(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     35644,       1243]
NotebookOptionsPosition[     31305,       1110]
NotebookOutlinePosition[     31697,       1127]
CellTagsIndexPosition[     31654,       1124]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Bernard Thermal Instability: Marginal Mode Analysis", "Title"],

Cell["\<\
APPH 4200 Physics of Fluids
Columbia University
November, 2007\
\>", "Subsubtitle"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell["\<\
This notebook solves for the marginal instability threshold for Bernard \
thermal instability. See Chapter 12, Section 3 of Kundu and Cohen's textbook.\
\
\>", "Text"],

Cell[TextData[{
 "Four steps are required. First, from the linearized equations for the fluid \
dynamics, a cubic equation is solved for the vertical variation of the \
vertical velocity. Second, these three solutions are used to satisfy three \
boundary conditions at the constant-temperature walls. This leads to a \
condition between Ra (the Rayleigh number) and ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["k", "2"], TraditionalForm]]],
 "for marginal instability. Finally, we plot this condition and examine the \
modes."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Vertical Velocity Equation", "Section"],

Cell["The equation for vertical velocity takes the form", "Text"],

Cell[BoxData[
 RowBox[{"eqVert", " ", "=", " ", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{
      SuperscriptBox["q", "2"], " ", "+", " ", 
      SuperscriptBox["k", "2"]}], ")"}], "3"], " ", "\[Equal]", "  ", 
   RowBox[{
    SuperscriptBox["k", "2"], "Ra"}]}]}]], "Input",
 CellLabel->"In[1]:="],

Cell[TextData[{
 "when the vertical velocity has the form, ",
 Cell[BoxData[
  FormBox[
   RowBox[{
    SubscriptBox["U", "z"], " ", "~", " ", 
    RowBox[{"Cos", "[", 
     RowBox[{"q", " ", "z"}], "]"}]}], TraditionalForm]]],
 "."
}], "Text"],

Cell[BoxData[
 RowBox[{"qSol", " ", "=", " ", 
  RowBox[{"Solve", "[", 
   RowBox[{
    RowBox[{"(", 
     RowBox[{
      RowBox[{"eqVert", " ", "/.", " ", 
       RowBox[{
        SuperscriptBox["q", "2"], " ", "\[Rule]", " ", "q2"}]}], " ", "/.", 
      " ", 
      RowBox[{
       SuperscriptBox["k", "2"], " ", "\[Rule]", " ", "k2"}]}], ")"}], ",", 
    " ", "q2"}], "]"}]}]], "Input",
 CellLabel->"In[2]:="],

Cell[CellGroupData[{

Cell["Cos[z] is Even", "Subsection"],

Cell[TextData[{
 "First, we'll look for ",
 StyleBox["even",
  FontSlant->"Italic"],
 " modes... "
}], "Text"],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Cos", "[", "z", "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"z", ",", 
     RowBox[{"-", 
      FractionBox["\[Pi]", "2"]}], ",", 
     FractionBox["\[Pi]", "2"]}], "}"}]}], "]"}]], "Input",
 CellLabel->"In[3]:="],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Evaluate", "[", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"Re", "[", "#1", "]"}], ",", 
         RowBox[{"Im", "[", "#1", "]"}]}], "}"}], "&"}], ")"}], "[", 
     RowBox[{"Cos", "[", 
      RowBox[{"z", " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", 
         RowBox[{"\[ImaginaryI]", " ", 
          SqrtBox["3"]}]}], ")"}]}], "]"}], "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"z", ",", 
     RowBox[{"-", 
      FractionBox["\[Pi]", "2"]}], ",", 
     FractionBox["\[Pi]", "2"]}], "}"}], ",", 
   RowBox[{"PlotStyle", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Blue", ",", "Red"}], "}"}]}]}], "]"}]], "Input",
 CellLabel->"In[4]:="],

Cell[TextData[{
 "An similiar analysis can be performed for the ",
 StyleBox["odd",
  FontSlant->"Italic"],
 " modes, but these modes have a higher critical Ra number."
}], "Text"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Satisfy Boundary Conditions", "Section"],

Cell[BoxData[
 RowBox[{
  RowBox[{"uz", "[", "z_", "]"}], " ", "=", " ", 
  RowBox[{
   RowBox[{"a", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"Cos", "[", 
       RowBox[{
        RowBox[{"Sqrt", "[", "q2", "]"}], " ", "z"}], "]"}], "/.", 
      RowBox[{
      "qSol", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], 
     ")"}]}], " ", "+", " ", 
   RowBox[{"b", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"Cos", "[", 
       RowBox[{
        RowBox[{"Sqrt", "[", "q2", "]"}], " ", "z"}], "]"}], "/.", 
      RowBox[{
      "qSol", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], 
     ")"}]}], "+", " ", 
   RowBox[{"c", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"Cos", "[", 
       RowBox[{
        RowBox[{"Sqrt", "[", "q2", "]"}], " ", "z"}], "]"}], "/.", 
      RowBox[{
      "qSol", "\[LeftDoubleBracket]", "3", "\[RightDoubleBracket]"}]}], 
     ")"}]}]}]}]], "Input",
 CellLabel->"In[5]:="],

