(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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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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[ 244979, 7434]*) (*NotebookOutlinePosition[ 260445, 7997]*) (* CellTagsIndexPosition[ 260281, 7988]*) (*WindowFrame->Normal*) Notebook[{ Cell["Linear stability of long waves in two-layer channel flow ", "Title"], Cell[TextData[{ StyleBox["This notebook has been written in ", FontSize->12], StyleBox["Mathematica ", FontSize->12, FontSlant->"Italic"], StyleBox["by \n\n", FontSize->12], StyleBox[ "Mark J. McCready\nProfessor and Chair of Chemical Engineering\nUniversity \ of Notre Dame\nNotre Dame IN 46556\nUSA", FontSize->14], StyleBox["\n\nMark.J.McCready.1@nd.edu\n", FontSize->12], ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\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, if you use it, this notice remain visible to other users. \nThere \ is no charge for copying and dissemination \n\n", StyleBox[ "Version: 5/14/99\nMore recent versions of this notebook should be \ available at the web site:\n", FontSize->12], ButtonBox["http://www.nd.edu/~mjm/longwave.nb", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/longwave.nb"], None}, ButtonStyle->"Hyperlink"] }], "Text", FontSize->10], Cell[CellGroupData[{ Cell["Summary", "Subtitle", Background->RGBColor[0.8, 0.919997, 0.919997]], Cell[TextData[{ "This notebook solves the longwave stability problem for two-layer, \ pressure driven channel flow. Through this example, the method for using a \ regular perturbation technique to solve a differential eigenvalue problem, \ with boundary conditions, is demonstrated. \n", StyleBox["\n", FontSize->9], StyleBox["References: \nC. -S Yih (1967) ", "SmallText", FontSize->10], StyleBox[ "\"Instability due to viscosity stratification\", J. Fluid Mech.,", "SmallText", FontSize->10, FontWeight->"Plain"], StyleBox[" ", "SmallText", FontSize->10], StyleBox["27", "SmallText", FontSize->10, FontWeight->"Bold"], StyleBox[" ", "SmallText", FontSize->10], StyleBox["pp337-352", "SmallText", FontSize->10, FontWeight->"Plain"], StyleBox[".\nS. G. Yiantsios and B. G. Higgins (1988) ", "SmallText", FontSize->10], StyleBox[ "\"Linear stability of plane Poiseullie flow of two-superposed fluids\", \ Phys. Fluids", "SmallText", FontSize->10, FontWeight->"Plain"], StyleBox[", ", "SmallText", FontSize->10], StyleBox["31", "SmallText", FontSize->10, FontWeight->"Bold"], StyleBox[" ", "SmallText", FontSize->10], StyleBox["pp3225-3238", "SmallText", FontSize->10, FontWeight->"Plain"], StyleBox[ ".\n\nThe 0 order and first order terms for the wave velocity for a \ pressure driven channel flow are obtained from a long wave expansion. All of \ the necessary manipulations are shown for this direct perturbation solution \ method. ", "SmallText", FontSize->10] }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->12], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " aside" }], "Subsection"], Cell[TextData[{ "In ", StyleBox["Mathematica", FontSlant->"Italic"], ", it is convenient to give all expressions a \"name\". I try to pick ones \ that are consistent with what is being done (but sometimes \"temp\" is used). \ This assignment is done with an \"=\" sign. To make an equation, a \"==\" \ is used. This distinction is very useful in computer algebra and is employed \ in all of the packages with which I am familiar ." }], "Text"], Cell[CellGroupData[{ Cell["Input notation", "Subsubsection"], Cell["\<\ I would enter the dynamic boundary condition from a key pad as\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(dc = \(-\ \[Gamma]\)\ D[\[Eta][x, t], {x, 2}]\ - \n\t\t \(\[Rho]\_1\) \((\(-D[\[Phi]\_1[x, y, t], t]\) - U1\ D[\[Phi]\_1[x, y, t], x] - g\ \[Eta][x, t])\) + \n\t\t\t \(\[Rho]\_2\) \((\(-D[\[Phi]\_2[x, y, t], t]\) - \n\t\t\t\t\t U2\ D[\[Phi]\_2[x, y, t], x] - g\ \[Eta][x, t])\)\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-\[Gamma]\), " ", RowBox[{ SuperscriptBox["\[Eta]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(x, t\), ")"}]}], "-", RowBox[{\(\[Rho]\_1\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U1", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}], "+", RowBox[{\(\[Rho]\_2\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U2", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "In ", StyleBox["Standard form", FontSlant->"Italic"], ", which is a little shorter, you would need the typeset window to make \ this practical. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(dc = \(-\((\[Gamma]\ \[PartialD]\_{x, 2}\[Eta][x, t])\)\) - \[Rho]\_1\ \((\(-\[PartialD]\_t \[Phi]\_1[x, y, t]\) - U1\ \[PartialD]\_x \[Phi]\_1[x, y, t] - g\ \[Eta][x, t])\) + \[Rho]\_2\ \((\(-\[PartialD]\_t \[Phi]\_2[x, y, t]\) - U2\ \[PartialD]\_x \[Phi]\_2[x, y, t] - g\ \[Eta][x, t])\)\)], "Input", CellTags->"standard"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-\[Gamma]\), " ", RowBox[{ SuperscriptBox["\[Eta]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(x, t\), ")"}]}], "-", RowBox[{\(\[Rho]\_1\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U1", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}], "+", RowBox[{\(\[Rho]\_2\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U2", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[ButtonBox["Back to start of calculations.", ButtonData:>"start", ButtonStyle->"Hyperlink"]], "Text"], Cell[TextData[{ "Finally it may be more easily ", StyleBox["read", FontVariations->{"Underline"->True}], " in ", StyleBox["Traditional form", FontSlant->"Italic"], " as shown here. Note that this particular cell is inactive and will not \ be evaluated. You can change this, if you really want to, by selecting the \ cell, going into the \"preferences\" under the Edit menu andunde Cell \ Options, Evaluation Options, set Evaluatable to True. " }], "Text", CellTags->"traditional"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"dc", "=", RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{"\[Gamma]", " ", FractionBox[\(\[PartialD]\^2\( \[Eta](x, t)\)\), \(\[PartialD]x\^2\), MultilineFunction->None]}], ")"}]}], "-", RowBox[{\(\[Rho]\_1\), " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[\(\[PartialD]\[Phi]\_1[x, y, t]\), \(\[PartialD]t\), MultilineFunction->None]}], "-", RowBox[{"U1", " ", FractionBox[\(\[PartialD]\[Phi]\_1[x, y, t]\), \(\[PartialD]x\), MultilineFunction->None]}], "-", \(g\ \(\[Eta](x, t)\)\)}], ")"}]}], "+", RowBox[{\(\[Rho]\_2\), " ", RowBox[{"(", RowBox[{ RowBox[{"-", FractionBox[\(\[PartialD]\[Phi]\_2[x, y, t]\), \(\[PartialD]t\), MultilineFunction->None]}], "-", RowBox[{"U2", " ", FractionBox[\(\[PartialD]\[Phi]\_2[x, y, t]\), \(\[PartialD]x\), MultilineFunction->None]}], "-", \(g\ \(\[Eta](x, t)\)\)}], ")"}]}]}]}], TraditionalForm]], "Input", Evaluatable->False, CellLabelAutoDelete->True], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(-\[Gamma]\), " ", RowBox[{ SuperscriptBox["\[Eta]", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "(", \(x, t\), ")"}]}], "-", RowBox[{\(\[Rho]\_1\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U1", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}], "+", RowBox[{\(\[Rho]\_2\), " ", RowBox[{"(", RowBox[{\(\(-g\)\ \(\[Eta](x, t)\)\), "-", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}], "-", RowBox[{"U2", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "2", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "(", \(x, y, t\), ")"}]}]}], ")"}]}]}], TraditionalForm]], "Output", Evaluatable->False, CellLabelAutoDelete->True] }, Open ]], Cell[TextData[ButtonBox["Back to start of calculations.", ButtonData:>"start", ButtonStyle->"Hyperlink"]], "Text"], Cell["\<\ Note that you can convert any \"cryptic\" input expression by \ selecting the cell, going