(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 56812, 1715]*) (*NotebookOutlinePosition[ 71532, 2252]*) (* CellTagsIndexPosition[ 71488, 2248]*) (*WindowFrame->Normal*) Notebook[{ Cell["Making a differential equation dimensionless", "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "This notebook has been written in ", StyleBox["Mathematica ", FontSlant->"Italic"], "by \n\nMark J. McCready\nProfessor and Chair of Chemical Engineering\n\ University of Notre Dame\nNotre Dame IN 46556\nUSA\n\n", ButtonBox["mjm@nd.edu", ButtonData:>{ URL[ "mailto:mjm@nd.edu"], None}, ButtonStyle->"Hyperlink"], "\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \ the World Wide Web and the Internet.\n\nI would hope that as a professional \ courtesy that this notice remain visible to other users. \nThere is no charge \ for copying and dissemination \n\nVersion: 11/06/00\nMore recent versions \ of this notebook should be available at the web site:\n", ButtonBox["http://www.nd.edu/~mjm/dimensionless.nb", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/dimensionless.nb"], None}, ButtonStyle->"Hyperlink"] }], "Text"], Cell[CellGroupData[{ Cell["Dimensionless equations", "Subtitle"], Cell[CellGroupData[{ Cell["How to make derivative terms nondimensional", "Subsection"], Cell[TextData[{ "Using the chain rule for derivatives \n\n\n", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\/\[PartialD]r\)]], "= ", Cell[BoxData[ \(TraditionalForm\`ds\/dr\)]], Cell[BoxData[ \(TraditionalForm\`\[PartialD]\/\[PartialD]s\)]], "= ", Cell[BoxData[ \(TraditionalForm\`1\/R\)]], Cell[BoxData[ \(TraditionalForm\`\[PartialD]\/\[PartialD]s\)]], "\n", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\^2\/\((\[PartialD]r)\)\^2\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\((ds\/dr)\)\^2\)]], Cell[BoxData[ \(TraditionalForm\`\[PartialD]\^2\/\((\[PartialD]s)\)\^2\)]], "= ", Cell[BoxData[ \(TraditionalForm\`1\/R\^2\)]], Cell[BoxData[ \(TraditionalForm\`\[PartialD]\^2\/\((\[PartialD]s)\)\^2\)]], "\n\nand making the substitutions:\n\ns= r/R,\n", Cell[BoxData[ \(u\&~\_\[Theta]\)], AspectRatioFixed->True], "= ", Cell[BoxData[ \(u\_\[Theta]\)], AspectRatioFixed->True], "/U, \n", Cell[BoxData[ \(u\&~\_r\)], AspectRatioFixed->True], "= ", Cell[BoxData[ \(u\_r\)], AspectRatioFixed->True], "/U\n" }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Let's try some substitutions.", "Subsection"], Cell[CellGroupData[{ Cell["Here is a first derivative term", "Subsubsection"], Cell[BoxData[ FormBox[ RowBox[{" ", FractionBox[\(\[PartialD]\(\(u\_r\)(r, \[Theta], t)\)\), \(\[PartialD]r\), MultilineFunction->None]}], TraditionalForm]], "Input", CellEditDuplicate->True, Evaluatable->False], Cell[TextData[{ "Here is a convenient input format for ", StyleBox["Mathematica", FontSlant->"Italic"], " " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(firstD\ = \ \ D[u\_r[r, \[Theta], t], {r, 1}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["u", "r", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta], t\), ")"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ Now we do the substitutions for the derivatives as described above. \ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"firstD", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[\(u\_r\), TagBox[\((a1_, a2_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta]\), "]"}], "\[RuleDelayed]", " ", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}\)u\&~\_r[ s, \[Theta]]\ R\^\(-a1\)\ U\)}], ",", RowBox[{ RowBox[{ SuperscriptBox[\(u\_\[Theta]\), TagBox[\((a1_, a2_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta]\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], \ a2}\)u\&~\_\[Theta][s, \[Theta]]\ R\^\(-a1\)\ U\)}], " ", ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ SuperscriptBox["p", TagBox[\((a1_, a2_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta]\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}\)p[ s, \[Theta]]\ R\^\(-a1\)\)}], ",", "\[IndentingNewLine]", \(u\_r[r, \[Theta]] \[Rule] u\&~\_r[s, \[Theta]]\ U\), ",", \(u\_\[Theta][r, \[Theta]] \[Rule] u\&~\_\[Theta][s, \[Theta]]\ U\), ",", \(r \[Rule] R\ s\)}], "}"}]}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["u", \(R\ s\), TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(R\ s, \[Theta], t\), ")"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"firstDans", "=", RowBox[{"firstD", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[\(u\_r\), TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", " ", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\[Tau], a3}\)u\ \&~\_r[s, \[Theta], \[Tau]]\ R\^\(-a1\)\ U\ \((R/U)\)\^\(-a3\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox[\(u\_\[Theta]\), TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\ \[Tau], a3}\)u\&~\_\[Theta][ s, \[Theta], \[Tau]]\ R\^\(-a1\)\ U \ \((R/U)\)\^\(-a3\)\)}], " ", ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ SuperscriptBox["p", TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\ \[Tau], a3}\)p[s, \[Theta], \[Tau]]\ \(R\^\(-a1\)\) \((R/U)\)\^\(-a3\)\)}], ",", "\[IndentingNewLine]", \(u\_r[r, \[Theta], t] \[Rule] u\&~\_r[s, \[Theta], t]\ U\), ",", \(u\_\[Theta][r, \[Theta], t\ ] \[Rule] u\&~\_\[Theta][s, \[Theta], t]\ U\), ",", \(r \[Rule] R\ s\)}], "}"}]}]}]], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"U", " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], "R"], TraditionalForm]], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Here is a time derivative term", "Subsubsection"], Cell[BoxData[ FormBox[ RowBox[{" ", FractionBox[\(\[PartialD]\(\(u\_r\)(r, \[Theta], t)\)\), \(\[PartialD]t\), MultilineFunction->None]}], TraditionalForm]], "Input", CellEditDuplicate->True, Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(timeD\ = \ \ D[u\_r[r, \[Theta], t\ ], {t, 1}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["u", "r", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta], t\), ")"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"timeDans", "=", RowBox[{"timeD", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[\(u\_r\), TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", " ", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\[Tau], a3}\)u\ \&~\_r[s, \[Theta], \[Tau]]\ R\^\(-a1\)\ U\ \((R/U)\)\^\(-a3\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox[\(u\_\[Theta]\), TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\ \[Tau], a3}\)u\&~\_\[Theta][ s, \[Theta], \[Tau]]\ R\^\(-a1\)\ U \ \((R/U)\)\^\(-a3\)\)}], " ", ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ SuperscriptBox["p", TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\ \[Tau], a3}\)p[s, \[Theta], \[Tau]]\ \(R\^\(-a1\)\) \((R/U)\)\^\(-a3\)\)}], ",", "\[IndentingNewLine]", \(u\_r[r, \[Theta], t] \[Rule] u\&~\_r[s, \[Theta], \[Tau]]\ U\), ",", \(u\_\[Theta][r, \[Theta], t\ ] \[Rule] u\&~\_\[Theta][s, \[Theta], \[Tau]]\ U\), ",", \(r \[Rule] R\ s\)}], "}"}]}]}]], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(U\^2\), " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], "R"], TraditionalForm]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Here is a second derivative term", "Subsubsection"], Cell[BoxData[ FormBox[ RowBox[{"\[Mu]", " ", FractionBox[\(\[PartialD]\^2\(\( u\_r\)(r, \[Theta], t)\)\), \(\[PartialD]r\^2\), MultilineFunction->None]}], TraditionalForm]], "Input", Evaluatable->False], Cell[TextData[{ "Here we write it in real ", StyleBox["Mathematica", FontSlant->"Italic"], " format. (The previous one would work also.)