Cell[BoxData[
 RowBox[{"eq1", " ", "=", " ", 
  RowBox[{
   RowBox[{"uz", "[", 
    RowBox[{"1", "/", "2"}], "]"}], " ", "\[Equal]", " ", "0"}]}]], "Input",
 CellLabel->"In[6]:="],

Cell[BoxData[
 RowBox[{"eq2", " ", "=", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{
     RowBox[{"D", "[", 
      RowBox[{
       RowBox[{"uz", "[", "z", "]"}], ",", "z"}], "]"}], " ", "\[Equal]", " ",
      "0"}], " ", "/.", " ", 
    RowBox[{"z", " ", "\[Rule]", " ", 
     RowBox[{"1", "/", "2"}]}]}], ")"}]}]], "Input",
 CellLabel->"In[7]:="],

Cell[BoxData[
 RowBox[{"eq3", " ", "=", " ", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{
     RowBox[{"Collect", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"D", "[", 
         RowBox[{
          RowBox[{"uz", "[", "z", "]"}], ",", 
          RowBox[{"{", 
           RowBox[{"z", ",", "4"}], "}"}]}], "]"}], " ", "-", " ", 
        RowBox[{"2", " ", "k2", " ", 
         RowBox[{"D", "[", 
          RowBox[{
           RowBox[{"uz", "[", "z", "]"}], ",", 
           RowBox[{"{", 
            RowBox[{"z", ",", "2"}], "}"}]}], "]"}]}], " ", "+", " ", 
        RowBox[{
         SuperscriptBox["k2", "2"], " ", 
         RowBox[{"uz", "[", "z", "]"}]}]}], ",", " ", 
       RowBox[{"{", 
        RowBox[{"a", ",", "b", ",", "c"}], "}"}], ",", " ", "Simplify"}], 
      "]"}], "\[Equal]", " ", "0"}], " ", "/.", " ", 
    RowBox[{"z", " ", "\[Rule]", " ", 
     RowBox[{"1", "/", "2"}]}]}], ")"}]}]], "Input",
 CellLabel->"In[8]:="],

Cell[TextData[{
 "These three equations can be satisfied simultaneous ",
 StyleBox["only if ",
  FontSlant->"Italic"],
 " the determinant of a characteristic matrix vanishes. This defines the \
marginal condition between ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["k", "2"], TraditionalForm]]],
 "and Ra."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Marginal Instability Condition", "Section"],

Cell["\<\
The marginal instability criterion is found by simultaneously solving three \
boundary conditions. This possible when the determinant of a characteristic \
matrix vanishes.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"First", "[", "eq1", "]"}]], "Input",
 CellLabel->"In[9]:="],

Cell[BoxData[
 RowBox[{"row1", " ", "=", " ", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq1", "]"}], ",", " ", "a"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq1", "]"}], ",", " ", "b"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq1", "]"}], ",", " ", "c"}], "]"}]}], 
   "}"}]}]], "Input",
 CellLabel->"In[10]:="],

Cell[BoxData[
 RowBox[{"row2", " ", "=", " ", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq2", "]"}], ",", " ", "a"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq2", "]"}], ",", " ", "b"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq2", "]"}], ",", " ", "c"}], "]"}]}], 
   "}"}]}]], "Input",
 CellLabel->"In[11]:="],

Cell[BoxData[
 RowBox[{"row3", " ", "=", " ", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq3", "]"}], ",", " ", "a"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq3", "]"}], ",", " ", "b"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq3", "]"}], ",", " ", "c"}], "]"}]}], 
   "}"}]}]], "Input",
 CellLabel->"In[12]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"Det", "[", 
   RowBox[{"{", 
    RowBox[{"row1", ",", "row2", ",", "row3"}], "}"}], "]"}], " ", "//", " ", 
  "Simplify"}]], "Input",
 CellLabel->"In[13]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"marginal", "[", 
   RowBox[{"k2_", ",", "Ra_"}], "]"}], " ", "=", " ", 
  RowBox[{
   RowBox[{"(", 
    RowBox[{
     RowBox[{
      RowBox[{"Det", "[", 
       RowBox[{"{", 
        RowBox[{"row1", ",", "row2", ",", "row3"}], "}"}], "]"}], "/", 
      SuperscriptBox["k2", 
       RowBox[{"2", "/", "3"}]]}], "/", 
     SuperscriptBox["Ra", 
      RowBox[{"2", "/", "3"}]]}], ")"}], " ", "//", " ", 
   "Simplify"}]}]], "Input",
 CellLabel->"In[14]:="],