up to the cell menu and selecting \"convert to \ traditional form\". This is \"shift curly t\" on the keypad. \ \>", "Text"] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["How to use this notebook", "Subtitle"], Cell[TextData[{ "The best way to use this notebook is to open it in ", StyleBox["Mathematica", FontSlant->"Italic"], " and work through the examples and make changes in parameters or \ procedural steps to ", StyleBox["explore", FontSlant->"Italic"], " these problems. On line help is available in ", StyleBox["Mathematica", FontSlant->"Italic"], " 3 so that a definition and in most cases examples for any unknown command \ can be obtained. If you do not have a license for ", StyleBox["Mathematica", FontSlant->"Italic"], ", you can download ", ButtonBox["MathReader", ButtonData:>{ URL[ "http://www.wolfram.com/mathreader/"], None}, ButtonStyle->"Hyperlink"], " (http://www.wolfram.com/mathreader/) free of charge. It does not let you \ change anything or run the calculations, but it does allow full access to the \ notebook. \n\nOther notebooks that cover a range of fluid dynamics problems \ are available at:\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\nThese can also be used to explore multifluid flows and learn more \ about using ", StyleBox["Mathematica", FontSlant->"Italic"], ". " }], "Text", CellTags->"howto"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Motivation: Importance of waves; Importance of long wave stability on flow regime transitions.\ \>", "Subtitle",\ Background->RGBColor[0.8, 0.919997, 0.919997]], Cell[TextData[{ "Waves form as the result of wind on natural bodies of water and contribute \ to increased gas transfer and droplet production. Despite the high Reynolds \ numbers involved, these originate as a hydrodynamic instability that requires \ a fully-viscous solution of the linearized Navier-Stokes equations to \ characterize. You may also observe waves on the windshield of your car \ during a rain storm or in the car wash or on the wing of an airplane -- \ perhaps as the deicing fluid is sheared of. These examples involve much \ lower liquid Reynolds numbers and (with some additional simplifications) can \ be analyzed with wave equations similar to the example used in the first part \ below. \n\nA big area of interest in interfacial waves are those that occur \ in gas-liquid (i.e., two-phase flow) pipelines and process equipment. In \ these situations they greatly increase the pressure drop and interface \ transport rates, and can lead to flow regime transitions. Some examples of \ these kinds of waves are available as still pictures and movies at ", ButtonBox["http://www.nd.edu/~mjm/waves.descrip.html", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/waves.descrip.html"], None}, ButtonStyle->"Hyperlink"], ". \n\nThe importance of flow regime is demonstrated in ", ButtonBox[ "Demonstration of the effect of flow regime on pressure drop in multifluid \ flows", ButtonData:>{ URL[ "http://www.mathsource.com/cgi-bin/msitem?0210-171"], None}, ButtonStyle->"Hyperlink"], ". For regime transitions where formation of waves are important, \ stratified to slug or stratified to annular, it is the growth of long waves \ that have been implicated in the transition. \n\nThis notebook also shows \ how to use regular perturbation theory to solve an eigenvalue problem that \ involves boundary conditions. " }], "Text"], Cell[CellGroupData[{ Cell["Examples of flow regimes (video clips)", "SectionFirst", CellTags->"movies"], Cell[TextData[ButtonBox["[BACK] to \"How to\"", ButtonData:>"howto", ButtonStyle->"Hyperlink"]], "Text"], Cell[TextData[{ "When a gas and a liquid are forced to flow together inside a pipe, there \ are at least 7 different geometrical configurations, or flow regimes, that \ are observed to occur. The regime depends on the fluid properties, the size \ of the conduit and the flow rates of each of the phases. The flow regime can \ also depend on the configuration of the inlet; the flow regime may take some \ distance to develop and it can change with distance as (perhaps) the \ pressure, which affects the gas density, changes. For fixed fluid properties \ and conduit, the flow rates are the independent variables that when adjusted \ will often lead to changes in the flow regime.