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(secondD\ = \ \[Mu]\ D[u\_r[r, \[Theta], t], {r, 2}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"\[Mu]", " ", RowBox[{ SubsuperscriptBox["u", "r", TagBox[\((2, 0, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta], t\), ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"secondDans", "=", RowBox[{"secondD", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[\(u\_r\), TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", " ", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\[Tau], a3}\)u\ \&~\_r[s, \[Theta], \[Tau]]\ R\^\(-a1\)\ U\ \((R/U)\)\^\(-a3\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox[\(u\_\[Theta]\), TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\ \[Tau], a3}\)u\&~\_\[Theta][ s, \[Theta], \[Tau]]\ R\^\(-a1\)\ U \ \((R/U)\)\^\(-a3\)\)}], " ", ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ SuperscriptBox["p", TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\ \[Tau], a3}\)p[s, \[Theta], \[Tau]]\ \(R\^\(-a1\)\) \((R/U)\)\^\(-a3\)\)}], ",", "\[IndentingNewLine]", \(u\_r[r, \[Theta], t] \[Rule] u\&~\_r[s, \[Theta], \[Tau]]\ U\), ",", \(u\_\[Theta][r, \[Theta], t\ ] \[Rule] u\&~\_\[Theta][s, \[Theta], \[Tau]]\ U\), ",", \(r \[Rule] R\ s\)}], "}"}]}]}]], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"U", " ", "\[Mu]", " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((2, 0, 0)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], \(R\^2\)], TraditionalForm]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Here is a nonlinear term", "Subsubsection"], Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", " ", \(\(u\_r\)(r, \[Theta])\), " ", FractionBox[\(\[PartialD]\(\(u\_r\)(r, \[Theta], t)\)\), \(\[PartialD]r\), MultilineFunction->None]}], TraditionalForm]], "Input", Evaluatable->False], Cell[CellGroupData[{ Cell[BoxData[ \(nonlinear\ = \ \[Rho]\ u\_r[r, \[Theta], t]\ D[ u\_r[r, \[Theta], t], {r, 1}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", " ", \(\(u\_r\)(r, \[Theta], t)\), " ", RowBox[{ SubsuperscriptBox["u", "r", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(r, \[Theta], t\), ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"nonlinearans", "=", RowBox[{"nonlinear", "/.", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ SuperscriptBox[\(u\_r\), TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", " ", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\[Tau], a3}\)u\ \&~\_r[s, \[Theta], \[Tau]]\ R\^\(-a1\)\ U\ \((R/U)\)\^\(-a3\)\)}], ",", RowBox[{ RowBox[{ SuperscriptBox[\(u\_\[Theta]\), TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\ \[Tau], a3}\)u\&~\_\[Theta][ s, \[Theta], \[Tau]]\ R\^\(-a1\)\ U \ \((R/U)\)\^\(-a3\)\)}], " ", ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ SuperscriptBox["p", TagBox[\((a1_, a2_, a3_)\), Derivative], MultilineFunction->None], "[", \(r, \[Theta], t\), "]"}], "\[RuleDelayed]", \(\[PartialD]\_\({s, a1}, {\[Theta], a2}, {\ \[Tau], a3}\)p[s, \[Theta], \[Tau]]\ \(R\^\(-a1\)\) \((R/U)\)\^\(-a3\)\)}], ",", "\[IndentingNewLine]", \(u\_r[r, \[Theta], t] \[Rule] u\&~\_r[s, \[Theta], \[Tau]]\ U\), ",", \(u\_\[Theta][r, \[Theta], t\ ] \[Rule] u\&~\_\[Theta][s, \[Theta], \[Tau]]\ U\), ",", \(r \[Rule] R\ s\)}], "}"}]}]}]], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{\(U\^2\), " ", "\[Rho]", " ", \(\(\(u\&~\)\_r\)(s, \[Theta], \[Tau])\), " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], "R"], TraditionalForm]], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ We can see how a typical nondimensionalization for the \ Navier-Stokes Equations would turn out. (Right side are positive \ terms.)\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(nondim1 = secondDans - nonlinearans - timeDans\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{ RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}], " ", \(U\^2\)}], "R"]}], "-", FractionBox[ RowBox[{ "\[Rho]", " ", \(\(\(u\&~\)\_r\)(s, \[Theta], \[Tau])\), " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}], " ", \(U\^2\)}], "R"], "+", FractionBox[ RowBox[{"\[Mu]", " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((2, 0, 0)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}], " ", "U"}], \(R\^2\)]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(nondim2 = Expand[\(nondim1/U\)/\[Mu]\ R^2]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"R", " ", "U", " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], "\[Mu]"]}], "-", FractionBox[ RowBox[{ "R", " ", "U", " ", "\[Rho]", " ", \(\(\(u\&~\)\_r\)(s, \[Theta], \[Tau])\), " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], "\[Mu]"], "+", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((2, 0, 0)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell["We substitute the Reynolds number into the equation.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(nondim3\ = \ nondim2 /. \[Mu] \[Rule] R\ U\ \[Rho]/Re\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", FractionBox[ RowBox[{"Re", " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], "\[Rho]"]}], "-", RowBox[{"Re", " ", \(\(\(u\&~\)\_r\)(s, \[Theta], \[Tau])\), " ", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], "+", RowBox[{ SubsuperscriptBox[\(u\&~\), "r", TagBox[\((2, 0, 0)\), Derivative], MultilineFunction->None], "(", \(s, \[Theta], \[Tau]\), ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Note that the Reynolds number, ", StyleBox["R U", FontSlant->"Italic"], " \[Rho]/\[Mu] is multiplying the inertia (nonlinear) term and the time \ derivative term in this nondimensional equation. What will happen if the \ Reynolds number is very small? \n\nYou can go back and do the pressure term \ on your own. " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["How big are the terms in a differential equation?", "Subsection"], Cell[CellGroupData[{ Cell["First try a dimensional equation", "Subsubsection"], Cell[CellGroupData[{ Cell["Here is the governing equation for pressure driven pipeflow", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(pipeDE = \(-dpdx\)\ + \ \[Mu]\ D[u[y], {y, 2}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Mu]", " ", RowBox[{ SuperscriptBox["u", "\[Prime]\[Prime]", MultilineFunction->None], "(", "y", ")"}]}], "-", "dpdx"}], TraditionalForm]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["It is readily solved. The term b is the channel half-height", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ans1 = \ DSolve[{pipeDE == 0, u[\(-b\)] == 0, u[b] == 0}, u[y], y]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{u( y) \[Rule] \(dpdx\ y\^2\)\/\(2\ \[Mu]\) - \(b\^2\ dpdx\)\/\(2\ \ \[Mu]\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ans = u[y] /. ans1[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(dpdx\ y\^2\)\/\(2\ \[Mu]\) - \(b\^2\ dpdx\)\/\(2\ \ \[Mu]\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["We can plot to be sure it behaves as advertised.