Cell[BoxData[
 RowBox[{"marginal", "[", 
  RowBox[{"2.0", ",", "4000.0"}], "]"}]], "Input",
 CellLabel->"In[15]:="],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Evaluate", "[", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"Re", "[", "#1", "]"}], ",", 
         RowBox[{"Im", "[", "#1", "]"}]}], "}"}], "&"}], ")"}], "[", 
     RowBox[{"marginal", "[", 
      RowBox[{"k2", ",", "4000.`"}], "]"}], "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"k2", ",", "0", ",", "50"}], "}"}], ",", 
   RowBox[{"PlotRange", "\[Rule]", "All"}], ",", 
   RowBox[{"AxesLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{
     "\"\<\\!\\(\\*SuperscriptBox[\\(k\\), \\(2\\)]\\)\>\"", ",", 
      "\"\<Det\>\""}], "}"}]}]}], "]"}]], "Input",
 CellLabel->"In[16]:="],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Evaluate", "[", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"Re", "[", "#1", "]"}], ",", 
         RowBox[{"Im", "[", "#1", "]"}]}], "}"}], "&"}], ")"}], "[", 
     RowBox[{"marginal", "[", 
      RowBox[{"1.`", ",", "Ra"}], "]"}], "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"Ra", ",", "0", ",", "5000"}], "}"}], ",", 
   RowBox[{"PlotRange", "\[Rule]", "All"}], ",", 
   RowBox[{"AxesLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"\"\<Ra\>\"", ",", "\"\<Det\>\""}], "}"}]}]}], "]"}]], "Input",
 CellLabel->"In[17]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{"marginal", "[", 
     RowBox[{"1.0", ",", "Ra"}], "]"}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"Ra", ",", " ", "2000.0"}], "}"}]}], "]"}], " ", "//", " ", 
  "Chop"}]], "Input",
 CellLabel->"In[18]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{"marginal", "[", 
     RowBox[{"5.0", ",", "Ra"}], "]"}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"Ra", ",", " ", "2000.0"}], "}"}]}], "]"}], " ", "//", " ", 
  "Chop"}]], "Input",
 CellLabel->"In[19]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{"marginal", "[", 
     RowBox[{"50.0", ",", "Ra"}], "]"}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"Ra", ",", " ", "2000.0"}], "}"}]}], "]"}], " ", "//", " ", 
  "Chop"}]], "Input",
 CellLabel->"In[20]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"marginalRa", "[", "k2_", "]"}], " ", ":=", " ", 
  RowBox[{"Ra", " ", "/.", " ", 
   RowBox[{"Chop", "[", 
    RowBox[{"FindRoot", "[", 
     RowBox[{
      RowBox[{"marginal", "[", 
       RowBox[{"k2", ",", "Ra"}], "]"}], ",", " ", 
      RowBox[{"{", 
       RowBox[{"Ra", ",", " ", "2000.0"}], "}"}]}], "]"}], "]"}]}]}]], "Input",\

 CellLabel->"In[21]:="],

Cell[BoxData[
 RowBox[{"marginalRa", "[", "2.0", "]"}]], "Input",
 CellLabel->"In[22]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"marginalBoundary", " ", "=", " ", 
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"marginalRa", "[", "k2", "]"}], ",", 
       RowBox[{"Sqrt", "[", "k2", "]"}]}], "}"}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"k2", ",", " ", "1.0", ",", " ", "50.0", ",", "1.0"}], "}"}]}], 
    "]"}]}], ";"}]], "Input",
 CellLabel->"In[23]:="],

Cell[BoxData[
 RowBox[{"evenMarginalPlt", "=", 
  RowBox[{"ListPlot", "[", 
   RowBox[{"marginalBoundary", ",", 
    RowBox[{"Joined", "\[Rule]", "True"}], ",", 
    RowBox[{"PlotRange", "\[Rule]", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "9000"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "9"}], "}"}]}], "}"}]}], ",", 
    RowBox[{"PlotLabel", "\[Rule]", "\"\<Marginal Instability\>\""}], ",", 
    RowBox[{"AxesLabel", "\[Rule]", 
     RowBox[{"{", 
      RowBox[{"\"\<Ra\>\"", ",", "\"\<k\>\""}], "}"}]}]}], "]"}]}]], "Input",
 CellLabel->"In[24]:="]
}, Closed]],

Cell[CellGroupData[{

Cell["Critical Mode", "Section"],

Cell["\<\
The critical mode and Rayleigh number were given in the textbook.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"critRa", " ", "=", " ", 
  RowBox[{"Min", "[", 
   RowBox[{"First", "[", 
    RowBox[{"Transpose", "[", "marginalBoundary", "]"}], "]"}], 
   "]"}]}]], "Input",
 CellLabel->"In[25]:="],

Cell[BoxData[
 RowBox[{"critk", " ", "=", " ", 
  RowBox[{
   RowBox[{"Select", "[", 
    RowBox[{"marginalBoundary", ",", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"First", "[", "#", "]"}], " ", "\[Equal]", " ", "critRa"}], 
       ")"}], "&"}]}], "]"}], "\[LeftDoubleBracket]", 
   RowBox[{"1", ",", "2"}], "\[RightDoubleBracket]"}]}]], "Input",
 CellLabel->"In[26]:="],

Cell[BoxData[
 RowBox[{"eq1Crit", " ", "=", " ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"eq1", " ", "/.", " ", 
      RowBox[{"k2", " ", "\[Rule]", " ", 
       SuperscriptBox["critk", "2"]}]}], " ", "/.", " ", 
     RowBox[{"Ra", " ", "\[Rule]", " ", "critRa"}]}], "  ", "/.", " ", 
    RowBox[{"a", " ", "\[Rule]", " ", "1"}]}], "//", " ", 
   "Simplify"}]}]], "Input",
 CellLabel->"In[27]:="],

