(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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). 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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[ \(TraditionalForm\`k\^2\)]], "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[ \(eqVert\ = \ \((q\^2\ + \ k\^2)\)\^3\ \[Equal] \ \ \(k\^2\) Ra\)], "Input", CellLabel->"In[1]:="], Cell[TextData[{ "when the vertical velocity has the form, ", Cell[BoxData[ \(TraditionalForm\`U\_z\ ~\ Cos[q\ z]\)]], "." }], "Text"], Cell[BoxData[ \(qSol\ = \ Solve[\((\(eqVert\ /. \ q\^2\ \[Rule] \ q2\)\ /. \ 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[ \(\(Plot[Cos[z], \ {z, \ \(-\[Pi]\)/2, \ \[Pi]/2}];\)\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \(\(Plot[ Evaluate[\({Re[#], Im[#]} &\)[ Cos[z\ \((1\ + \ \[ImaginaryI]\ Sqrt[ 3])\)]]], \ {z, \ \(-\[Pi]\)/2, \ \[Pi]/2}, \ PlotStyle\ \[Rule] \ {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[ \(uz[z_]\ = \ a\ \((Cos[Sqrt[q2]\ z] /. qSol\[LeftDoubleBracket]1\[RightDoubleBracket])\)\ + \ b\ \((Cos[Sqrt[q2]\ z] /. qSol\[LeftDoubleBracket]2\[RightDoubleBracket])\) + \ c\ \((Cos[Sqrt[q2]\ z] /. qSol\[LeftDoubleBracket]3\[RightDoubleBracket])\)\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \(eq1\ = \ uz[1/2]\ \[Equal] \ 0\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \(eq2\ = \((D[uz[z], z]\ \[Equal] \ 0\ /. \ z\ \[Rule] \ 1/2)\)\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(eq3\ = \ \((Collect[ D[uz[z], {z, 4}]\ - \ 2\ k2\ D[uz[z], {z, 2}]\ + \ k2\^2\ uz[z], \ {a, b, c}, \ Simplify] \[Equal] \ 0\ /. \ z\ \[Rule] \ 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[ \(TraditionalForm\`k\^2\)]], "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[ \(First[eq1]\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \(row1\ = \ {Coefficient[First[eq1], \ a], Coefficient[First[eq1], \ b], Coefficient[First[eq1], \ c]}\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(row2\ = \ {Coefficient[First[eq2], \ a], Coefficient[First[eq2], \ b], Coefficient[First[eq2], \ c]}\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(row3\ = \ {Coefficient[First[eq3], \ a], Coefficient[First[eq3], \ b], Coefficient[First[eq3], \ c]}\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(Det[{row1, row2, row3}]\ // \ Simplify\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \(marginal[k2_, Ra_]\ = \ \((\(Det[{row1, row2, row3}]/k2\^\(2/3\)\)/ Ra\^\(2/3\))\)\ // \ Simplify\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \(marginal[2.0, 4000.0]\)], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \(\(Plot[ Evaluate[\({Re[#], Im[#]} &\)[marginal[k2, 4000.0]]], \ {k2, \ 0, \ 50}, \ PlotRange\ \[Rule] \ All, \ AxesLabel\ \[Rule] \ {\*"\"\<\!\(k\^2\)\>\"", "\"}];\)\)], \ "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(\(Plot[ Evaluate[\({Re[#], Im[#]} &\)[marginal[1.0, Ra]]], \ {Ra, \ 0, \ 5000}, \ PlotRange\ \[Rule] \ All, \ AxesLabel\ \[Rule] \ {"\", "\"}];\)\)], "Input", CellLabel->"In[17]:="], Cell[BoxData[ \(FindRoot[marginal[1.0, Ra], \ {Ra, \ 2000.0}]\ // \ Chop\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \(FindRoot[marginal[5.0, Ra], \ {Ra, \ 2000.0}]\ // \ Chop\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \(FindRoot[marginal[50.0, Ra], \ {Ra, \ 2000.0}]\ // \ Chop\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ \(marginalRa[k2_]\ := \ Ra\ /. \ Chop[FindRoot[marginal[k2, Ra], \ {Ra, \ 2000.0}]]\)], "Input",\ CellLabel->"In[21]:="], Cell[BoxData[ \(marginalRa[2.0]\)], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \(\(marginalBoundary\ = \ Table[{marginalRa[k2], Sqrt[k2]}, \ {k2, \ 1.0, \ 50.0, 1.0}];\)\)], "Input", CellLabel->"In[23]:="], Cell[BoxData[ \(\(evenMarginalPlt\ = \ ListPlot[marginalBoundary, PlotJoined\ \[Rule] \ True, \ PlotRange\ \[Rule] \ {{0, 9000}, {0, 9}}, \ PlotLabel\ \[Rule] \ "\", \ AxesLabel\ \[Rule] \ {"\", "\"}];\)\)], "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[ \(critRa\ = \ Min[First[Transpose[marginalBoundary]]]\)], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \(critk\ = \ \(Select[ marginalBoundary, \((First[#]\ \[Equal] \ critRa)\) &]\)\[LeftDoubleBracket]1, 2\[RightDoubleBracket]\)], "Input", CellLabel->"In[26]:="], Cell[BoxData[ \(eq1Crit\ = \ \(\(eq1\ /. \ k2\ \[Rule] \ critk\^2\)\ /. \ Ra\ \[Rule] \ critRa\)\ \ /. \ a\ \[Rule] \ 1 // \ Simplify\)], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \(eq2Crit\ = \(\(eq2\ /. \ k2\ \[Rule] \ critk\^2\)\ /. \ Ra\ \[Rule] \ critRa\)\ /. \ a\ \[Rule] \ 1\ // \ Simplify\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \(modeCrit\ = \ First[Solve[{eq1Crit, eq2Crit}, \ {b, \ c}]]\)], "Input",\ CellLabel->"In[29]:="], Cell[BoxData[ \(uzCrit[ z_]\ = \ \(\(\(uz[z]\ /. \ a\ \[Rule] \ 1\) /. \ modeCrit\) /. \ k2\ \[Rule] \ critk\^2\)\ /. \ \(\(Ra\)\(\ \)\(\[Rule]\)\(\ \)\(critRa\)\(\ \)\ \)\)], "Input", CellLabel->"In[30]:="], Cell[BoxData[ \(uzCrit[0.0]\)], "Input", CellLabel->"In[31]:="], Cell[BoxData[ \(\(Plot[Evaluate[Chop[uzCrit[z]]], \ {z, \ \(-0.499\), \ 0.499}, \ PlotLabel \[Rule] \ "\", \ AxesLabel\ \[Rule] \ {"\", "\<\>"}];\)\)], "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[ \(\(Plot[Sin[z], \ {z, \ \(-\[Pi]\), \ \[Pi]}];\)\)], "Input", CellLabel->"In[33]:="], Cell[BoxData[ \(\(Plot[ Evaluate[\({Re[#], Im[#]} &\)[ Sin[z\ \((1\ + \ \[ImaginaryI]\ Sqrt[ 3])\)]]], \ {z, \ \(-\[Pi]\), \ \[Pi]}, \ PlotStyle\ \[Rule] \ {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[ \(uz[z_]\ = \ a\ \((Sin[Sqrt[q2]\ z] /. qSol\[LeftDoubleBracket]1\[RightDoubleBracket])\)\ + \ b\ \((Sin[Sqrt[q2]\ z] /. qSol\[LeftDoubleBracket]2\[RightDoubleBracket])\) + \ c\ \((Sin[Sqrt[q2]\ z] /. qSol\[LeftDoubleBracket]3\[RightDoubleBracket])\)\)], "Input", CellLabel->"In[35]:="], Cell[BoxData[ \(eq1\ = \ uz[1/2]\ \[Equal] \ 0\)], "Input", CellLabel->"In[36]:="], Cell[BoxData[ \(eq2\ = \((D[uz[z], z]\ \[Equal] \ 0\ /. \ z\ \[Rule] \ 1/2)\)\)], "Input", CellLabel->"In[37]:="], Cell[BoxData[ \(eq3\ = \ \((Collect[ D[uz[z], {z, 4}]\ - \ 2\ k2\ D[uz[z], {z, 2}]\ + \ k2\^2\ uz[z], \ {a, b, c}, \ Simplify] \[Equal] \ 0\ /. \ z\ \[Rule] \ 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[ \(TraditionalForm\`k\^2\)]], "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[ \(First[eq1]\)], "Input", CellLabel->"In[39]:="], Cell[BoxData[ \(row1\ = \ {Coefficient[First[eq1], \ a], Coefficient[First[eq1], \ b], Coefficient[First[eq1], \ c]}\)], "Input", CellLabel->"In[40]:="], Cell[BoxData[ \(row2\ = \ {Coefficient[First[eq2], \ a], Coefficient[First[eq2], \ b], Coefficient[First[eq2], \ c]}\)], "Input", CellLabel->"In[41]:="], Cell[BoxData[ \(row3\ = \ {Coefficient[First[eq3], \ a], Coefficient[First[eq3], \ b], Coefficient[First[eq3], \ c]}\)], "Input", CellLabel->"In[42]:="], Cell[BoxData[ \(Det[{row1, row2, row3}]\ // \ Simplify\)], "Input", CellLabel->"In[43]:="], Cell[BoxData[ \(marginal[k2_, Ra_]\ = \ \((\(Det[{row1, row2, row3}]/k2\^\(2/3\)\)/ Ra\^\(2/3\))\)\ // \ Simplify\)], "Input", CellLabel->"In[44]:="], Cell[BoxData[{ \(marginal[50.0, 4000.0]\), "\[IndentingNewLine]", \(marginal[50.0, 1000.0]\)}], "Input", CellLabel->"In[45]:="], Cell[BoxData[ \(\(Plot[ Evaluate[\({Re[#], Im[#]} &\)[marginal[k2, 4000.0]]], \ {k2, \ 0, \ 70}, \ PlotRange\ \[Rule] \ All, \ AxesLabel\ \[Rule] \ {\*"\"\<\!