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Plot[ ans /. {b -> 10, \ \[Mu] -> .01, dpdx -> \(- .01\)}, {y, \(-10\), 10}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.0476191 0.0147151 0.0117721 [ [.02381 .00222 -9 -9 ] [.02381 .00222 9 0 ] [.2619 .00222 -6 -9 ] [.2619 .00222 6 0 ] [.7381 .00222 -3 -9 ] [.7381 .00222 3 0 ] [.97619 .00222 -6 -9 ] [.97619 .00222 6 0 ] [.4875 .13244 -12 -4.5 ] [.4875 .13244 0 4.5 ] [.4875 .25016 -12 -4.5 ] [.4875 .25016 0 4.5 ] [.4875 .36788 -12 -4.5 ] [.4875 .36788 0 4.5 ] [.4875 .4856 -12 -4.5 ] [.4875 .4856 0 4.5 ] [.4875 .60332 -12 -4.5 ] [.4875 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .01472 m .02381 .02097 L s [(-10)] .02381 .00222 0 1 Mshowa .2619 .01472 m .2619 .02097 L s [(-5)] .2619 .00222 0 1 Mshowa .7381 .01472 m .7381 .02097 L s [(5)] .7381 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(10)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .21429 .01472 m .21429 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .40476 .01472 m .40476 .01847 L s .45238 .01472 m .45238 .01847 L s .54762 .01472 m .54762 .01847 L s .59524 .01472 m .59524 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .78571 .01472 m .78571 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .5 .13244 m .50625 .13244 L s [(10)] .4875 .13244 1 0 Mshowa .5 .25016 m .50625 .25016 L s [(20)] .4875 .25016 1 0 Mshowa .5 .36788 m .50625 .36788 L s [(30)] .4875 .36788 1 0 Mshowa .5 .4856 m .50625 .4856 L s [(40)] .4875 .4856 1 0 Mshowa .5 .60332 m .50625 .60332 L s [(50)] .4875 .60332 1 0 Mshowa .125 Mabswid .5 .03826 m .50375 .03826 L s .5 .0618 m .50375 .0618 L s .5 .08535 m .50375 .08535 L s .5 .10889 m .50375 .10889 L s .5 .15598 m .50375 .15598 L s .5 .17952 m .50375 .17952 L s .5 .20307 m .50375 .20307 L s .5 .22661 m .50375 .22661 L s .5 .2737 m .50375 .2737 L s .5 .29725 m .50375 .29725 L s .5 .32079 m .50375 .32079 L s .5 .34433 m .50375 .34433 L s .5 .39142 m .50375 .39142 L s .5 .41497 m .50375 .41497 L s .5 .43851 m .50375 .43851 L s .5 .46205 m .50375 .46205 L s .5 .50914 m .50375 .50914 L s .5 .53269 m .50375 .53269 L s .5 .55623 m .50375 .55623 L s .5 .57978 m .50375 .57978 L s .25 Mabswid .5 0 m .5 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .06244 .10635 L .10458 .19746 L .14415 .27463 L .18221 .34117 L .22272 .40374 L .26171 .45593 L .30316 .50274 L .34309 .53941 L .3815 .56687 L .40095 .57785 L .42237 .58767 L .44268 .59479 L .45178 .59728 L .46172 .59952 L .4671 .60051 L .4721 .6013 L .47727 .60198 L .48196 .60248 L .48658 .60285 L .4887 .60299 L .49093 .60311 L .49332 .6032 L .49438 .60324 L .49552 .60327 L .49675 .60329 L .49789 .60331 L .49859 .60331 L .49925 .60332 L .50049 .60332 L .50163 .60331 L .50286 .6033 L .50401 .60328 L .50508 .60325 L .50754 .60317 L .51014 .60305 L .51268 .6029 L .51504 .60273 L .5204 .60224 L .5293 .60109 L .53882 .59941 L .54906 .59707 L .56016 .59392 L .58032 .58658 L .60019 .57726 L .62123 .56517 L .65912 .5376 L .69946 .50005 L .73829 .45593 L .77956 .40045 L Mistroke .81932 .33864 L .85757 .27144 L .89827 .19159 L .93745 .1066 L .97619 .01472 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgo8F>o000=Hkl3 0004Hkl2000kHkl3001kHkl3000kHkl30004Hkl2000:Hkl000iS_`03001S_f>o009S_`04001S_f>o 000iHkl01@00HkmS_f>o0000NF>o00D006>oHkmS_`0003]S_`03001S_f>o009S_`04001S_f>o0009 Hkl000IS_`D000=S_`03001S_f>o009S_`04001S_f>o000cHkl50005Hkl00`00HkmS_`1kHkl00`00 HkmS_`0iHkl00`00HkmS_`02Hkl01000HkmS_`002F>o000>Hkl00`00HkmS_`02Hkl01000HkmS_`00 ?F>o00<006>oHkl0Nf>o00<006>oHkl0>F>o00<006>oHkl00V>o00@006>oHkl000US_`003V>o00<0 06>oHkl00V>o00@006>oHkl003US_`@007YS_`@003aS_`03001S_f>o009S_`04001S_f>o0009Hkl0 00eS_`8000AS_`04001S_f>o000iHkl00`00HkmS_`1kHkl00`00HkmS_`0lHkl20004Hkl01000HkmS _`002F>o000>Hkl00`00HkmS_`03Hkl2000jHkl5001iHkl5000kHkl00`00HkmS_`03Hkl2000:Hkl0 091S_`03001S_f>o08eS_`00T6>o00<006>oHkl0SF>o002@Hkl00`00HkmS_`2=Hkl0091S_`03001S _f>o08eS_`0026>oo`004P001f>o000?Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0: Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`09Hkl00`00HkmS_`0:Hkl00`00HkmS_`0: Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0: Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0: Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0;Hkl000mS_`8003mS_`03001S_f>o03eS _`03001S_f>o03iS_`03001S_f>o03eS_`8000eS_`0046>o00<006>oHkl0OF>o00<006>oHkl0OV>o 00<006>oHkl036>o000@Hkl00`00HkmS_`1mHkl00`00HkmS_`1mHkl00`00HkmS_`0=Hkl0015S_`03 001S_f>o07aS_`03001S_f>o07eS_`03001S_f>o00eS_`004F>o00<006>oHkl0O6>o0P00OV>o00<0 06>oHkl03F>o000BHkl00`00HkmS_`1kHkl00`00HkmS_`1lHkl00`00HkmS_`0>Hkl0019S_`03001S _f>o07]S_`03001S_f>o07aS_`03001S_f>o00iS_`004V>o00<006>oHkl0Nf>o00<006>oHkl0Nf>o 00<006>oHkl03f>o000CHkl00`00HkmS_`1jHkl00`00HkmS_`1kHkl00`00HkmS_`0?Hkl001=S_`03 001S_f>o07YS_`03001S_f>o07YS_`03001S_f>o011S_`0056>o00<006>oHkl0NF>o00<006>oHkl0 NV>o00<006>oHkl046>o000DHkl00`00HkmS_`1iHkl2001jHkl00`00HkmS_`0AHkl001AS_`03001S _f>o07US_`03001S_f>o07US_`03001S_f>o015S_`005F>o00<006>oHkl0N6>o00<006>oHkl0NF>o 00<006>oHkl04F>o000EHkl00`00HkmS_`1hHkl00`00HkmS_`1hHkl00`00HkmS_`0BHkl001IS_`03 001S_f>o07MS_`03001S_f>o07QS_`03001S_f>o019S_`005V>o00<006>oHkl0Mf>o00<006>oHkl0 Mf>o00<006>oHkl04f>o000FHkl00`00HkmS_`1gHkl2001hHkl00`00HkmS_`0CHkl001MS_`03001S _f>o07IS_`03001S_f>o07IS_`03001S_f>o01AS_`005f>o00<006>oHkl0MV>o00<006>oHkl0MV>o 00<006>oHkl056>o000HHkl00`00HkmS_`1eHkl00`00HkmS_`1fHkl00`00HkmS_`0DHkl001QS_`03 001S_f>o07ES_`03001S_f>o07ES_`03001S_f>o01ES_`0066>o00<006>oHkl0MF>o00<006>oHkl0 MF>o00<006>oHkl05F>o000IHkl00`00HkmS_`1dHkl00`00HkmS_`1dHkl00`00HkmS_`0FHkl001US _`03001S_f>o07AS_`8007ES_`03001S_f>o01IS_`006V>o00<006>oHkl0Lf>o00<006>oHkl0Lf>o 00<006>oHkl05f>o000JHkl00`00HkmS_`1cHkl00`00HkmS_`1cHkl00`00HkmS_`0GHkl001]S_`03 001S_f>o06AS_`<000AS_`8000ES_`03001S_f>o079S_`03001S_f>o01QS_`006f>o00<006>oHkl0 IF>o00<006>oHkl00V>o00@006>oHkl000AS_`03001S_f>o079S_`03001S_f>o01QS_`0076>o00<0 06>oHkl0I6>o00<006>oHkl00V>o00@006>oHkl000AS_`03001S_f>o079S_`03001S_f>o01QS_`00 76>o00<006>oHkl0I6>o00<006>oHkl00V>o00@006>oHkl000AS_`<0075S_`03001S_f>o01US_`00 7F>o00<006>oHkl0Hf>o00<006>oHkl00V>o00@006>oHkl000AS_`03001S_f>o075S_`03001S_f>o 01US_`007F>o00<006>oHkl0HV>o0P0016>o00@006>oHkl000AS_`03001S_f>o071S_`03001S_f>o 01YS_`007V>o00<006>oHkl0HV>o00<006>oHkl00f>o0P001F>o00<006>oHkl0L6>o00<006>oHkl0 6V>o000NHkl00`00HkmS_`1_Hkl00`00HkmS_`1_Hkl00`00HkmS_`0KHkl001mS_`03001S_f>o06iS _`03001S_f>o06mS_`03001S_f>o01]S_`007f>o00<006>oHkl0KV>o0P00L6>o00<006>oHkl06f>o 000OHkl00`00HkmS_`1^Hkl00`00HkmS_`1^Hkl00`00HkmS_`0LHkl0021S_`03001S_f>o06eS_`03 001S_f>o06iS_`03001S_f>o01aS_`0086>o00<006>oHkl0KF>o00<006>oHkl0KF>o00<006>oHkl0 7F>o000QHkl00`00HkmS_`1/Hkl00`00HkmS_`1]Hkl00`00HkmS_`0MHkl0025S_`03001S_f>o06aS _`03001S_f>o06aS_`03001S_f>o01iS_`008V>o00<006>oHkl0Jf>o00<006>oHkl0K6>o00<006>o Hkl07V>o000RHkl00`00HkmS_`1[Hkl2001]Hkl00`00HkmS_`0NHkl002=S_`03001S_f>o06YS_`03 001S_f>o06]S_`03001S_f>o01mS_`008f>o00<006>oHkl0JV>o00<006>oHkl0Jf>o00<006>oHkl0 7f>o000THkl00`00HkmS_`1YHkl00`00HkmS_`1ZHkl00`00HkmS_`0PHkl002AS_`03001S_f>o06US _`03001S_f>o06YS_`03001S_f>o021S_`009F>o00<006>oHkl0J6>o00<006>oHkl0JF>o00<006>o Hkl08F>o000UHkl00`00HkmS_`1XHkl2001ZHkl00`00HkmS_`0QHkl002IS_`03001S_f>o06MS_`03 001S_f>o06QS_`03001S_f>o029S_`009V>o00<006>oHkl0If>o00<006>oHkl0J6>o00<006>oHkl0 8V>o000WHkl00`00HkmS_`1VHkl00`00HkmS_`1WHkl00`00HkmS_`0SHkl002MS_`03001S_f>o06IS _`03001S_f>o06IS_`03001S_f>o02AS_`00:6>o00<006>oHkl0IF>o00<006>oHkl0IV>o00<006>o Hkl096>o000XHkl00`00HkmS_`1UHkl00`00HkmS_`1UHkl00`00HkmS_`0UHkl002US_`03001S_f>o 06AS_`8006IS_`03001S_f>o02ES_`00:F>o00<006>oHkl0I6>o00<006>oHkl0I6>o00<006>oHkl0 9V>o000ZHkl00`00HkmS_`1SHkl00`00HkmS_`1THkl00`00HkmS_`0VHkl002]S_`03001S_f>o05=S _`D000=S_`8000ES_`03001S_f>o06=S_`03001S_f>o02MS_`00:f>o00<006>oHkl0E6>o00@006>o Hkl0009S_`04001S_f>o0004Hkl00`00HkmS_`1SHkl00`00HkmS_`0WHkl002aS_`03001S_f>o05AS _`03001S_f>o009S_`04001S_f>o0004Hkl00`00HkmS_`1RHkl00`00HkmS_`0XHkl002aS_`03001S _f>o05ES_`05001S_f>oHkl00002Hkl00`00HkmS_`02Hkl3001RHkl00`00HkmS_`0XHkl002eS_`03 001S_f>o055S_`05001S_f>oHkl00002Hkl01000HkmS_`0016>o00<006>oHkl0HF>o00<006>oHkl0 :F>o000]Hkl00`00HkmS_`1AHkl01@00HkmS_f>o00000V>o00@006>oHkl000AS_`03001S_f>o061S _`03001S_f>o02YS_`00;V>o00<006>oHkl0DF>o0`0016>o0P001F>o00<006>oHkl0H6>o00<006>o Hkl0:V>o000^Hkl00`00HkmS_`1OHkl00`00HkmS_`1OHkl00`00HkmS_`0[Hkl002mS_`03001S_f>o 05iS_`03001S_f>o05mS_`03001S_f>o02]S_`00;f>o00<006>oHkl0GV>o0P00Gf>o00<006>oHkl0 ;6>o000`Hkl00`00HkmS_`1MHkl00`00HkmS_`1NHkl00`00HkmS_`0/Hkl0031S_`03001S_f>o05eS _`03001S_f>o05eS_`03001S_f>o02eS_`00o00<006>oHkl0G6>o00<006>oHkl0GF>o00<006>o Hkl0;F>o000aHkl00`00HkmS_`1LHkl00`00HkmS_`1LHkl00`00HkmS_`0^Hkl0039S_`03001S_f>o 05]S_`03001S_f>o05]S_`03001S_f>o02mS_`00o00<006>oHkl0FV>o00<006>oHkl0Ff>o00<0 06>oHkl0;f>o000cHkl00`00HkmS_`1JHkl2001KHkl00`00HkmS_`0`Hkl003AS_`03001S_f>o05US _`03001S_f>o05YS_`03001S_f>o031S_`00=6>o00<006>oHkl0FF>o00<006>oHkl0FF>o00<006>o Hkl0o000eHkl00`00HkmS_`1HHkl00`00HkmS_`1IHkl00`00HkmS_`0aHkl003ES_`03001S_f>o 05QS_`03001S_f>o05QS_`03001S_f>o039S_`00=V>o00<006>oHkl0Ef>o00<006>oHkl0F6>o00<0 06>oHkl0o000fHkl00`00HkmS_`1GHkl2001HHkl00`00HkmS_`0cHkl003MS_`03001S_f>o05IS _`03001S_f>o05IS_`03001S_f>o03AS_`00>6>o00<006>oHkl0EF>o00<006>oHkl0EV>o00<006>o Hkl0=6>o000hHkl00`00HkmS_`1EHkl00`00HkmS_`1EHkl00`00HkmS_`0eHkl003US_`03001S_f>o 05AS_`03001S_f>o05ES_`03001S_f>o03ES_`00>F>o00<006>oHkl0E6>o00<006>oHkl0E6>o00<0 06>oHkl0=V>o000jHkl00`00HkmS_`1CHkl00`00HkmS_`1DHkl00`00HkmS_`0fHkl003YS_`03001S _f>o05=S_`8005AS_`03001S_f>o03MS_`00>f>o00<006>oHkl0DV>o00<006>oHkl0DV>o00<006>o Hkl0>6>o000lHkl00`00HkmS_`1AHkl00`00HkmS_`1BHkl00`00HkmS_`0hHkl003aS_`03001S_f>o 04=S_`<000AS_`8000ES_`03001S_f>o055S_`03001S_f>o03US_`00?F>o00<006>oHkl0@F>o00D0 06>oHkmS_`00009S_`04001S_f>o0004Hkl00`00HkmS_`1@Hkl00`00HkmS_`0jHkl003iS_`03001S _f>o04AS_`04001S_f>o0002Hkl00`00HkmS_`02Hkl00`00HkmS_`1@Hkl00`00HkmS_`0jHkl003iS _`03001S_f>o049S_`8000=S_`04001S_f>o0004Hkl3001?Hkl00`00HkmS_`0kHkl003mS_`03001S _f>o04=S_`04001S_f>o0002Hkl00`00HkmS_`02Hkl00`00HkmS_`1>Hkl00`00HkmS_`0lHkl0041S _`03001S_f>o03iS_`05001S_f>oHkl00002Hkl01000HkmS_`0016>o00<006>oHkl0CV>o00<006>o Hkl0?6>o0010Hkl00`00HkmS_`0oHkl30004Hkl20005Hkl00`00HkmS_`1=Hkl00`00HkmS_`0mHkl0 045S_`03001S_f>o04aS_`03001S_f>o04eS_`03001S_f>o03eS_`00@F>o00<006>oHkl0C6>o00<0 06>oHkl0C6>o00<006>oHkl0?V>o0012Hkl00`00HkmS_`1;Hkl2001o04YS_`03001S_f>o04]S_`03001S_f>o03mS_`00@f>o00<006>oHkl0BV>o00<006>o Hkl0BV>o00<006>oHkl0@6>o0014Hkl00`00HkmS_`19Hkl00`00HkmS_`19Hkl00`00HkmS_`11Hkl0 04ES_`03001S_f>o04QS_`03001S_f>o04US_`03001S_f>o045S_`00AF>o00<006>oHkl0B6>o00<0 06>oHkl0B6>o00<006>oHkl0@V>o0016Hkl00`00HkmS_`17Hkl00`00HkmS_`17Hkl00`00HkmS_`13 Hkl004MS_`03001S_f>o04IS_`8004MS_`03001S_f>o04AS_`00B6>o00<006>oHkl0AF>o00<006>o Hkl0AV>o00<006>oHkl0A6>o0018Hkl00`00HkmS_`15Hkl00`00HkmS_`15Hkl00`00HkmS_`15Hkl0 04US_`03001S_f>o04AS_`03001S_f>o04AS_`03001S_f>o04IS_`00BV>o00<006>oHkl0@f>o00<0 06>oHkl0@f>o00<006>oHkl0Af>o001:Hkl00`00HkmS_`13Hkl00`00HkmS_`13Hkl00`00HkmS_`17 Hkl004]S_`03001S_f>o049S_`8004=S_`03001S_f>o04QS_`00C6>o00<006>oHkl0@F>o00<006>o Hkl0@F>o00<006>oHkl0BF>o001=Hkl00`00HkmS_`10Hkl00`00HkmS_`11Hkl00`00HkmS_`19Hkl0 04eS_`03001S_f>o041S_`03001S_f>o041S_`03001S_f>o04YS_`00CV>o00<006>oHkl0?f>o00<0 06>oHkl0?f>o00<006>oHkl0Bf>o001?Hkl00`00HkmS_`0nHkl00`00HkmS_`0nHkl00`00HkmS_`1< Hkl004mS_`03001S_f>o03iS_`03001S_f>o03iS_`03001S_f>o04aS_`00D6>o00<006>oHkl0?F>o 0P00?V>o00<006>oHkl0CF>o001AHkl00`00HkmS_`0lHkl00`00HkmS_`0lHkl00`00HkmS_`1>Hkl0 059S_`03001S_f>o03]S_`03001S_f>o03]S_`03001S_f>o04mS_`00Df>o00<006>oHkl0;F>o0`00 0f>o0P001F>o00<006>oHkl0>V>o00<006>oHkl0D6>o001DHkl00`00HkmS_`0]Hkl01@00HkmS_f>o 00000V>o00<006>oHkl00V>o00<006>oHkl0>F>o00<006>oHkl0DF>o001EHkl00`00HkmS_`0YHkl5 0002Hkl01000HkmS_`0016>o00<006>oHkl0>6>o00<006>oHkl0DV>o001FHkl00`00HkmS_`0XHkl0 1000HkmS_`000f>o00@006>oHkl000AS_`<003MS_`03001S_f>o05=S_`00Ef>o00<006>oHkl0:6>o 00<006>o00000f>o00@006>oHkl000AS_`03001S_f>o03IS_`03001S_f>o05AS_`00F6>o00<006>o Hkl0:6>o0P000f>o00@006>oHkl000AS_`03001S_f>o03ES_`03001S_f>o05ES_`00FF>o00<006>o Hkl0:6>o00<006>oHkl00V>o0P001F>o00<006>oHkl0=6>o00<006>oHkl0EV>o001JHkl00`00HkmS _`0cHkl00`00HkmS_`0cHkl00`00HkmS_`1GHkl005]S_`8003=S_`03001S_f>o039S_`03001S_f>o 05QS_`00GF>o00<006>oHkl0<6>o0P00o00<006>oHkl0FF>o001NHkl00`00HkmS_`0_Hkl00`00 HkmS_`0`Hkl00`00HkmS_`1JHkl005mS_`03001S_f>o02iS_`03001S_f>o02mS_`03001S_f>o05]S _`00H6>o00<006>oHkl0;F>o00<006>oHkl0;V>o00<006>oHkl0G6>o001QHkl00`00HkmS_`0/Hkl0 0`00HkmS_`0]Hkl00`00HkmS_`1MHkl0069S_`03001S_f>o02]S_`03001S_f>o02aS_`03001S_f>o 05iS_`00Hf>o00<006>oHkl0:V>o00<006>oHkl0:f>o00<006>oHkl0Gf>o001THkl00`00HkmS_`0Y Hkl2000[Hkl00`00HkmS_`1PHkl006ES_`03001S_f>o02QS_`03001S_f>o02QS_`8006=S_`00IV>o 0P00:6>o00<006>oHkl09f>o00<006>oHkl0Hf>o001XHkl00`00HkmS_`0UHkl00`00HkmS_`0VHkl0 0`00HkmS_`1THkl006US_`8002ES_`03001S_f>o02AS_`8006MS_`00Jf>o00<006>oHkl08V>o00<0 06>oHkl08f>o00<006>oHkl0If>o001/Hkl00`00HkmS_`0QHkl2000RHkl2001ZHkl006eS_`80025S _`03001S_f>o021S_`03001S_f>o06YS_`00Kf>o00<006>oHkl07V>o00<006>oHkl07f>o00<006>o Hkl0Jf>o001`Hkl2000NHkl00`00HkmS_`0MHkl2001^Hkl0079S_`8001aS_`03001S_f>o01]S_`80 071S_`00M6>o00<006>oHkl06F>o00<006>oHkl06V>o00<006>oHkl0L6>o001eHkl2000IHkl00`00 HkmS_`0HHkl2001cHkl007MS_`8001MS_`8001IS_`<007ES_`00NF>o0P005F>o00<006>oHkl04V>o 0`00N6>o001kHkl3000BHkl00`00HkmS_`0?Hkl3001kHkl007iS_`<00004Hkl000000004Hkl20005 Hkl00`00HkmS_`0=Hkl2001nHkl0085S_`D0009S_`04001S_f>o0004Hkl00`00HkmS_`0:Hkl30020 Hkl008ES_`D00003Hkl006>o00=S_`03001S_f>o00AS_`H008=S_`00QF>o00D006>oHkl006>o00d0 08US_`00PF>o10000f>o00@006>oHkl000AS_`03001S_f>o08eS_`00PF>o00<006>oHkl016>o00@0 06>oHkl000AS_`03001S_f>o08eS_`00PF>o1@000f>o0P001F>o00<006>oHkl0SF>o002@Hkl00`00 HkmS_`2=Hkl00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-11.1809, -3.87349, \ 0.0772357, 0.312425}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["How big is the second derivative term?", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\((\[Mu]\ D[ans, {y, 2}])\) /. {b -> 10, \ \[Mu] -> .01}\)], "Input"], Cell[BoxData[ \(TraditionalForm\`dpdx\)], "Output"] }, Open ]] }, Open ]], Cell["\<\ We see that while the velocity is \"50\" in some units, the second \ derivative terms is \"-.01\" in corresponding units. How could we guess \ this?\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Now do a nondimensional equation", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(eq = \(-dpdx\)\ + \ \[Mu]\ D[u[y], {y, 2}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Mu]", " ", RowBox[{ SuperscriptBox["u", "\[Prime]\[Prime]", MultilineFunction->None], "(", "y", ")"}]}], "-", "dpdx"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ueq1", "=", RowBox[{"eq", "/.", RowBox[{"{", RowBox[{ RowBox[{ SuperscriptBox["u", TagBox[\((a1_)\), Derivative], MultilineFunction->None], "[", "y", "]"}], "\[RuleDelayed]", " ", \(\[PartialD]\_{s, a1}u\&~[s]\ B\^\(-a1\)\ U\)}], " ", "}"}]}]}]], "Input", AspectRatioFixed->True], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"U", " ", "\[Mu]", " ", RowBox[{ SuperscriptBox[\(u\&~\), "\[Prime]\[Prime]", MultilineFunction->None], "(", "s", ")"}]}], \(B\^2\)], "-", "dpdx"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ We now choose a pressure driven velocity. We can then divide out \ the pressure gradient.