Cell[BoxData[
 RowBox[{"eq2Crit", " ", "=", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"eq2", " ", "/.", " ", 
      RowBox[{"k2", " ", "\[Rule]", " ", 
       SuperscriptBox["critk", "2"]}]}], " ", "/.", " ", 
     RowBox[{"Ra", " ", "\[Rule]", " ", "critRa"}]}], " ", "/.", " ", 
    RowBox[{"a", " ", "\[Rule]", " ", "1"}]}], " ", "//", " ", 
   "Simplify"}]}]], "Input",
 CellLabel->"In[28]:="],

Cell[BoxData[
 RowBox[{"modeCrit", " ", "=", " ", 
  RowBox[{"First", "[", 
   RowBox[{"Solve", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"eq1Crit", ",", "eq2Crit"}], "}"}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"b", ",", " ", "c"}], "}"}]}], "]"}], "]"}]}]], "Input",
 CellLabel->"In[29]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"uzCrit", "[", "z_", "]"}], " ", "=", " ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"uz", "[", "z", "]"}], " ", "/.", " ", 
      RowBox[{"a", " ", "\[Rule]", " ", "1"}]}], "/.", " ", "modeCrit"}], "/.",
     " ", 
    RowBox[{"k2", " ", "\[Rule]", " ", 
     SuperscriptBox["critk", "2"]}]}], " ", "/.", " ", 
   RowBox[{"Ra", " ", "\[Rule]", " ", "critRa", " "}]}]}]], "Input",
 CellLabel->"In[30]:="],

Cell[BoxData[
 RowBox[{"uzCrit", "[", "0.0", "]"}]], "Input",
 CellLabel->"In[31]:="],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Evaluate", "[", 
    RowBox[{"Chop", "[", 
     RowBox[{"uzCrit", "[", "z", "]"}], "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"z", ",", 
     RowBox[{"-", "0.499`"}], ",", "0.499`"}], "}"}], ",", 
   RowBox[{"PlotLabel", "\[Rule]", "\"\<Critical Uz[z]\>\""}], ",", 
   RowBox[{"AxesLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"\"\<z\>\"", ",", "\"\<\>\""}], "}"}]}]}], "]"}]], "Input",
 CellLabel->"In[32]:="]
}, Closed]],

Cell[CellGroupData[{

Cell["Odd Mode (?)", "Section"],

Cell[CellGroupData[{

Cell["Sin[z] is Even", "Subsection"],

Cell[TextData[{
 "First, we'll look for ",
 StyleBox["even",
  FontSlant->"Italic"],
 " modes... "
}], "Text"],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Sin", "[", "z", "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"z", ",", 
     RowBox[{"-", "\[Pi]"}], ",", "\[Pi]"}], "}"}]}], "]"}]], "Input",
 CellLabel->"In[33]:="],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Evaluate", "[", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"Re", "[", "#1", "]"}], ",", 
         RowBox[{"Im", "[", "#1", "]"}]}], "}"}], "&"}], ")"}], "[", 
     RowBox[{"Sin", "[", 
      RowBox[{"z", " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", 
         RowBox[{"\[ImaginaryI]", " ", 
          SqrtBox["3"]}]}], ")"}]}], "]"}], "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"z", ",", 
     RowBox[{"-", "\[Pi]"}], ",", "\[Pi]"}], "}"}], ",", 
   RowBox[{"PlotStyle", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Blue", ",", "Red"}], "}"}]}]}], "]"}]], "Input",
 CellLabel->"In[34]:="],

Cell[TextData[{
 "An similiar analysis can be performed for the ",
 StyleBox["odd",
  FontSlant->"Italic"],
 " modes, but these modes have a higher critical Ra number."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Satisfy Boundary Conditions", "Subsection"],

Cell[BoxData[
 RowBox[{
  RowBox[{"uz", "[", "z_", "]"}], " ", "=", " ", 
  RowBox[{
   RowBox[{"a", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"Sin", "[", 
       RowBox[{
        RowBox[{"Sqrt", "[", "q2", "]"}], " ", "z"}], "]"}], "/.", 
      RowBox[{
      "qSol", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], 
     ")"}]}], " ", "+", " ", 
   RowBox[{"b", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"Sin", "[", 
       RowBox[{
        RowBox[{"Sqrt", "[", "q2", "]"}], " ", "z"}], "]"}], "/.", 
      RowBox[{
      "qSol", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], 
     ")"}]}], "+", " ", 
   RowBox[{"c", " ", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"Sin", "[", 
       RowBox[{
        RowBox[{"Sqrt", "[", "q2", "]"}], " ", "z"}], "]"}], "/.", 
      RowBox[{
      "qSol", "\[LeftDoubleBracket]", "3", "\[RightDoubleBracket]"}]}], 
     ")"}]}]}]}]], "Input",
 CellLabel->"In[35]:="],

Cell[BoxData[
 RowBox[{"eq1", " ", "=", " ", 
  RowBox[{
   RowBox[{"uz", "[", 
    RowBox[{"1", "/", "2"}], "]"}], " ", "\[Equal]", " ", "0"}]}]], "Input",
 CellLabel->"In[36]:="],

