098428*^9}, {3.6083907704127207`*^9, 3.6083908515203595`*^9}, {3.62963230032441*^9, 3.629632331055293*^9}, { 3.629632387941766*^9, 3.629632399575097*^9}, {3.629635654925077*^9, 3.629635655949827*^9}, {3.629635688738286*^9, 3.62963569077806*^9}, { 3.629715196629582*^9, 3.629715197389764*^9}, {3.681473096222622*^9, 3.681473103316493*^9}, {3.681473149874251*^9, 3.68147319243223*^9}, { 3.692114304204277*^9, 3.692114308779739*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["3.6 Force and Moment on a Body", "Section", CellChangeTimes->{{3.5701920212760305`*^9, 3.5701920701197166`*^9}, { 3.570193223076954*^9, 3.5701932287553635`*^9}, {3.5916277766508684`*^9, 3.5916277779019403`*^9}, {3.5973992069644384`*^9, 3.5973992076684785`*^9}, 3.6297148087889338`*^9}, CellTags->"force"], Cell[CellGroupData[{ Cell["3.6.1 Introduction", "Subsection", CellChangeTimes->{{3.629717805442626*^9, 3.629717814412092*^9}}, CellTags->"force: introduction"], Cell[TextData[{ StyleBox["This section demonstrates the power of complex analysis to provide \ remarkably simple results for the force and moment per unit span for \ potential flow over a two-dimensional object.", FontWeight->"Bold"], " The approach is to write the force and moment on a body in potential flow \ as closed contour integrals of analytic functions of a complex variable with \ isolated pole-type singularities inside the object. 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The shape of the object \ is specified by the contour ", Cell[BoxData[ FormBox["C", TraditionalForm]]], " with ", Cell[BoxData[ FormBox[ OverscriptBox["n", "^"], TraditionalForm]]], " denoting the outward unit normal and ", Cell[BoxData[ FormBox[ RowBox[{"\[DifferentialD]", "s"}], TraditionalForm]]], ", the infinitesimal segment of the contour located at ", Cell[BoxData[ FormBox[ OverscriptBox["r", "\[RightVector]"], TraditionalForm]]], ". 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SubscriptBox["\[ScriptF]", "y"]}]}], "=", RowBox[{ FractionBox["1", "2"], "\[ImaginaryI]", " ", "\[Rho]", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{"w", " ", OverscriptBox["w", "_"], RowBox[{"\[DifferentialD]", OverscriptBox["z", "_"]}]}]}]}]}], TraditionalForm]]], "\nLook at the moment equation and note that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"x", " ", RowBox[{"\[DifferentialD]", "x"}]}], "+", RowBox[{"y", " ", RowBox[{"\[DifferentialD]", "y"}]}]}], TraditionalForm]]], " in the moment equation can be written in terms of the real part of ", Cell[BoxData[ FormBox[ RowBox[{"z", " ", RowBox[{"\[DifferentialD]", OverscriptBox["z", "_"]}]}], TraditionalForm]]], ".\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"z", " ", RowBox[{"\[DifferentialD]", OverscriptBox["z", "_"]}]}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", RowBox[{"\[ImaginaryI]", " ", "y"}]}], ")"}], RowBox[{"(", RowBox[{ RowBox[{"\[DifferentialD]", "x"}], "-", RowBox[{"\[ImaginaryI]", " ", 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RowBox[{ SubscriptBox["\[ScriptF]", "x"], "-", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["\[ScriptF]", "y"]}]}], "=", RowBox[{ FractionBox["1", "2"], "\[ImaginaryI]", " ", "\[Rho]", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ RowBox[{"w", "(", "z", ")"}], " ", RowBox[{ OverscriptBox["w", "_"], "(", OverscriptBox["z", "_"], ")"}], RowBox[{"\[DifferentialD]", OverscriptBox["z", "_"]}]}]}]}]}], TraditionalForm]]], "\n\t\t", Cell[BoxData[ FormBox[ RowBox[{"\[ScriptM]", "=", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], RowBox[{"\[Rho]Re", "[", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ RowBox[{"w", "(", "z", ")"}], " ", OverscriptBox["w", "_"], " ", RowBox[{"(", OverscriptBox["z", "_"], ")"}], "z", " ", RowBox[{"\[DifferentialD]", OverscriptBox["z", "_"]}]}]}]}]}]}], TraditionalForm]]], "\nThe complex potential is analytic in ", Cell[BoxData[ FormBox["z", TraditionalForm]]], ", and the ", ButtonBox["conjugate of the complex velocity is analytic in the conjugate of \ z", BaseStyle->"Hyperlink", ButtonData->{ "IntroductionToComplexAnalysis.cdf", "functions: analytic: conjugate"}], "." }], "Text", CellChangeTimes->{ 3.629645519393959*^9, 3.629645588377735*^9, {3.629645641636661*^9, 3.6296458618984623`*^9}, {3.629646109803866*^9, 3.629646110652257*^9}, { 3.62964796744829*^9, 3.6296479674497633`*^9}, 3.629715242054878*^9, { 3.679404046149764*^9, 3.6794040465605707`*^9}, {3.687622688452715*^9, 3.6876228005678167`*^9}, {3.692115006614563*^9, 3.69211500662217*^9}, { 3.6921151021837063`*^9, 3.692115102190888*^9}, 3.692115136565813*^9, { 3.6921151805729322`*^9, 3.692115180580489*^9}, {3.692115395674864*^9, 3.692115410951725*^9}, {3.6921154691394663`*^9, 3.692115469144567*^9}, { 3.6928037514778023`*^9, 3.692803751949525*^9}}], Cell[TextData[{ "Write the complex velocity as the derivative of the complex potential.\n\t\t\ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[ScriptF]", "x"], "-", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["\[ScriptF]", "y"]}]}], "=", RowBox[{ FractionBox["1", "2"], "\[ImaginaryI]", " ", "\[Rho]", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "z", ")"}], " ", RowBox[{ OverscriptBox["f", "_"], "'"}], RowBox[{"(", OverscriptBox["z", "_"], ")"}], RowBox[{"\[DifferentialD]", OverscriptBox["z", "_"]}]}]}]}]}], TraditionalForm]]], "\n\t\t", Cell[BoxData[ FormBox[ RowBox[{"\[ScriptM]", "=", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], RowBox[{"\[Rho]Re", "[", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "z", ")"}], " ", "z", " ", RowBox[{ OverscriptBox["f", "_"], "'"}], " ", RowBox[{"(", OverscriptBox["z", "_"], ")"}], " ", RowBox[{"\[DifferentialD]", OverscriptBox["z", "_"]}]}]}], "]"}]}]}], TraditionalForm]]], "\nSubstitute ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ OverscriptBox["f", "_"], "'"}], RowBox[{"(", OverscriptBox["z", "_"], ")"}], RowBox[{"\[DifferentialD]", OverscriptBox["z", "_"]}]}], "=", RowBox[{"\[DifferentialD]", RowBox[{ OverscriptBox["f", "_"], "(", OverscriptBox["z", "_"], ")"}]}]}], TraditionalForm]]], " into the force and moment equations.\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[ScriptF]", "x"], "-", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["\[ScriptF]", "y"]}]}], "=", RowBox[{ FractionBox["1", "2"], "\[ImaginaryI]", " ", "\[Rho]", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "z", ")"}], RowBox[{"\[DifferentialD]", OverscriptBox["f", "_"]}]}]}]}]}], TraditionalForm]]], "\n\t\t", Cell[BoxData[ FormBox[ RowBox[{"\[ScriptM]", "=", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], RowBox[{"\[Rho]Re", "[", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], " ", RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "z", ")"}], " ", "z", " ", RowBox[{"\[DifferentialD]", OverscriptBox["f", "_"]}]}]}], "]"}]}]}], TraditionalForm]]], "\n", StyleBox["The contour of integration is taken as the closed curve specifying \ the object.", FontWeight->"Bold"] }], "Text", CellChangeTimes->{{3.629645319503476*^9, 3.629645323749362*^9}, { 3.62964595431767*^9, 3.629645958030326*^9}, {3.629646044630052*^9, 3.629646099722775*^9}, {3.679404081289275*^9, 3.679404092309308*^9}, { 3.687622854125527*^9, 3.687622856941815*^9}, {3.692629816344713*^9, 3.692629816344893*^9}, {3.692803756023308*^9, 3.692803763196992*^9}}], Cell[TextData[{ "The object itself is a streamline of the flow so ", Cell[BoxData[ FormBox[ RowBox[{"\[Psi]", "=", "constant"}], TraditionalForm]]], " on the object. Since the stream function is undetermined by an additive \ constant, this constant can be set to zero. Thus, the following statement is \ valid on the contour specifying the object.