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ueq2 = ueq1 /. U -> \ \(-dpdx\)\ B^2/\[Mu]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{ SuperscriptBox[\(u\&~\), "\[Prime]\[Prime]", MultilineFunction->None], "(", "s", ")"}]}], " ", "dpdx"}], "-", "dpdx"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ueq3 = Cancel[ueq2/dpdx]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"-", RowBox[{ SuperscriptBox[\(u\&~\), "\[Prime]\[Prime]", MultilineFunction->None], "(", "s", ")"}]}], "-", "1"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(DSolve[{ueq3 == 0, u\&~[\(-1\)] == 0, u\&~[1] == 0}, u\&~[s], s]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{\(u\&~\)(s) \[Rule] 1\/2 - s\^2\/2}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ans2 = u\&~[s] /. %[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`1\/2 - s\^2\/2\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(Plot[ans2, {s, \(-1\), 1}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.476191 0.0147151 1.17721 [ [.02381 .00222 -6 -9 ] [.02381 .00222 6 0 ] [.2619 .00222 -12 -9 ] [.2619 .00222 12 0 ] [.7381 .00222 -9 -9 ] [.7381 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.4875 .13244 -18 -4.5 ] [.4875 .13244 0 4.5 ] [.4875 .25016 -18 -4.5 ] [.4875 .25016 0 4.5 ] [.4875 .36788 -18 -4.5 ] [.4875 .36788 0 4.5 ] [.4875 .4856 -18 -4.5 ] [.4875 .4856 0 4.5 ] [.4875 .60332 -18 -4.5 ] [.4875 .60332 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .01472 m .02381 .02097 L s [(-1)] .02381 .00222 0 1 Mshowa .2619 .01472 m .2619 .02097 L s [(-0.5)] .2619 .00222 0 1 Mshowa .7381 .01472 m .7381 .02097 L s [(0.5)] .7381 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(1)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .21429 .01472 m .21429 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .40476 .01472 m .40476 .01847 L s .45238 .01472 m .45238 .01847 L s .54762 .01472 m .54762 .01847 L s .59524 .01472 m .59524 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .78571 .01472 m .78571 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .5 .13244 m .50625 .13244 L s [(0.1)] .4875 .13244 1 0 Mshowa .5 .25016 m .50625 .25016 L s [(0.2)] .4875 .25016 1 0 Mshowa .5 .36788 m .50625 .36788 L s [(0.3)] .4875 .36788 1 0 Mshowa .5 .4856 m .50625 .4856 L s [(0.4)] .4875 .4856 1 0 Mshowa .5 .60332 m .50625 .60332 L s [(0.5)] .4875 .60332 1 0 Mshowa .125 Mabswid .5 .03826 m .50375 .03826 L s .5 .0618 m .50375 .0618 L s .5 .08535 m .50375 .08535 L s .5 .10889 m .50375 .10889 L s .5 .15598 m .50375 .15598 L s .5 .17952 m .50375 .17952 L s .5 .20307 m .50375 .20307 L s .5 .22661 m .50375 .22661 L s .5 .2737 m .50375 .2737 L s .5 .29725 m .50375 .29725 L s .5 .32079 m .50375 .32079 L s .5 .34433 m .50375 .34433 L s .5 .39142 m .50375 .39142 L s .5 .41497 m .50375 .41497 L s .5 .43851 m .50375 .43851 L s .5 .46205 m .50375 .46205 L s .5 .50914 m .50375 .50914 L s .5 .53269 m .50375 .53269 L s .5 .55623 m .50375 .55623 L s .5 .57978 m .50375 .57978 L s .25 Mabswid .5 0 m .5 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .06244 .10635 L .10458 .19746 L .14415 .27463 L .18221 .34117 L .22272 .40374 L .26171 .45593 L .30316 .50274 L .34309 .53941 L .3815 .56687 L .40095 .57785 L .42237 .58767 L .44268 .59479 L .45178 .59728 L .46172 .59952 L .4671 .60051 L .4721 .6013 L .47727 .60198 L .48196 .60248 L .48658 .60285 L .4887 .60299 L .49093 .60311 L .49332 .6032 L .49438 .60324 L .49552 .60327 L .49675 .60329 L .49789 .60331 L .49859 .60331 L .49925 .60332 L .50049 .60332 L .50163 .60331 L .50286 .6033 L .50401 .60328 L .50508 .60325 L .50754 .60317 L .51014 .60305 L .51268 .6029 L .51504 .60273 L .5204 .60224 L .5293 .60109 L .53882 .59941 L .54906 .59707 L .56016 .59392 L .58032 .58658 L .60019 .57726 L .62123 .56517 L .65912 .5376 L .69946 .50005 L .73829 .45593 L .77956 .40045 L Mistroke .81932 .33864 L .85757 .27144 L .89827 .19159 L .93745 .1066 L .97619 .01472 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgo8F>o000?Hkl3 000hHkl20004Hkl00`00HkmS_`02Hkl3001aHkl20004Hkl00`00HkmS_`02Hkl3000gHkl3000?Hkl0 011S_`03001S_f>o03IS_`04001S_f>o0007Hkl01@00HkmS_f>o0000Kf>o00@006>oHkl000MS_`05 001S_f>oHkl0000gHkl00`00HkmS_`0>Hkl000QS_`D000=S_`03001S_f>o02mS_`D0009S_`04001S _f>o000;Hkl00`00HkmS_`1]Hkl01000HkmS_`002f>o00<006>oHkl0=F>o00<006>oHkl03V>o000@ Hkl00`00HkmS_`0fHkl01000HkmS_`002f>o00<006>oHkl0KF>o00@006>oHkl000]S_`03001S_f>o 03ES_`03001S_f>o00iS_`0046>o00<006>oHkl0=V>o00@006>oHkl000MS_`@0071S_`04001S_f>o 0007Hkl4000hHkl00`00HkmS_`0>Hkl000mS_`8003QS_`04001S_f>o0007Hkl00`00HkmS_`1aHkl0 1000HkmS_`001f>o00<006>oHkl0>6>o0P0046>o000@Hkl00`00HkmS_`0gHkl20008Hkl5001`Hkl2 0008Hkl5000gHkl00`00HkmS_`0>Hkl008mS_`03001S_f>o08iS_`00Sf>o00<006>oHkl0SV>o002? Hkl00`00HkmS_`2>Hkl008mS_`03001S_f>o08iS_`001f>oo`004P0026>o000>Hkl00`00HkmS_`09 Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0: Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0: Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0: Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`0:Hkl00`00HkmS_`09Hkl00`00HkmS_`0= Hkl000iS_`8003iS_`03001S_f>o03iS_`03001S_f>o03iS_`03001S_f>o03aS_`8000mS_`003f>o 00<006>oHkl0OF>o00<006>oHkl0OF>o00<006>oHkl03V>o000?Hkl00`00HkmS_`1mHkl00`00HkmS _`1mHkl00`00HkmS_`0>Hkl0011S_`03001S_f>o07aS_`03001S_f>o07aS_`03001S_f>o00mS_`00 46>o00<006>oHkl0O6>o0P00OF>o00<006>oHkl03f>o000AHkl00`00HkmS_`1kHkl00`00HkmS_`1k Hkl00`00HkmS_`0@Hkl0015S_`03001S_f>o07]S_`03001S_f>o07]S_`03001S_f>o011S_`004F>o 00<006>oHkl0Nf>o00<006>oHkl0Nf>o00<006>oHkl046>o000BHkl00`00HkmS_`1jHkl00`00HkmS _`1jHkl00`00HkmS_`0AHkl0019S_`03001S_f>o07YS_`03001S_f>o07YS_`03001S_f>o015S_`00 4f>o00<006>oHkl0NF>o00<006>oHkl0NF>o00<006>oHkl04V>o000CHkl00`00HkmS_`1iHkl2001j Hkl00`00HkmS_`0BHkl001=S_`03001S_f>o07US_`03001S_f>o07US_`03001S_f>o019S_`0056>o 00<006>oHkl0N6>o00<006>oHkl0N6>o00<006>oHkl04f>o000DHkl00`00HkmS_`1hHkl00`00HkmS _`1hHkl00`00HkmS_`0CHkl001ES_`03001S_f>o07MS_`03001S_f>o07MS_`03001S_f>o01AS_`00 5F>o00<006>oHkl0Mf>o00<006>oHkl0Mf>o00<006>oHkl056>o000EHkl00`00HkmS_`1gHkl2001h Hkl00`00HkmS_`0DHkl001IS_`03001S_f>o07IS_`03001S_f>o07IS_`03001S_f>o01ES_`005V>o 00<006>oHkl0MV>o00<006>oHkl0MV>o00<006>oHkl05F>o000GHkl00`00HkmS_`1eHkl00`00HkmS _`1eHkl00`00HkmS_`0FHkl001MS_`03001S_f>o07ES_`03001S_f>o07ES_`03001S_f>o01IS_`00 5f>o00<006>oHkl0MF>o00<006>oHkl0MF>o00<006>oHkl05V>o000HHkl00`00HkmS_`1dHkl00`00 HkmS_`1dHkl00`00HkmS_`0GHkl001QS_`03001S_f>o07AS_`8007ES_`03001S_f>o01MS_`006F>o 00<006>oHkl0Lf>o00<006>oHkl0Lf>o00<006>oHkl066>o000IHkl00`00HkmS_`1cHkl00`00HkmS _`1cHkl00`00HkmS_`0HHkl001YS_`03001S_f>o05mS_`8000AS_`03001S_f>o009S_`<000ES_`03 001S_f>o079S_`03001S_f>o01US_`006V>o00<006>oHkl0GV>o00@006>oHkl000US_`03001S_f>o 00AS_`03001S_f>o079S_`03001S_f>o01US_`006V>o00<006>oHkl0GV>o00@006>oHkl000US_`03 001S_f>o00AS_`03001S_f>o075S_`03001S_f>o01YS_`006f>o00<006>oHkl0GF>o00@006>oHkl0 00US_`03001S_f>o00AS_`<0075S_`03001S_f>o01YS_`006f>o00<006>oHkl0GF>o00@006>oHkl0 00US_`03001S_f>o00AS_`03001S_f>o071S_`03001S_f>o01]S_`0076>o00<006>oHkl0G6>o00@0 06>oHkl000QS_`8000IS_`03001S_f>o071S_`03001S_f>o01]S_`0076>o00<006>oHkl0GF>o0P00 2V>o00<006>oHkl016>o00<006>oHkl0Kf>o00<006>oHkl076>o000MHkl00`00HkmS_`1_Hkl00`00 HkmS_`1_Hkl00`00HkmS_`0LHkl001eS_`03001S_f>o06mS_`03001S_f>o06iS_`03001S_f>o01eS _`007V>o00<006>oHkl0KV>o0P00Kf>o00<006>oHkl07F>o000NHkl00`00HkmS_`1^Hkl00`00HkmS _`1^Hkl00`00HkmS_`0MHkl001iS_`03001S_f>o06iS_`03001S_f>o06eS_`03001S_f>o01iS_`00 7f>o00<006>oHkl0KF>o00<006>oHkl0KF>o00<006>oHkl07V>o000OHkl00`00HkmS_`1]Hkl00`00 HkmS_`1/Hkl00`00HkmS_`0OHkl0021S_`03001S_f>o06aS_`03001S_f>o06aS_`03001S_f>o01mS _`0086>o00<006>oHkl0K6>o00<006>oHkl0Jf>o00<006>oHkl086>o000QHkl00`00HkmS_`1[Hkl2 001/Hkl00`00HkmS_`0PHkl0025S_`03001S_f>o06]S_`03001S_f>o06YS_`03001S_f>o025S_`00 8F>o00<006>oHkl0Jf>o00<006>oHkl0JV>o00<006>oHkl08F>o000RHkl00`00HkmS_`1ZHkl00`00 HkmS_`1YHkl00`00HkmS_`0RHkl0029S_`03001S_f>o06YS_`03001S_f>o06US_`03001S_f>o029S _`008f>o00<006>oHkl0JF>o00<006>oHkl0J6>o00<006>oHkl08f>o000SHkl00`00HkmS_`1YHkl2 001YHkl00`00HkmS_`0SHkl002AS_`03001S_f>o06QS_`03001S_f>o06MS_`03001S_f>o02AS_`00 96>o00<006>oHkl0J6>o00<006>oHkl0If>o00<006>oHkl096>o000UHkl00`00HkmS_`1WHkl00`00 HkmS_`1VHkl00`00HkmS_`0UHkl002ES_`03001S_f>o06MS_`03001S_f>o06IS_`03001S_f>o02ES _`009V>o00<006>oHkl0IV>o00<006>oHkl0IF>o00<006>oHkl09V>o000VHkl00`00HkmS_`1VHkl0 0`00HkmS_`1UHkl00`00HkmS_`0VHkl002MS_`03001S_f>o06ES_`8006ES_`03001S_f>o02MS_`00 9f>o00<006>oHkl0IF>o00<006>oHkl0I6>o00<006>oHkl09f>o000XHkl00`00HkmS_`1THkl00`00 HkmS_`1SHkl00`00HkmS_`0XHkl002US_`03001S_f>o051S_`8000AS_`04001S_f>oHkl50004Hkl0 