Cell[BoxData[
 RowBox[{"eq2", " ", "=", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{
     RowBox[{"D", "[", 
      RowBox[{
       RowBox[{"uz", "[", "z", "]"}], ",", "z"}], "]"}], " ", "\[Equal]", " ",
      "0"}], " ", "/.", " ", 
    RowBox[{"z", " ", "\[Rule]", " ", 
     RowBox[{"1", "/", "2"}]}]}], ")"}]}]], "Input",
 CellLabel->"In[37]:="],

Cell[BoxData[
 RowBox[{"eq3", " ", "=", " ", 
  RowBox[{"(", 
   RowBox[{
    RowBox[{
     RowBox[{"Collect", "[", 
      RowBox[{
       RowBox[{
        RowBox[{"D", "[", 
         RowBox[{
          RowBox[{"uz", "[", "z", "]"}], ",", 
          RowBox[{"{", 
           RowBox[{"z", ",", "4"}], "}"}]}], "]"}], " ", "-", " ", 
        RowBox[{"2", " ", "k2", " ", 
         RowBox[{"D", "[", 
          RowBox[{
           RowBox[{"uz", "[", "z", "]"}], ",", 
           RowBox[{"{", 
            RowBox[{"z", ",", "2"}], "}"}]}], "]"}]}], " ", "+", " ", 
        RowBox[{
         SuperscriptBox["k2", "2"], " ", 
         RowBox[{"uz", "[", "z", "]"}]}]}], ",", " ", 
       RowBox[{"{", 
        RowBox[{"a", ",", "b", ",", "c"}], "}"}], ",", " ", "Simplify"}], 
      "]"}], "\[Equal]", " ", "0"}], " ", "/.", " ", 
    RowBox[{"z", " ", "\[Rule]", " ", 
     RowBox[{"1", "/", "2"}]}]}], ")"}]}]], "Input",
 CellLabel->"In[38]:="],

Cell[TextData[{
 "These three equations can be satisfied simultaneous ",
 StyleBox["only if ",
  FontSlant->"Italic"],
 " the determinant of a characteristic matrix vanishes. This defines the \
marginal condition between ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["k", "2"], TraditionalForm]]],
 "and Ra."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Marginal Instability Condition", "Subsection"],

Cell["\<\
The marginal instability criterion is found by simultaneously solving three \
boundary conditions. This possible when the determinant of a characteristic \
matrix vanishes.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"First", "[", "eq1", "]"}]], "Input",
 CellLabel->"In[39]:="],

Cell[BoxData[
 RowBox[{"row1", " ", "=", " ", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq1", "]"}], ",", " ", "a"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq1", "]"}], ",", " ", "b"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq1", "]"}], ",", " ", "c"}], "]"}]}], 
   "}"}]}]], "Input",
 CellLabel->"In[40]:="],

Cell[BoxData[
 RowBox[{"row2", " ", "=", " ", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq2", "]"}], ",", " ", "a"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq2", "]"}], ",", " ", "b"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq2", "]"}], ",", " ", "c"}], "]"}]}], 
   "}"}]}]], "Input",
 CellLabel->"In[41]:="],

Cell[BoxData[
 RowBox[{"row3", " ", "=", " ", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq3", "]"}], ",", " ", "a"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq3", "]"}], ",", " ", "b"}], "]"}], ",", 
    RowBox[{"Coefficient", "[", 
     RowBox[{
      RowBox[{"First", "[", "eq3", "]"}], ",", " ", "c"}], "]"}]}], 
   "}"}]}]], "Input",
 CellLabel->"In[42]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"Det", "[", 
   RowBox[{"{", 
    RowBox[{"row1", ",", "row2", ",", "row3"}], "}"}], "]"}], " ", "//", " ", 
  "Simplify"}]], "Input",
 CellLabel->"In[43]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"marginal", "[", 
   RowBox[{"k2_", ",", "Ra_"}], "]"}], " ", "=", " ", 
  RowBox[{
   RowBox[{"(", 
    RowBox[{
     RowBox[{
      RowBox[{"Det", "[", 
       RowBox[{"{", 
        RowBox[{"row1", ",", "row2", ",", "row3"}], "}"}], "]"}], "/", 
      SuperscriptBox["k2", 
       RowBox[{"2", "/", "3"}]]}], "/", 
     SuperscriptBox["Ra", 
      RowBox[{"2", "/", "3"}]]}], ")"}], " ", "//", " ", 
   "Simplify"}]}]], "Input",
 CellLabel->"In[44]:="],