\n\nAir-water flow in a 1.27 cm \ diameter pipe oriented horizontally. \n\nThese videos are from ", StyleBox[ "\"Two Phase Flow Regimes in Reduced Gravity\", NASA Lewis Research Center \ Motion Picture Directory 1704", FontWeight->"Bold"], ".\n\n By ", StyleBox["J. B. McQuillen, R. Vernon and A. E. Dukler. \n \n", FontWeight->"Bold"], "The video was taken at 400 fps and the projection is at 29.97 fps\n\n", StyleBox["Earth gravity", FontWeight->"Bold"], "\n\n", ButtonBox["Bubbly flow(.mov movie)", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/1gbubbly.mov"], None}, ButtonStyle->"Hyperlink"], "\n\nSuperficial gas velocity = .16 m/s, Superficial liquid velocity = .90 \ m/s.\n\nIn this example of bubbly flow, the liquid flow rate is high enough \ to break up the gas into bubbles, but it is not high enough to cause the \ bubbles to become mixed well within the liquid phase. The Froude number, \n\n\ ", Cell[BoxData[ \(TraditionalForm\`F\)]], StyleBox[" = ", FontSize->15], Cell[BoxData[ \(TraditionalForm\`\(g\ d\)\/U\_L\^2\)], FontSize->15], " = .15\n\nwhere ", StyleBox["g", FontSlant->"Italic"], " is gravitation acceleration, ", Cell[BoxData[ \(TraditionalForm\`d\)]], " is the pipe diameter and ", Cell[BoxData[ \(TraditionalForm\`U\_L\)]], " is liquid superficial velocity, shows that while inertia is slightly \ larger than gravity forces, the experiments show that there is not enough \ mixing to scatter the bubbles. \n\nIf the pipe were oriented vertically, the \ phase orientation would be symmetric, but there would likely be \"slip\" \ between the phases and the gas would not move at the same speed as the \ liquid. \n\n", ButtonBox["Annular Flow (.mov movie)", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/1gannular.mov"], None}, ButtonStyle->"Hyperlink"], "\n\nSuperficial gas velocity = 7.4 m/s, Superficial liquid velocity = .08 \ m/s.\n\nIn annular flow, the liquid coats the walls. However, because of \ gravity, the liquid distribution is not symmetric. There is much more liquid \ on the bottom of the pipe than the top. The velocity of the gas is large \ enough to cause waves to form in the liquid and also to atomize some liquid. \ The maximum possible wave amplitudes scale, for liquid layers that are not \ too thin, as roughly the liquid thickness. \n\n", ButtonBox["Slug flow (.mov movie)", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/1gslug.mov"], None}, ButtonStyle->"Hyperlink"], "\n\nSuperficial gas velocity = .17 m/s, Superficial liquid velocity = .08 \ m/s.\n\nThe slug regime, is characterized by the presence of liquid rich ", StyleBox["slugs", FontSlant->"Italic"], " that span the entire channel or pipe diameter. These travel at a speed \ that is a substantial fraction of the gas velocity and they occur \ intermittently. Slugs cause large pressure and liquid flow rate \ fluctuations. The movie shows the approach of first a large wave and later a \ long slug. Other movies of slugs would show much more gas entrainment and a \ flow that looks much more violent. The length to diameter ratio of slugs \ varies greatly with flow rates, pipe diameter and fluid properties. If the \ diameter is very large, ", StyleBox["F", FontSlant->"Italic"], " can always be large and slug flow, where the entire diameter is bridged, \ will not form. Instead roll waves waves, which are breaking traveling waves, \ will be seen. Liquid may or may not coat the entire pipe because there will \ be substantial atomization. \n\n\n", StyleBox[ "Here are the same flow rates if gravity is reduced to an insignificant \ level.", FontWeight->"Bold"], " \n\n", ButtonBox["Bubbly flow", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/0gbubbly.mov"], None}, ButtonStyle->"Hyperlink"], "\n\nSuperficial gas velocity = .13 m/s, Superficial liquid velocity = .89 \ m/s.