0`00HkmS_`1SHkl00`00HkmS_`0XHkl002US_`03001S_f>o04mS_`04001S_f>o0008Hkl01000HkmS _`0016>o00<006>oHkl0HV>o00<006>oHkl0:F>o000ZHkl00`00HkmS_`1>Hkl01000HkmS_`002F>o 00<006>oHkl016>o00<006>oHkl0HV>o00<006>oHkl0:F>o000ZHkl00`00HkmS_`1>Hkl01000HkmS _`002V>o00<006>oHkl00f>o0`00HF>o00<006>oHkl0:V>o000[Hkl00`00HkmS_`1=Hkl01000HkmS _`001f>o00D006>oHkmS_`0000AS_`03001S_f>o065S_`03001S_f>o02YS_`00:f>o00<006>oHkl0 CF>o00@006>oHkl000MS_`05001S_f>oHkl00004Hkl00`00HkmS_`1PHkl00`00HkmS_`0[Hkl002aS _`03001S_f>o04eS_`8000US_`<000ES_`03001S_f>o061S_`03001S_f>o02]S_`00;6>o00<006>o Hkl0H6>o00<006>oHkl0Gf>o00<006>oHkl0;6>o000]Hkl00`00HkmS_`1OHkl00`00HkmS_`1OHkl0 0`00HkmS_`0/Hkl002eS_`03001S_f>o05mS_`8005mS_`03001S_f>o02eS_`00;V>o00<006>oHkl0 GV>o00<006>oHkl0GV>o00<006>oHkl0;F>o000^Hkl00`00HkmS_`1NHkl00`00HkmS_`1MHkl00`00 HkmS_`0^Hkl002mS_`03001S_f>o05eS_`03001S_f>o05eS_`03001S_f>o02iS_`00<6>o00<006>o Hkl0G6>o00<006>oHkl0G6>o00<006>oHkl0;f>o000`Hkl00`00HkmS_`1LHkl00`00HkmS_`1KHkl0 0`00HkmS_`0`Hkl0035S_`03001S_f>o05]S_`03001S_f>o05]S_`03001S_f>o031S_`00o00<0 06>oHkl0Ff>o0P00Ff>o00<006>oHkl0o000bHkl00`00HkmS_`1JHkl00`00HkmS_`1JHkl00`00 HkmS_`0aHkl003=S_`03001S_f>o05US_`03001S_f>o05US_`03001S_f>o039S_`00o00<006>o Hkl0FF>o00<006>oHkl0FF>o00<006>oHkl0o000dHkl00`00HkmS_`1HHkl00`00HkmS_`1HHkl0 0`00HkmS_`0cHkl003AS_`03001S_f>o05QS_`03001S_f>o05QS_`03001S_f>o03=S_`00=F>o00<0 06>oHkl0Ef>o0P00F6>o00<006>oHkl0=6>o000fHkl00`00HkmS_`1FHkl00`00HkmS_`1FHkl00`00 HkmS_`0eHkl003IS_`03001S_f>o05IS_`03001S_f>o05IS_`03001S_f>o03ES_`00=f>o00<006>o Hkl0EF>o00<006>oHkl0EF>o00<006>oHkl0=V>o000gHkl00`00HkmS_`1EHkl00`00HkmS_`1EHkl0 0`00HkmS_`0fHkl003QS_`03001S_f>o05AS_`03001S_f>o05AS_`03001S_f>o03MS_`00>F>o00<0 06>oHkl0Df>o00<006>oHkl0E6>o00<006>oHkl0=f>o000iHkl00`00HkmS_`1CHkl2001DHkl00`00 HkmS_`0hHkl003YS_`03001S_f>o059S_`03001S_f>o059S_`03001S_f>o03US_`00>f>o00<006>o Hkl0DF>o00<006>oHkl0DV>o00<006>oHkl0>F>o000kHkl00`00HkmS_`0nHkl20004Hkl00`00HkmS _`02Hkl30005Hkl00`00HkmS_`1AHkl00`00HkmS_`0jHkl003aS_`03001S_f>o03aS_`04001S_f>o 0007Hkl01@00HkmS_f>o000016>o00<006>oHkl0D6>o00<006>oHkl0>f>o000mHkl00`00HkmS_`0k Hkl01000HkmS_`002f>o00<006>oHkl00V>o00<006>oHkl0D6>o00<006>oHkl0>f>o000mHkl00`00 HkmS_`0kHkl01000HkmS_`002F>o0P001F>o0`00Cf>o00<006>oHkl0?6>o000nHkl00`00HkmS_`0j Hkl01000HkmS_`002f>o00<006>oHkl00V>o00<006>oHkl0CV>o00<006>oHkl0?F>o000oHkl00`00 HkmS_`0iHkl01000HkmS_`001f>o00D006>oHkmS_`0000AS_`03001S_f>o04iS_`03001S_f>o03eS _`00?f>o00<006>oHkl0>V>o0P002F>o0`001F>o00<006>oHkl0CF>o00<006>oHkl0?V>o0010Hkl0 0`00HkmS_`1o04aS_`03001S_f>o 04aS_`03001S_f>o03mS_`00@F>o00<006>oHkl0Bf>o0P00C6>o00<006>oHkl0@6>o0012Hkl00`00 HkmS_`1:Hkl00`00HkmS_`1;Hkl00`00HkmS_`10Hkl0049S_`03001S_f>o04YS_`03001S_f>o04YS _`03001S_f>o045S_`00@f>o00<006>oHkl0BF>o00<006>oHkl0BF>o00<006>oHkl0@V>o0014Hkl0 0`00HkmS_`18Hkl00`00HkmS_`19Hkl00`00HkmS_`12Hkl004AS_`03001S_f>o04QS_`03001S_f>o 04QS_`03001S_f>o04=S_`00AF>o00<006>oHkl0Af>o00<006>oHkl0Af>o00<006>oHkl0A6>o0016 Hkl00`00HkmS_`16Hkl20017Hkl00`00HkmS_`15Hkl004MS_`03001S_f>o04ES_`03001S_f>o04IS _`03001S_f>o04ES_`00Af>o00<006>oHkl0AF>o00<006>oHkl0AF>o00<006>oHkl0AV>o0018Hkl0 0`00HkmS_`14Hkl00`00HkmS_`14Hkl00`00HkmS_`17Hkl004US_`03001S_f>o04=S_`03001S_f>o 04=S_`03001S_f>o04QS_`00BF>o00<006>oHkl0@f>o00<006>oHkl0@f>o00<006>oHkl0B6>o001: Hkl00`00HkmS_`12Hkl20013Hkl00`00HkmS_`19Hkl004]S_`03001S_f>o045S_`03001S_f>o045S _`03001S_f>o04YS_`00C6>o00<006>oHkl0@6>o00<006>oHkl0@F>o00<006>oHkl0BV>o001o03mS_`03001S_f>o 03mS_`03001S_f>o04aS_`00CV>o00<006>oHkl0?V>o00<006>oHkl0?V>o00<006>oHkl0CF>o001> Hkl00`00HkmS_`0nHkl00`00HkmS_`0nHkl00`00HkmS_`1=Hkl004mS_`03001S_f>o03eS_`8003iS _`03001S_f>o04iS_`00D6>o00<006>oHkl0?6>o00<006>oHkl0?6>o00<006>oHkl0Cf>o001AHkl0 0`00HkmS_`0kHkl00`00HkmS_`0kHkl00`00HkmS_`1@Hkl0059S_`03001S_f>o02MS_`8000AS_`03 001S_f>o00=S_`<000AS_`03001S_f>o03YS_`03001S_f>o055S_`00Df>o00<006>oHkl09F>o00@0 06>oHkl000YS_`03001S_f>o00=S_`03001S_f>o03US_`03001S_f>o059S_`00E6>o00<006>oHkl0 96>o00@006>oHkl000MS_`D000AS_`03001S_f>o03QS_`03001S_f>o05=S_`00E6>o00<006>oHkl0 96>o00@006>oHkl000MS_`04001S_f>o0005Hkl3000gHkl00`00HkmS_`1DHkl005ES_`03001S_f>o 02=S_`04001S_f>o0008Hkl00`00Hkl00005Hkl00`00HkmS_`0fHkl00`00HkmS_`1EHkl005IS_`03 001S_f>o029S_`04001S_f>o0009Hkl20005Hkl00`00HkmS_`0eHkl00`00HkmS_`1FHkl005MS_`03 001S_f>o029S_`8000]S_`03001S_f>o00=S_`03001S_f>o03AS_`03001S_f>o05MS_`00F6>o00<0 06>oHkl0=6>o00<006>oHkl0o0P00FV>o001IHkl2000dHkl00`00HkmS_`0aHkl00`00HkmS_`1J Hkl005]S_`03001S_f>o035S_`80035S_`03001S_f>o05]S_`00G6>o00<006>oHkl0<6>o00<006>o Hkl0;f>o00<006>oHkl0G6>o001MHkl00`00HkmS_`0_Hkl00`00HkmS_`0^Hkl00`00HkmS_`1MHkl0 05iS_`03001S_f>o02iS_`03001S_f>o02eS_`03001S_f>o05iS_`00Gf>o00<006>oHkl0;F>o00<0 06>oHkl0;6>o00<006>oHkl0Gf>o001PHkl00`00HkmS_`0/Hkl00`00HkmS_`0[Hkl00`00HkmS_`1P Hkl0065S_`03001S_f>o02]S_`03001S_f>o02YS_`03001S_f>o065S_`00HV>o00<006>oHkl0:V>o 0P00:V>o00<006>oHkl0HV>o001SHkl00`00HkmS_`0YHkl00`00HkmS_`0WHkl2001UHkl006AS_`80 02US_`03001S_f>o02IS_`03001S_f>o06ES_`00IV>o00<006>oHkl09V>o00<006>oHkl09F>o00<0 06>oHkl0IV>o001WHkl2000VHkl00`00HkmS_`0THkl00`00HkmS_`1WHkl006US_`8002AS_`03001S _f>o029S_`8006YS_`00Jf>o00<006>oHkl08F>o0P008V>o00<006>oHkl0JV>o001/Hkl2000QHkl0 0`00HkmS_`0PHkl00`00HkmS_`1[Hkl006iS_`03001S_f>o01iS_`03001S_f>o01mS_`03001S_f>o 06aS_`00Kf>o0P007V>o00<006>oHkl07F>o0P00Kf>o001aHkl2000LHkl00`00HkmS_`0KHkl2001a Hkl007=S_`03001S_f>o01US_`03001S_f>o01US_`8007=S_`00M6>o0P006F>o00<006>oHkl05f>o 0P00MF>o001fHkl2000GHkl2000EHkl3001gHkl007QS_`8001ES_`03001S_f>o019S_`8007YS_`00 NV>o0P004f>o00<006>oHkl03f>o0`00O6>o001lHkl30003Hkl00`00HkmS_`02Hkl30005Hkl00`00 HkmS_`0o00L00003Hkl006>o009S_`03001S_f>o009S_`03001S _f>o00US_`<0089S_`00Nf>o00@006>oHkl000IS_`@00003Hkl006>o00=S_`03001S_f>o00AS_`D0 08ES_`00Nf>o00@006>oHkl000YS_`d008YS_`00Nf>o00@006>oHkl000MS_`@000ES_`03001S_f>o 08iS_`00Nf>o00@006>oHkl000MS_`03001S_f>o00IS_`03001S_f>o08iS_`00O6>o0P0026>o1@00 16>o00<006>oHkl0SV>o002?Hkl00`00HkmS_`2>Hkl00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.10833, -0.0387349, \ 0.00772357, 0.00312425}}] }, Open ]], Cell[CellGroupData[{ Cell["How big is the second derivative term?", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(D[ans2, {s, 2}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-1\)\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ We see that for the nondimensionalized equation the (maximum) \ velocity is 0.5, the pressure gradient is -1. Both of these have an order of \ magnitude of unity. From \"ueq2\" above, the choice of the nondimensionalization of the pressure \ gradient tell us how changing the dimensional pressure gradient will change \ the dimensional velocity. These are related linearly (with a minus sign on \ dpdx). We did not have to solve anything to figure this out. In case you are interested, the average velocity is 1/3. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ans2, {s, \(-1\), 1}]/Integrate[1, {s, \(-1\), 1}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`1\/3\)], "Output"] }, Open ]] }, Open ]] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"4.0 for Macintosh", ScreenRectangle->{{0, 1024}, {0, 748}}, CellGrouping->Manual, WindowSize->{520, 677}, WindowMargins->{{Automatic, 66}, {Automatic, 12}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, CellLabelAutoDelete->True, StyleDefinitions -> Notebook[{ Cell[CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell["\<\ Modify the definitions below to change the default appearance of \ all cells in a given style. Make modifications to any definition using commands in the Format menu.