Cell[BoxData[{
 RowBox[{"marginal", "[", 
  RowBox[{"50.0", ",", "4000.0"}], "]"}], "\[IndentingNewLine]", 
 RowBox[{"marginal", "[", 
  RowBox[{"50.0", ",", "1000.0"}], "]"}]}], "Input",
 CellLabel->"In[45]:="],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Evaluate", "[", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"Re", "[", "#1", "]"}], ",", 
         RowBox[{"Im", "[", "#1", "]"}]}], "}"}], "&"}], ")"}], "[", 
     RowBox[{"marginal", "[", 
      RowBox[{"k2", ",", "4000.`"}], "]"}], "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"k2", ",", "0", ",", "70"}], "}"}], ",", 
   RowBox[{"PlotRange", "\[Rule]", "All"}], ",", 
   RowBox[{"AxesLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{
     "\"\<\\!\\(\\*SuperscriptBox[\\(k\\), \\(2\\)]\\)\>\"", ",", 
      "\"\<Det\>\""}], "}"}]}]}], "]"}]], "Input",
 CellLabel->"In[47]:="],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Evaluate", "[", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"Re", "[", "#1", "]"}], ",", 
         RowBox[{"Im", "[", "#1", "]"}]}], "}"}], "&"}], ")"}], "[", 
     RowBox[{"marginal", "[", 
      RowBox[{"50.`", ",", "Ra"}], "]"}], "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"Ra", ",", "0", ",", "5000"}], "}"}], ",", 
   RowBox[{"PlotRange", "\[Rule]", "All"}], ",", 
   RowBox[{"AxesLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"\"\<Ra\>\"", ",", "\"\<Det\>\""}], "}"}]}]}], "]"}]], "Input",
 CellLabel->"In[48]:="],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Evaluate", "[", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{
         RowBox[{"Re", "[", "#1", "]"}], ",", 
         RowBox[{"Im", "[", "#1", "]"}]}], "}"}], "&"}], ")"}], "[", 
     RowBox[{"marginal", "[", 
      RowBox[{"25.`", ",", "Ra"}], "]"}], "]"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"Ra", ",", "0", ",", "5000"}], "}"}], ",", 
   RowBox[{"PlotRange", "\[Rule]", "All"}], ",", 
   RowBox[{"AxesLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"\"\<Ra\>\"", ",", "\"\<Det\>\""}], "}"}]}]}], "]"}]], "Input",
 CellLabel->"In[49]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{"marginal", "[", 
     RowBox[{"25.0", ",", "Ra"}], "]"}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"Ra", ",", " ", "3000.0"}], "}"}]}], "]"}], " ", "//", " ", 
  "Chop"}]], "Input",
 CellLabel->"In[50]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{"marginal", "[", 
     RowBox[{"50.0", ",", "Ra"}], "]"}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"Ra", ",", " ", "3000.0"}], "}"}]}], "]"}], " ", "//", " ", 
  "Chop"}]], "Input",
 CellLabel->"In[51]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{"marginal", "[", 
     RowBox[{"100.0", ",", "Ra"}], "]"}], ",", " ", 
    RowBox[{"{", 
     RowBox[{"Ra", ",", " ", "3000.0"}], "}"}]}], "]"}], " ", "//", " ", 
  "Chop"}]], "Input",
 CellLabel->"In[52]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"marginalRa", "[", 
   RowBox[{"k2_", ",", 
    RowBox[{"start_:", "3000.0"}]}], "]"}], " ", ":=", " ", 
  RowBox[{"Ra", " ", "/.", " ", 
   RowBox[{"Chop", "[", 
    RowBox[{"FindRoot", "[", 
     RowBox[{
      RowBox[{"marginal", "[", 
       RowBox[{"k2", ",", "Ra"}], "]"}], ",", " ", 
      RowBox[{"{", 
       RowBox[{"Ra", ",", " ", "start"}], "}"}]}], "]"}], "]"}]}]}]], "Input",\

 CellLabel->"In[53]:="],

Cell[BoxData[
 RowBox[{"globalStart", " ", "=", " ", 
  RowBox[{"marginalRa", "[", 
   RowBox[{"25.0", ",", "500.0"}], "]"}]}]], "Input",
 CellLabel->"In[54]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"marginalBoundary", " ", "=", " ", 
   RowBox[{"Table", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{
       RowBox[{"globalStart", "=", 
        RowBox[{"marginalRa", "[", 
         RowBox[{"k2", ",", "globalStart"}], "]"}]}], ",", 
       RowBox[{"Sqrt", "[", "k2", "]"}]}], "}"}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"k2", ",", " ", "25.0", ",", " ", "100.0", ",", "2.0"}], 
      "}"}]}], "]"}]}], ";"}]], "Input",
 CellLabel->"In[55]:="],

Cell[BoxData[
 RowBox[{"oddMarginalPlt", "=", 
  RowBox[{"ListPlot", "[", 
   RowBox[{"marginalBoundary", ",", 
    RowBox[{"Joined", "\[Rule]", "True"}], ",", 
    RowBox[{"PlotRange", "\[Rule]", 
     RowBox[{"{", 
      RowBox[{
       RowBox[{"{", 
        RowBox[{"0", ",", "9000"}], "}"}], ",", 
       RowBox[{"{", 
        RowBox[{"0", ",", "9"}], "}"}]}], "}"}]}], ",", 
    RowBox[{"PlotLabel", "\[Rule]", "\"\<Marginal Instability\>\""}], ",", 
    RowBox[{"AxesLabel", "\[Rule]", 
     RowBox[{"{", 
      RowBox[{"\"\<Ra\>\"", ",", "\"\<k\>\""}], "}"}]}]}], "]"}]}]], "Input",
 CellLabel->"In[56]:="],

Cell[BoxData[
 RowBox[{"Show", "[", 
  RowBox[{"evenMarginalPlt", ",", "oddMarginalPlt"}], "]"}]], "Input",
 CellLabel->"In[57]:="]
}, Closed]],