\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[DifferentialD]", OverscriptBox["f", "_"]}], "=", RowBox[{ RowBox[{ RowBox[{"\[DifferentialD]", "\[Phi]"}], "-", RowBox[{"\[ImaginaryI]", RowBox[{"\[DifferentialD]", "\[Psi]"}]}]}], "=", RowBox[{"\[DifferentialD]", "\[Phi]"}]}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[DifferentialD]", "f"}], "=", RowBox[{ RowBox[{ RowBox[{"\[DifferentialD]", "\[Phi]"}], "+", RowBox[{"\[ImaginaryI]", RowBox[{"\[DifferentialD]", "\[Psi]"}]}]}], "=", RowBox[{"\[DifferentialD]", "\[Phi]"}]}]}], TraditionalForm]]], " so ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[DifferentialD]", OverscriptBox["f", "_"]}], "=", RowBox[{"\[DifferentialD]", "f"}]}], TraditionalForm]]], "\nCollect the equations for computing force and moment on the object using \ the contour specifying the object, writing ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[DifferentialD]", "f"}], "=", RowBox[{ RowBox[{"f", "'"}], RowBox[{"\[DifferentialD]", "z"}]}]}], TraditionalForm]]], ". ", StyleBox["These are known as Blasius\[CloseCurlyQuote] equations and can be \ written in terms of either complex potential or complex velocity.", FontWeight->"Bold"], "\n\t\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["\[ScriptF]", "x"], "-", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["\[ScriptF]", "y"]}]}], "=", RowBox[{ RowBox[{ FractionBox["1", "2"], "\[ImaginaryI]", " ", "\[Rho]", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "z", ")"}]}], ")"}], "2"], " ", RowBox[{"\[DifferentialD]", "z"}]}]}]}], "=", RowBox[{ FractionBox["1", "2"], "\[ImaginaryI]", " ", "\[Rho]", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"w", "(", "z", ")"}], ")"}], "2"], " ", RowBox[{"\[DifferentialD]", "z"}]}]}]}]}]}], TraditionalForm]]], "\n\t\t", Cell[BoxData[ FormBox[ RowBox[{"\[ScriptM]", "=", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], "\[Rho]", " ", RowBox[{"Re", "[", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"f", "'"}], RowBox[{"(", "z", ")"}]}], ")"}], "2"], " ", "z", " ", RowBox[{"\[DifferentialD]", "z"}]}]}], "]"}]}], "=", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], "\[Rho]", " ", RowBox[{"Re", "[", RowBox[{ SubscriptBox["\[ContourIntegral]", "C"], RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"w", "(", "z", ")"}], ")"}], "2"], " ", "z", " ", RowBox[{"\[DifferentialD]", "z"}]}]}], "]"}]}]}]}], TraditionalForm]]] }], "Text", CellChangeTimes->{ 3.576247100104289*^9, {3.5763325478692226`*^9, 3.5763326234113293`*^9}, { 3.5763327027034416`*^9, 3.576332716763461*^9}, {3.576332747205505*^9, 3.5763328037975855`*^9}, {3.576332953549796*^9, 3.576332953549796*^9}, { 3.5765073920582247`*^9, 3.5765074170282593`*^9}, {3.576507462004326*^9, 3.576507480684352*^9}, {3.5765076533485966`*^9, 3.5765077570087414`*^9}, { 3.576507895980937*^9, 3.5765079655010347`*^9}, {3.5765086991440916`*^9, 3.576508703504098*^9}, {3.5770261910964327`*^9, 3.577026194376437*^9}, { 3.598805419636475*^9, 3.598805419636475*^9}, {3.600598759497881*^9, 3.6005989226622133`*^9}, {3.6005990430080967`*^9, 3.6005991207375426`*^9}, {3.600599835170406*^9, 3.600599835170406*^9}, { 3.6005998666672072`*^9, 3.600599929697812*^9}, {3.6005999657168727`*^9, 3.600600037946004*^9}, 3.600600074734108*^9, {3.607954036548479*^9, 3.607954084020194*^9}, {3.607954211124464*^9, 3.607954271266904*^9}, 3.627295620882227*^9, {3.6273122488141737`*^9, 3.627312260301672*^9}, 3.629646181948642*^9, 3.629714837587738*^9, {3.6794041618499804`*^9, 3.6794043307669153`*^9}, {3.6795865950259447`*^9, 3.67958660028965*^9}, { 3.68666943961554*^9, 3.686669442103513*^9}, {3.69262997744602*^9, 3.692630012396565*^9}, {3.692630368342341*^9, 3.692630406138983*^9}, { 3.69263109183502*^9, 3.692631093104374*^9}, {3.692803769812459*^9, 3.692803770301166*^9}, 3.69306517792214*^9}], Cell[TextData[{ "The derivative of an analytic function is analytic, and the product of two \ analytic functions is analytic. Thus, the integrands of the two integrals \ above are analytic with possible pole-type singularities interior to the \ contour ", Cell[BoxData[ FormBox["\[ScriptCapitalC]", TraditionalForm]]], " specifying the object. The powerful technique of the ", ButtonBox["residue theorem", BaseStyle->"Hyperlink", ButtonData->{"ReviewOfComplexAnalysis.cdf", "residue"}], " can be used to evaluate them. If there are no poles associated with the \ integrands inside the contour specifying the object, the integral is zero. If \ there are poles inside the contour, the force and moment are determined by \ the sums of the residues associated with the poles of the integrands. Note \ the simplicity of this result, there is no need for numerical integration no \ matter how