\ \>", "Text"], Cell[CellGroupData[{ Cell["Style Environment Names", "Section"], Cell[StyleData[All, "Working"], ScriptMinSize->9], Cell[StyleData[All, "Printout"], PageWidth->PaperWidth, ShowCellLabel->False, ImageSize->{200, 200}, PrivateFontOptions->{"FontType"->"Outline"}] }, Closed]], Cell[CellGroupData[{ Cell["Notebook Options", "Section"], Cell["\<\ The options defined for the style below will be used at the \ Notebook level.\ \>", "Text"], Cell[StyleData["Notebook"], PageHeaders->{{Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"], None, Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"]}, {Cell[ TextData[ { ValueBox[ "FileName"]}], "Header"], None, Cell[ TextData[ { CounterBox[ "Page"]}], "PageNumber"]}}, PageHeaderLines->{True, True}, PrintingOptions->{"FirstPageHeader"->False, "FacingPages"->True}, CellLabelAutoDelete->False, CellFrameLabelMargins->6, StyleMenuListing->None] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Headings", "Section"], Cell[CellGroupData[{ Cell[StyleData["Title"], CellFrame->{{0, 0}, {0, 0.25}}, CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"TitleGrouping", 0}, PageBreakBelow->False, CellFrameMargins->9, LineSpacing->{0.95, 0}, CounterIncrements->"Title", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->36, Background->RGBColor[0.750011, 1, 0.750011]], Cell[StyleData["Title", "Printout"], CellMargins->{{18, 30}, {4, 0}}, CellFrameMargins->4, FontSize->30] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Subtitle"], CellMargins->{{18, 30}, {0, 10}}, CellGroupingRules->{"TitleGrouping", 10}, PageBreakBelow->False, LineSpacing->{1, 0}, CounterIncrements->"Subtitle", CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->24, FontSlant->"Italic", Background->RGBColor[0.8, 0.920012, 0.920012]], Cell[StyleData["Subtitle", "Printout"], CellMargins->{{18, 30}, {0, 10}}, FontSize->18] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["SectionFirst"], CellFrame->{{0, 0}, {0, 3}}, CellMargins->{{18, 30}, {4, 30}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CellFrameMargins->3, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontSize->18, FontWeight->"Bold"], Cell[StyleData["SectionFirst", "Printout"], FontSize->14] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Section"], CellMargins->{{18, 30}, {4, 30}}, CellGroupingRules->{"SectionGrouping", 40}, PageBreakBelow->False, CounterIncrements->"Section", CounterAssignments->{{"Subsection", 0}, {"Subsubsection", 0}}, FontSize->18, FontWeight->"Bold", Background->RGBColor[0.920012, 0.870024, 0.770016]], Cell[StyleData["Section", "Printout"], FontSize->14] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Subsection"], CellDingbat->"\[FilledSquare]", CellMargins->{{18, 30}, {4, 20}}, CellGroupingRules->{"SectionGrouping", 50}, PageBreakBelow->False, CounterIncrements->"Subsection", CounterAssignments->{{"Subsubsection", 0}}, FontSize->14, FontWeight->"Bold", Background->RGBColor[0.970001, 0.890013, 0.850004]], Cell[StyleData["Subsection", "Printout"], FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Subsubsection"], CellDingbat->"\[FilledSmallSquare]", CellMargins->{{18, 30}, {4, 12}}, CellGroupingRules->{"SectionGrouping", 60}, PageBreakBelow->False, CounterIncrements->"Subsubsection", FontSize->12, FontWeight->"Bold", Background->RGBColor[0.870008, 1, 0.960006]], Cell[StyleData["Subsubsection", "Printout"], FontSize->10] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[CellGroupData[{ Cell[StyleData["Text"], CellMargins->{{18, 10}, {Inherited, 6}}, TextJustification->1, LineSpacing->{1, 2}, CounterIncrements->"Text", Background->RGBColor[0.900008, 1, 0.940002]], Cell[StyleData["Text", "Printout"], CellMargins->{{18, 30}, {Inherited, 4}}, LineSpacing->{1, 3}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Caption"], CellMargins->{{55, 50}, {5, 5}}, PageBreakAbove->False, FontSize->10], Cell[StyleData["Caption", "Printout"], CellMargins->{{55, 55}, {5, 2}}, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell["\<\ The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["Input"], CellMargins->{{55, 10}, {5, 8}}, Evaluatable->True, CellGroupingRules->"InputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultInputFormatType, AutoItalicWords->{}, FormatType->InputForm, ShowStringCharacters->True, NumberMarks->True, CounterIncrements->"Input", FontSize->12, FontWeight->"Bold"], Cell[StyleData["Input", "Printout"], CellMargins->{{55, 55}, {0, 10}}, ShowCellLabel->False, FontSize->9.5] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Output"], CellMargins->{{55, 10}, {8, 5}}, CellEditDuplicate->True, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, CellLabelPositioning->Left, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Output", Background->RGBColor[0.940017, 0.890013, 0.990005]], Cell[StyleData["Output", "Printout"], CellMargins->{{55, 55}, {10, 10}}, ShowCellLabel->False, FontSize->9.5] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["Message"], CellDingbat->"\[LongDash]", CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Message", StyleMenuListing->None, FontSize->10, FontSlant->"Italic"], Cell[StyleData["Message", "Printout"], CellMargins->{{55, 55}, {0, 3}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Print"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"OutputGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GroupPageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, CellLabelMargins->{{26, Inherited}, {Inherited, Inherited}}, DefaultFormatType->DefaultOutputFormatType, TextAlignment->Left, AutoItalicWords->{}, FormatType->InputForm, CounterIncrements->"Print", StyleMenuListing->None], Cell[StyleData["Print", "Printout"], CellMargins->{{54, 72}, {2, 10}}, FontSize->8] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Graphics"], CellMargins->{{55, Inherited}, {Inherited, Inherited}}, CellGroupingRules->"GraphicsGrouping", CellHorizontalScrolling->True, PageBreakWithin->False, GeneratedCell->True, CellAutoOverwrite->True, ShowCellLabel->False, DefaultFormatType->DefaultOutputFormatType, FormatType->InputForm, CounterIncrements->"Graphics", StyleMenuListing->None, Background->RGBColor[0.780011, 0.920012, 1], ButtonBoxOptions->{Background->RGBColor[0.750011, 0.630014, 0.760021]}], Cell[StyleData["Graphics", "Printout"], CellMargins->{{55, 55}, {0, 15}}, ImageSize->{0.0625, 0.0625}, ImageMargins->{{35, Inherited}, {Inherited, 0}}, FontSize->8] }, Open ]], Cell[CellGroupData[{ Cell[StyleData["CellLabel"], CellMargins->{{9, Inherited}, {Inherited, Inherited}}, StyleMenuListing->None, FontFamily->"Helvetica", FontSize->9, FontSlant->"Oblique"], Cell[StyleData["CellLabel", "Printout"], CellMargins->{{0, Inherited}, {Inherited, Inherited}}, FontSize->8] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Unique Styles", "Section"], Cell[CellGroupData[{ Cell[StyleData["Author"], CellMargins->{{45, Inherited}, {2, 20}}, CellGroupingRules->{"TitleGrouping", 20}, PageBreakBelow->False, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->14, FontWeight->"Bold"], Cell[StyleData["Author", "Printout"], CellMargins->{{36, Inherited}, {2, 30}}, FontSize->12] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Address"], CellMargins->{{45, Inherited}, {2, 2}}, CellGroupingRules->{"TitleGrouping", 30}, PageBreakBelow->False, LineSpacing->{1, 1}, CounterAssignments->{{"Section", 0}, {"Equation", 0}, {"Figure", 0}}, FontSize->12, FontSlant->"Italic"], Cell[StyleData["Address", "Printout"], CellMargins->{{36, Inherited}, {2, 2}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Abstract"], CellMargins->{{45, 75}, {Inherited, 30}}, LineSpacing->{1, 0}], Cell[StyleData["Abstract", "Printout"], CellMargins->{{36, 67}, {Inherited, 50}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Reference"], CellMargins->{{18, 40}, {2, 2}}, TextJustification->1, LineSpacing->{1, 0}], Cell[StyleData["Reference", "Printout"], CellMargins->{{18, 40}, {Inherited, 0}}, FontSize->8] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell["\<\ The following styles are useful for numbered equations, figures, \ etc. They automatically give the cell a FrameLabel containing a reference to \ a particular counter, and also increment that counter.