Cell[CellGroupData[{

Cell["Critical Mode", "Subsection"],

Cell["\<\
The critical mode and Rayleigh number were given in the textbook.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"critRa", " ", "=", " ", 
  RowBox[{"Min", "[", 
   RowBox[{"First", "[", 
    RowBox[{"Transpose", "[", "marginalBoundary", "]"}], "]"}], 
   "]"}]}]], "Input",
 CellLabel->"In[58]:="],

Cell[BoxData[
 RowBox[{"critk", " ", "=", " ", 
  RowBox[{
   RowBox[{"Select", "[", 
    RowBox[{"marginalBoundary", ",", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"First", "[", "#", "]"}], " ", "\[Equal]", " ", "critRa"}], 
       ")"}], "&"}]}], "]"}], "\[LeftDoubleBracket]", 
   RowBox[{"1", ",", "2"}], "\[RightDoubleBracket]"}]}]], "Input",
 CellLabel->"In[59]:="],

Cell[BoxData[
 RowBox[{"eq1Crit", " ", "=", " ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"eq1", " ", "/.", " ", 
      RowBox[{"k2", " ", "\[Rule]", " ", 
       SuperscriptBox["critk", "2"]}]}], " ", "/.", " ", 
     RowBox[{"Ra", " ", "\[Rule]", " ", "critRa"}]}], "  ", "/.", " ", 
    RowBox[{"a", " ", "\[Rule]", " ", "1"}]}], "//", " ", 
   "Simplify"}]}]], "Input",
 CellLabel->"In[60]:="],

Cell[BoxData[
 RowBox[{"eq2Crit", " ", "=", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{"eq2", " ", "/.", " ", 
      RowBox[{"k2", " ", "\[Rule]", " ", 
       SuperscriptBox["critk", "2"]}]}], " ", "/.", " ", 
     RowBox[{"Ra", " ", "\[Rule]", " ", "critRa"}]}], " ", "/.", " ", 
    RowBox[{"a", " ", "\[Rule]", " ", "1"}]}], " ", "//", " ", 
   "Simplify"}]}]], "Input",
 CellLabel->"In[61]:="],

Cell[BoxData[
 RowBox[{"modeCrit", " ", "=", " ", 
  RowBox[{"First", "[", 
   RowBox[{"Solve", "[", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"eq1Crit", ",", "eq2Crit"}], "}"}], ",", " ", 
     RowBox[{"{", 
      RowBox[{"b", ",", " ", "c"}], "}"}]}], "]"}], "]"}]}]], "Input",
 CellLabel->"In[62]:="],

Cell[BoxData[
 RowBox[{
  RowBox[{"uzCrit", "[", "z_", "]"}], " ", "=", " ", 
  RowBox[{
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"uz", "[", "z", "]"}], " ", "/.", " ", 
      RowBox[{"a", " ", "\[Rule]", " ", "1"}]}], "/.", " ", "modeCrit"}], "/.",
     " ", 
    RowBox[{"k2", " ", "\[Rule]", " ", 
     SuperscriptBox["critk", "2"]}]}], " ", "/.", " ", 
   RowBox[{"Ra", " ", "\[Rule]", " ", "critRa", " "}]}]}]], "Input",
 CellLabel->"In[63]:="],

Cell[BoxData[
 RowBox[{"uzCrit", "[", "0.0", "]"}]], "Input",
 CellLabel->"In[64]:="],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"Evaluate", "[", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"Re", "[", 
       RowBox[{"uzCrit", "[", "z", "]"}], "]"}], ",", 
      RowBox[{"Im", "[", 
       RowBox[{"uzCrit", "[", "z", "]"}], "]"}]}], "}"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"z", ",", 
     RowBox[{"-", "0.499`"}], ",", "0.499`"}], "}"}], ",", 
   RowBox[{"PlotLabel", "\[Rule]", "\"\<Critical Uz[z]\>\""}], ",", 
   RowBox[{"AxesLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"\"\<z\>\"", ",", "\"\<\>\""}], "}"}]}], ",", 
   RowBox[{"PlotRange", "\[Rule]", "All"}], ",", 
   RowBox[{"PlotStyle", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"Blue", ",", "Red"}], "}"}]}]}], "]"}]], "Input",
 CellLabel->"In[65]:="]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Summary", "Section"],