complicated the shape of the object. Also note that, since there \ are no singularities outside ", Cell[BoxData[ FormBox["\[ScriptCapitalC]", TraditionalForm]]], ", Cauchy\[CloseCurlyQuote]s theorem implies that the integral gives the \ same value for any contour outside the object." }], "Text", CellChangeTimes->{{3.576507803668807*^9, 3.5765078533108773`*^9}, { 3.57650798378106*^9, 3.5765079969210787`*^9}, {3.5765081536213055`*^9, 3.576508158611312*^9}, {3.5765082956555223`*^9, 3.5765083309655714`*^9}, { 3.5765084149696913`*^9, 3.5765084149796915`*^9}, {3.57650851306183*^9, 3.5765085295218534`*^9}, {3.5765085675939074`*^9, 3.5765085860239334`*^9}, {3.576508616333976*^9, 3.576508663604042*^9}, { 3.5765087189141197`*^9, 3.5765087189141197`*^9}, {3.576508772610201*^9, 3.5765088120902557`*^9}, {3.576508846610304*^9, 3.576509152810739*^9}, 3.576509192800795*^9, {3.577454710391821*^9, 3.5774547288018465`*^9}, { 3.6005991495351896`*^9, 3.6005991793668957`*^9}, {3.600599228356698*^9, 3.600599510509836*^9}, {3.600599552416233*^9, 3.6005995657979984`*^9}, { 3.6006001181725926`*^9, 3.6006003230483108`*^9}, 3.600600361161491*^9, { 3.6006007837456613`*^9, 3.6006008413689575`*^9}, {3.6006008777260365`*^9, 3.60060088512646*^9}, {3.600616169059963*^9, 3.6006161989106708`*^9}, { 3.6006162309575033`*^9, 3.600616286432677*^9}, {3.600616323587802*^9, 3.6006166220958757`*^9}, {3.607954096389901*^9, 3.607954104162346*^9}, { 3.6079543007555904`*^9, 3.6079543145573797`*^9}, {3.6079543518245115`*^9, 3.607954380245137*^9}, {3.6079544117599397`*^9, 3.6079544365333567`*^9}, { 3.6079544757325983`*^9, 3.6079545140867925`*^9}, {3.6254125296669083`*^9, 3.625412615333489*^9}, {3.625412677505014*^9, 3.625412832646778*^9}, 3.627301807835766*^9, 3.629646459974978*^9, {3.6296464993395853`*^9, 3.629646500827684*^9}, {3.629714838735014*^9, 3.629714841648308*^9}, { 3.692115670938157*^9, 3.692115675032455*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["3.6.5 Force and Moment for Potential Flow Over a Circle", "Subsection", CellChangeTimes->{{3.66955630239194*^9, 3.669556345842512*^9}, { 3.669556379501614*^9, 3.66955639969403*^9}}, CellTags->"force: circle"], Cell[TextData[{ "Compute the force and moment per unit span on the circle of radius ", Cell[BoxData[ FormBox["a", TraditionalForm]]], " in uniform flow as illustrated in Figure 3.1. Start with the dimensional \ form for the complex potential. Compute complex velocity and note that there \ is a pole-type singularity at ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", "0"}], TraditionalForm]]], ". 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The complex force is defined as ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[ScriptF]", "x"], "-", RowBox[{"\[ImaginaryI]", " ", SubscriptBox["\[ScriptF]", "y"]}]}], TraditionalForm]]], ", so ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["f", "x"], "=", "0"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["f", "y"], "=", RowBox[{"\[Rho]", " ", SubscriptBox["V", "\[Infinity]"], " ", "\[CapitalGamma]"}]}], TraditionalForm]]], ". In the terminology of aerodynamics, potential flow over a circle computes \ the lift per unit span to be ", Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", " ", SubscriptBox["V", "\[Infinity]"], " ", "\[CapitalGamma]"}], TraditionalForm]]], " and the drag per unit span to be zero." }], "Text", CellChangeTimes->{{3.6296469413532248`*^9, 3.629647098761767*^9}, { 3.62964713047806*^9, 3.6296472235734797`*^9}, {3.6296472590432987`*^9, 3.629647295958475*^9}, {3.669557528815496*^9, 3.669557547663001*^9}, { 3.681475215290806*^9, 3.681475283624782*^9}, {3.687623128242484*^9, 3.6876231289706993`*^9}, 3.692115795067487*^9}], Cell["\<\ Use the Blasius\[CloseCurlyQuote] equations to compute the pitching moment \ per unit span on the circle.\ \>", "Text", CellChangeTimes->{{3.6082842263670783`*^9, 3.6082842540536623`*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[ScriptM]", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox["1", "2"]}], "\[Rho]"}], " ", ")"}], RowBox[{"Re", "[", 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