\ \>", "Text"], Cell[CellGroupData[{ Cell[StyleData["NumberedEquation"], CellMargins->{{55, 10}, {0, 10}}, CellFrameLabels->{{None, Cell[ TextData[ {"(", CounterBox[ "NumberedEquation"], ")"}]]}, {None, None}}, DefaultFormatType->DefaultInputFormatType, CounterIncrements->"NumberedEquation", FormatTypeAutoConvert->False], Cell[StyleData["NumberedEquation", "Printout"], CellMargins->{{55, 55}, {0, 10}}, FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedFigure"], CellMargins->{{55, 145}, {2, 10}}, CellHorizontalScrolling->True, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Figure ", CounterBox[ "NumberedFigure"]}], FontWeight -> "Bold"], None}}, CounterIncrements->"NumberedFigure", FormatTypeAutoConvert->False], Cell[StyleData["NumberedFigure", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["NumberedTable"], CellMargins->{{55, 145}, {2, 10}}, CellFrameLabels->{{None, None}, {Cell[ TextData[ {"Table ", CounterBox[ "NumberedTable"]}], FontWeight -> "Bold"], None}}, TextAlignment->Center, CounterIncrements->"NumberedTable", FormatTypeAutoConvert->False], Cell[StyleData["NumberedTable", "Printout"], CellMargins->{{18, Inherited}, {Inherited, Inherited}}, FontSize->10] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[CellGroupData[{ Cell[StyleData["DisplayFormula"], CellMargins->{{55, 10}, {2, 10}}, CellHorizontalScrolling->True, DefaultFormatType->DefaultInputFormatType, ScriptLevel->0, SingleLetterItalics->True, UnderoverscriptBoxOptions->{LimitsPositioning->True}], Cell[StyleData["DisplayFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["ChemicalFormula"], CellMargins->{{55, 10}, {2, 10}}, DefaultFormatType->DefaultInputFormatType, AutoSpacing->False, ScriptLevel->1, ScriptBaselineShifts->{0.6, Automatic}, SingleLetterItalics->False, ZeroWidthTimes->True], Cell[StyleData["ChemicalFormula", "Printout"], FontSize->10] }, Closed]], Cell[CellGroupData[{ Cell[StyleData["Program"], CellMargins->{{18, 10}, {Inherited, 6}}, FontFamily->"Courier"], Cell[StyleData["Program", "Printout"], CellMargins->{{18, 30}, {Inherited, 4}}, FontSize->9.5] }, Closed]] }, Closed]] }, Open ]] }], MacintoshSystemPageSetup->"\<\ 00<0004/0B`000002mT8o?mooh<" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1717, 49, 109, 2, 114, "Title", Evaluatable->False], Cell[1829, 53, 1048, 24, 322, "Text"], Cell[CellGroupData[{ Cell[2902, 81, 43, 0, 53, "Subtitle"], Cell[CellGroupData[{ Cell[2970, 85, 65, 0, 58, "Subsection"], Cell[3038, 87, 1161, 44, 212, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[4236, 136, 51, 0, 42, "Subsection"], Cell[CellGroupData[{ Cell[4312, 140, 56, 0, 48, "Subsubsection"], Cell[4371, 142, 259, 7, 44, "Input", Evaluatable->False], Cell[4633, 151, 129, 5, 42, "Text"], Cell[CellGroupData[{ Cell[4787, 160, 79, 1, 28, "Input"], Cell[4869, 163, 239, 7, 46, "Output"] }, Open ]], Cell[5123, 173, 95, 3, 40, "Text"], Cell[CellGroupData[{ Cell[5243, 180, 1454, 34, 105, "Input"], Cell[6700, 216, 247, 7, 47, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6984, 228, 1691, 36, 161, "Input"], Cell[8678, 266, 325, 9, 63, "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[9052, 281, 55, 0, 40, "Subsubsection"], Cell[9110, 283, 259, 7, 44, "Input", Evaluatable->False], Cell[CellGroupData[{ Cell[9394, 294, 80, 1, 28, "Input"], Cell[9477, 297, 239, 7, 46, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[9753, 309, 1701, 36, 161, "Input"], Cell[11457, 347, 330, 9, 63, "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[11836, 362, 57, 0, 40, "Subsubsection"], Cell[11896, 364, 248, 6, 47, "Input", Evaluatable->False], Cell[12147, 372, 159, 5, 42, "Text"], Cell[CellGroupData[{ Cell[12331, 381, 85, 1, 28, "Input"], Cell[12419, 384, 280, 8, 46, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[12736, 397, 1705, 36, 161, "Input"], Cell[14444, 435, 348, 9, 63, "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[14841, 450, 49, 0, 40, "Subsubsection"], Cell[14893, 452, 274, 6, 44, "Input", Evaluatable->False], Cell[CellGroupData[{ Cell[15192, 462, 121, 2, 28, "Input"], Cell[15316, 466, 316, 8, 46, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[15669, 479, 1709, 36, 161, "Input"], Cell[17381, 517, 403, 10, 63, "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[17845, 534, 163, 4, 60, "Subsection"], Cell[CellGroupData[{ Cell[18033, 542, 79, 1, 28, "Input"], Cell[18115, 545, 1092, 28, 63, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[19244, 578, 75, 1, 28, "Input"], Cell[19322, 581, 1022, 27, 65, "Output"] }, Open ]], Cell[20359, 611, 68, 0, 40, "Text"], Cell[CellGroupData[{ Cell[20452, 615, 87, 1, 28, "Input"], Cell[20542, 618, 917, 24, 65, "Output"] }, Open ]], Cell[21474, 645, 355, 8, 98, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[21866, 658, 71, 0, 42, "Subsection"], Cell[CellGroupData[{ Cell[21962, 662, 57, 0, 48, "Subsubsection"], Cell[CellGroupData[{ Cell[22044, 666, 75, 0, 40, "Text"], Cell[CellGroupData[{ Cell[22144, 670, 80, 1, 28, "Input"], Cell[22227, 673, 253, 7, 44, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[22529, 686, 75, 0, 40, "Text"], Cell[CellGroupData[{ Cell[22629, 690, 106, 2, 44, "Input"], Cell[22738, 694, 142, 3, 64, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[22917, 702, 60, 1, 28, "Input"], Cell[22980, 705, 112, 2, 64, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[23141, 713, 64, 0, 40, "Text"], Cell[CellGroupData[{ Cell[23230, 717, 134, 3, 44, "Input"], Cell[23367, 722, 14588, 408, 202, 3721, 269, "GraphicsData", "PostScript", \ "Graphics"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[38004, 1136, 54, 0, 40, "Text"], Cell[CellGroupData[{ Cell[38083, 1140, 90, 1, 28, "Input"], Cell[38176, 1143, 55, 1, 43, "Output"] }, Open ]] }, Open ]], Cell[38258, 1148, 173, 4, 54, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[38468, 1157, 57, 0, 40, "Subsubsection"], Cell[CellGroupData[{ Cell[38550, 1161, 76, 1, 28, "Input"], Cell[38629, 1164, 253, 7, 44, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[38919, 1176, 437, 12, 32, "Input"], Cell[39359, 1190, 310, 8, 61, "Output"] }, Open ]], Cell[39684, 1201, 114, 3, 40, "Text"], Cell[CellGroupData[{ Cell[39823, 1208, 75, 1, 28, "Input"], Cell[39901, 1211, 291, 8, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[40229, 1224, 57, 1, 28, "Input"], Cell[40289, 1227, 246, 7, 45, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[40572, 1239, 104, 2, 29, "Input"], Cell[40679, 1243, 89, 1, 62, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[40805, 1249, 61, 1, 29, "Input"], Cell[40869, 1252, 65, 1, 62, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[40971, 1258, 64, 1, 28, "Input"], Cell[41038, 1261, 14712, 409, 202, 3727, 269, "GraphicsData", "PostScript", \ "Graphics"] }, Open ]], Cell[CellGroupData[{ Cell[55787, 1675, 54, 0, 40, "Text"], Cell[CellGroupData[{ Cell[55866, 1679, 48, 1, 28, "Input"], Cell[55917, 1682, 57, 1, 43, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[56023, 1689, 549, 11, 152, "Text"], Cell[CellGroupData[{ Cell[56597, 1704, 93, 1, 44, "Input"], Cell[56693, 1707, 55, 1, 59, "Output"] }, Open ]] }, Open ]] }, Closed]] }, Closed]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)