Cell["\<\
The linearized fluid dynamics equations were solved to find the marginal \
instability boundary for Bernard thermal instability.\
\>", "Text"]
}, Closed]]
}, Open  ]]
},
WindowSize->{599, 765},
WindowMargins->{{4, Automatic}, {Automatic, 4}},
PrintingCopies->1,
PrintingPageRange->{1, Automatic},
FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (May 21, 2008)",
StyleDefinitions->"TutorialBook.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[590, 23, 68, 0, 97, "Title"],
Cell[661, 25, 93, 4, 55, "Subsubtitle"],
Cell[CellGroupData[{
Cell[779, 33, 31, 0, 86, "Section"],
Cell[813, 35, 177, 4, 41, "Text"],
Cell[993, 41, 541, 11, 95, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[1571, 57, 45, 0, 54, "Section"],
Cell[1619, 59, 65, 0, 23, "Text"],
Cell[1687, 61, 316, 10, 36, "Input"],
Cell[2006, 73, 244, 9, 25, "Text"],
Cell[2253, 84, 412, 13, 34, "Input"],
Cell[CellGroupData[{
Cell[2690, 101, 36, 0, 34, "Subsection"],
Cell[2729, 103, 110, 5, 23, "Text"],
Cell[2842, 110, 271, 9, 44, "Input"],
Cell[3116, 121, 768, 25, 114, "Input"],
Cell[3887, 148, 180, 5, 41, "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[4116, 159, 46, 0, 54, "Section"],
Cell[4165, 161, 949, 31, 64, "Input"],
Cell[5117, 194, 179, 5, 26, "Input"],
Cell[5299, 201, 344, 11, 26, "Input"],
Cell[5646, 214, 939, 27, 102, "Input"],
Cell[6588, 243, 321, 10, 41, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[6946, 258, 49, 0, 54, "Section"],
Cell[6998, 260, 196, 4, 41, "Text"],
Cell[7197, 266, 83, 2, 26, "Input"],
Cell[7283, 270, 474, 14, 63, "Input"],
Cell[7760, 286, 474, 14, 63, "Input"],
Cell[8237, 302, 474, 14, 63, "Input"],
Cell[8714, 318, 192, 6, 26, "Input"],
Cell[8909, 326, 488, 16, 54, "Input"],
Cell[9400, 344, 115, 3, 26, "Input"],
Cell[9518, 349, 704, 21, 112, "Input"],
Cell[10225, 372, 648, 19, 77, "Input"],
Cell[10876, 393, 282, 9, 44, "Input"],
Cell[11161, 404, 282, 9, 44, "Input"],
Cell[11446, 415, 283, 9, 44, "Input"],
Cell[11732, 426, 395, 12, 64, "Input"],
Cell[12130, 440, 89, 2, 26, "Input"],
Cell[12222, 444, 408, 12, 62, "Input"],
Cell[12633, 458, 614, 16, 97, "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[13284, 479, 32, 0, 54, "Section"],
Cell[13319, 481, 89, 2, 23, "Text"],
Cell[13411, 485, 208, 6, 26, "Input"],
Cell[13622, 493, 396, 11, 62, "Input"],
Cell[14021, 506, 406, 11, 66, "Input"],
Cell[14430, 519, 405, 11, 66, "Input"],
Cell[14838, 532, 308, 9, 26, "Input"],
Cell[15149, 543, 461, 13, 67, "Input"],
Cell[15613, 558, 85, 2, 26, "Input"],
Cell[15701, 562, 482, 13, 62, "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[16220, 580, 31, 0, 54, "Section"],
Cell[CellGroupData[{
Cell[16276, 584, 36, 0, 34, "Subsection"],
Cell[16315, 586, 110, 5, 23, "Text"],
Cell[16428, 593, 223, 7, 26, "Input"],
Cell[16654, 602, 720, 23, 110, "Input"],
Cell[17377, 627, 180, 5, 41, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[17594, 637, 49, 0, 34, "Subsection"],
Cell[17646, 639, 950, 31, 64, "Input"],
Cell[18599, 672, 180, 5, 26, "Input"],
Cell[18782, 679, 345, 11, 26, "Input"],
Cell[19130, 692, 940, 27, 102, "Input"],
Cell[20073, 721, 321, 10, 41, "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell[20431, 736, 52, 0, 34, "Subsection"],
Cell[20486, 738, 196, 4, 41, "Text"],
Cell[20685, 744, 84, 2, 26, "Input"],
Cell[20772, 748, 474, 14, 63, "Input"],
Cell[21249, 764, 474, 14, 63, "Input"],
Cell[21726, 780, 474, 14, 63, "Input"],
Cell[22203, 796, 192, 6, 26, "Input"],
Cell[22398, 804, 488, 16, 54, "Input"],
Cell[22889, 822, 211, 5, 45, "Input"],
Cell[23103, 829, 704, 21, 112, "Input"],
Cell[23810, 852, 649, 19, 77, "Input"],
Cell[24462, 873, 649, 19, 77, "Input"],
Cell[25114, 894, 283, 9, 44, "Input"],
Cell[25400, 905, 283, 9, 44, "Input"],
Cell[25686, 916, 284, 9, 44, "Input"],
Cell[25973, 927, 449, 14, 64, "Input"],
Cell[26425, 943, 161, 4, 26, "Input"],
Cell[26589, 949, 491, 14, 62, "Input"],
Cell[27083, 965, 613, 16, 97, "Input"],
Cell[27699, 983, 131, 3, 26, "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[27867, 991, 35, 0, 34, "Subsection"],
Cell[27905, 993, 89, 2, 23, "Text"],
Cell[27997, 997, 208, 6, 26, "Input"],
Cell[28208, 1005, 396, 11, 62, "Input"],
Cell[28607, 1018, 406, 11, 66, "Input"],
Cell[29016, 1031, 405, 11, 66, "Input"],
Cell[29424, 1044, 308, 9, 45, "Input"],
Cell[29735, 1055, 461, 13, 67, "Input"],
Cell[30199, 1070, 85, 2, 26, "Input"],
Cell[30287, 1074, 760, 21, 79, "Input"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[31096, 1101, 26, 0, 54, "Section"],
Cell[31125, 1103, 152, 3, 41, "Text"]
}, Closed]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)