\n\nIn this example of bubbly flow, there is no gravity so that there is \ no buoyancy force on the gas bubbles. Thus they mix freely within the \ liquid. While there is definitely a continuous phase (liquid) and a \ dispersed phase (gas), this is close to the idealized ", StyleBox["homogenous", FontSlant->"Italic"], " or ", StyleBox["dispersed", FontSlant->"Italic"], " flow that will be used for comparison in the examples below. \n\n", ButtonBox["Annular flow", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/0gannular.mov"], None}, ButtonStyle->"Hyperlink"], "\n\nSuperficial gas velocity = 8.1 m/s, Superficial liquid velocity = .08 \ m/s.\n\nNow that gravity is removed, the liquid distribution is uniform \ around the pipe. Large disturbance waves still occur but they are seen as \ \"ring - like\" disturbances. The absence of gravity also increases the \ amplitude to film depth ratio of traveling waves because there is no liquid \ drainage from the wave caused by gravity. \n\n", ButtonBox["Slug Flow", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/0gslug.mov"], None}, ButtonStyle->"Hyperlink"], "\n\nSuperficial gas velocity = .16 m/s, Superficial liquid velocity = .08 \ m/s.\n\nIn the absence of gravity, the liquid distribution is uniform and the \ slugs are now liquid \"trapped\" between traveling \"Taylor Bubbles\". This \ flow will not experience large pressure fluctuations and the flow rate \ fluctuations occur only on the size of the bubbles. This is close to the \ idealized ", StyleBox["slug", FontSlant->"Italic"], " regime that is considered in the calculations below. \n\nThe calculations \ below show directly that for the easily calculable case of laminar flow, the \ flow regime greatly influences the pressure - drop flow rate relation. Thus, \ if design or operation of a device requires accurate knowledge of flow rate \ and pressure drop, there is a need to know the flow regime. The rates of \ heat and mass transfer are also often important in process equipment and the \ movies suggest that these will also depend significantly on the flow regime!\n\ " }], "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["The base state flow", "Subtitle", CellLabelAutoDelete->True], Cell["\<\ This notebook addresses the case of Poiseulle flow so that the \ walls are fixed. The upper fluid is 1 and the lower is 2, the velocity at the interface, y=0, \ is used to normalize the velocity so that we have\ \>", "Text", Evaluatable->False, CellLabelAutoDelete->True, AspectRatioFixed->True], Cell[BoxData[ \(\(u1\ = 1 + a1\ y + b1\ y\^2; \)\)], "Input", CellLabelAutoDelete->True, AspectRatioFixed->True], Cell[BoxData[ \(\(u2\ = 1 + a2\ y + b2\ y\^2; \)\)], "Input", CellLabelAutoDelete->True, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(The\ constants\ are\ given\ here\ but\ we\ will\ substitute\ these\ as\ needed\ later\)], "Text", Evaluatable->False, CellLabelAutoDelete->True, AspectRatioFixed->True], Cell[BoxData[ \(\(a1 = \(m - n\^2\)\/\(n\^2 + n\); \)\)], "Text", CellLabelAutoDelete->True, AspectRatioFixed->True], Cell[BoxData[ \(\(b1 = \(-\(\(m + n\)\/\(n\^2 + n\)\)\); \)\)], "Text", CellLabelAutoDelete->True, AspectRatioFixed->True], Cell[BoxData[ \(\(a2 = a1\/m; \)\)], "Text", CellLabelAutoDelete->True, AspectRatioFixed->True], Cell[BoxData[ \(\(b2 = b1\/m; \)\)], "Text", CellLabelAutoDelete->True, AspectRatioFixed->True] }, Open ]], Cell[TextData[{ "The m's and n's are the viscosity ratio, m= ", Cell[BoxData[ \(\[Mu]\_2\/\[Mu]\_1\)]], " and the thickness ratio, n=", Cell[BoxData[ \(d\_2\/d\_1\)]], ". 