\(k\^2\)\>\"", "\"}];\)\)], \ "Input", CellLabel->"In[47]:="], Cell[BoxData[ \(\(Plot[ Evaluate[\({Re[#], Im[#]} &\)[marginal[50.0, Ra]]], \ {Ra, \ 0, \ 5000}, \ PlotRange\ \[Rule] \ All, \ AxesLabel\ \[Rule] \ {"\", "\"}];\)\)], "Input", CellLabel->"In[48]:="], Cell[BoxData[ \(\(Plot[ Evaluate[\({Re[#], Im[#]} &\)[marginal[25.0, Ra]]], \ {Ra, \ 0, \ 5000}, \ PlotRange\ \[Rule] \ All, \ AxesLabel\ \[Rule] \ {"\", "\"}];\)\)], "Input", CellLabel->"In[49]:="], Cell[BoxData[ \(FindRoot[marginal[25.0, Ra], \ {Ra, \ 3000.0}]\ // \ Chop\)], "Input", CellLabel->"In[50]:="], Cell[BoxData[ \(FindRoot[marginal[50.0, Ra], \ {Ra, \ 3000.0}]\ // \ Chop\)], "Input", CellLabel->"In[51]:="], Cell[BoxData[ \(FindRoot[marginal[100.0, Ra], \ {Ra, \ 3000.0}]\ // \ Chop\)], "Input",\ CellLabel->"In[52]:="], Cell[BoxData[ \(marginalRa[k2_, start_: 3000.0]\ := \ Ra\ /. \ Chop[FindRoot[marginal[k2, Ra], \ {Ra, \ start}]]\)], "Input",\ CellLabel->"In[53]:="], Cell[BoxData[ \(globalStart\ = \ marginalRa[25.0, 500.0]\)], "Input", CellLabel->"In[54]:="], Cell[BoxData[ \(\(marginalBoundary\ = \ Table[{globalStart = marginalRa[k2, globalStart], Sqrt[k2]}, \ {k2, \ 25.0, \ 100.0, 2.0}];\)\)], "Input", CellLabel->"In[55]:="], Cell[BoxData[ \(\(oddMarginalPlt\ = \ ListPlot[marginalBoundary, PlotJoined\ \[Rule] \ True, \ PlotRange\ \[Rule] \ {{0, 9000}, {0, 9}}, \ PlotLabel\ \[Rule] \ "\", \ AxesLabel\ \[Rule] \ {"\", "\"}];\)\)], "Input", CellLabel->"In[56]:="], Cell[BoxData[ \(\(Show[evenMarginalPlt, oddMarginalPlt];\)\)], "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[ \(critRa\ = \ Min[First[Transpose[marginalBoundary]]]\)], "Input", CellLabel->"In[58]:="], Cell[BoxData[ \(critk\ = \ \(Select[ marginalBoundary, \((First[#]\ \[Equal] \ critRa)\) &]\)\[LeftDoubleBracket]1, 2\[RightDoubleBracket]\)], "Input", CellLabel->"In[59]:="], Cell[BoxData[ \(eq1Crit\ = \ \(\(eq1\ /. \ k2\ \[Rule] \ critk\^2\)\ /. \ Ra\ \[Rule] \ critRa\)\ \ /. \ a\ \[Rule] \ 1 // \ Simplify\)], "Input", CellLabel->"In[60]:="], Cell[BoxData[ \(eq2Crit\ = \(\(eq2\ /. \ k2\ \[Rule] \ critk\^2\)\ /. \ Ra\ \[Rule] \ critRa\)\ /. \ a\ \[Rule] \ 1\ // \ Simplify\)], "Input", CellLabel->"In[61]:="], Cell[BoxData[ \(modeCrit\ = \ First[Solve[{eq1Crit, eq2Crit}, \ {b, \ c}]]\)], "Input",\ CellLabel->"In[62]:="], Cell[BoxData[ \(uzCrit[ z_]\ = \ \(\(\(uz[z]\ /. \ a\ \[Rule] \ 1\) /. \ modeCrit\) /. \ k2\ \[Rule] \ critk\^2\)\ /. \ \(\(Ra\)\(\ \)\(\[Rule]\)\(\ \)\(critRa\)\(\ \)\ \)\)], "Input", CellLabel->"In[63]:="], Cell[BoxData[ \(uzCrit[0.0]\)], "Input", CellLabel->"In[64]:="], Cell[BoxData[ \(\(Plot[ Evaluate[{Re[uzCrit[z]], Im[uzCrit[z]]}], \ {z, \ \(-0.499\), \ 0.499}, \ PlotLabel \[Rule] \ "\", \ AxesLabel\ \[Rule] \ {"\", "\<\>"}, \ PlotRange\ \[Rule] \ All, \ PlotStyle\ \[Rule] \ {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 ]] }, FrontEndVersion->"5.2 for Macintosh", ScreenRectangle->{{0, 1600}, {0, 998}}, WindowSize->{599, 765}, WindowMargins->{{4, Automatic}, {Automatic, 4}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, StyleDefinitions -> "TutorialBook.nb" ] (******************************************************************* Cached data follows. 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