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Thus we must expand \ the velocity to include the effect of the average velocity\n \n ***\t \ \[Phi]'[0] + \[Eta] ", Cell[BoxData[ \(U1\&_\)]], "'[y] = \[Psi]'[0] + \[Eta] ", Cell[BoxData[ \(U2\&_\)]], "'[y]\n \n this is\n \n\[Eta] is given from the Kinematic condition D[\ \[Eta],t] = ", Cell[BoxData[ \(v\_y\)]], "[0]\n\nsubstituting gives:\n\n*** \t\[Phi]'[0]-\[Psi]'[0] = \ \[Phi][0](a2-a1)/(c-", Cell[BoxData[ \(U\&_\)]], "[0])\n\nNext the normal velocity at the interface \n\nd[\[Psi],x] = d[\ \[Phi],x] which gives simply\n\n*** \t\[Psi][0] = \[Phi][0]\n\nNext, the \ tangential stress is continuous across the interface\n\n", Cell[BoxData[ \(\[Mu]\_1\)]], " (\[PartialD]u1/\[PartialD]y + \[PartialD]v1/\[PartialD]x) = ", Cell[BoxData[ \(\[Mu]\_2\)]], " (\[PartialD]u2/\[PartialD]y + \[PartialD]v2/\[PartialD]x)\n\t\t\t\nwhich \ gives\n\t\t\t\n*** \t\[Phi]''[0] + ", Cell[BoxData[ \(\[Alpha]\^2\)]], " \[Phi][0] = m (\[Psi]''[0] + ", Cell[BoxData[ \(\[Alpha]\^2\)]], " \[Psi][0])\n\t\t\t\nFinally, we have the continuity of Normal stress. \ This is comprised of, in dimensional terms,\n\n(p - \[Rho] g a - \[Mu] dv/dy \ ) = \[Sigma] \[PartialD]", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], " \[Eta]/\[PartialD]x", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], " \n\t\t\nwhich says that the pressure change differs by the surface \ tension jump. \n\t\t\nWe substitute for the pressure from the governing \ equation and make some other simplifications and get\n\t\t\n\t\t\ -\[ImaginaryI] \[Alpha] R ((c-", Cell[BoxData[ \(U\&_\)]], "[0]) \[Phi]'[0] + a1 \[Phi][0]) - \[Phi]'''[0] + 3 ", Cell[BoxData[ \(\[Alpha]\^2\)]], " \[Phi]'[0] +\n\t\t \[ImaginaryI] \[Alpha] R r ((c-", Cell[BoxData[ \(U\&_\)]], "[0]) \[Psi]'[0] + a2 \[Psi][0]) + m( \[Psi]'''[0] - 2 ", Cell[BoxData[ \(\[Alpha]\^2\)]], " \[Psi]'[0]) = \n\t\t \t\t I \[Alpha] R (F + ", Cell[BoxData[ \(\[Alpha]\^2\)]], " S) \[Phi][0]/(c-", Cell[BoxData[ \(U\&_\)]], "[0])" }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Expansions for the stream functions and wave speed", "Subsubsection"], Cell[TextData[ "The problem will now be solved using the long wave expansion, i.e. \ \[Alpha]-->0 so that that we can write the stream functions and the wave \ speed as perturbation expansions in \[Alpha]. 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(9)", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c00\ = 1 + \ \((2\ \((m - n^2)\)\ \((m - 1)\)\ \((n^3 + n^2)\))\)/\n\t \((\((n^2 + n)\) \((n^4 + 4\ n^3\ m\ + \ 6\ n^2\ m + \ 4\ n\ m\ + \ m^2)\)) \)\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(2\ \((m - 1)\)\ \((m - n\^2)\)\ \((n\^3 + n\^2)\)\)\/\(\((n\^2 + n)\)\ \((n\^4 + 4\ m\ n\^3 + 6\ m\ n\^2 + 4\ m\ n + m\^2)\)\) + 1\)], "Output"] }, Open ]], Cell["It works!!", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(check = Simplify[c00 - c0eig]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`0\)], "Output"] }, Open ]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Here is the solution to the first order problem", "Subtitle"], Cell[CellGroupData[{ Cell["Here are the equations for the first order problem", "Subsubsection"], Cell[CellGroupData[{ Cell["os1O1", "Input"], Cell[BoxData[ FormBox[ RowBox[{\(-\[ImaginaryI]\), " ", RowBox[{"(", RowBox[{\(\(-2\)\ b1\ R\ \(\(\[Phi]\_0\)(y)\)\), "+", RowBox[{"R", " ", \((b1\ y\^2 + a1\ y - c0 + 1)\), " ", RowBox[{ SubsuperscriptBox["\[Phi]", "0", "\[DoublePrime]", MultilineFunction->None], "(", "y", ")"}]}], "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "y", ")"}]}]}], ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["os2O1", "Input"], Cell[BoxData[ FormBox[ RowBox[{"-", FractionBox[ RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{\(\(-2\)\ b2\ r\ R\ \(\(\[Psi]\_0\)(y)\)\), "+", RowBox[{ "r", " ", "R", " ", \((b2\ y\^2 + a2\ y - c0 + 1)\), " ", RowBox[{ SubsuperscriptBox["\[Psi]", "0", "\[DoublePrime]", MultilineFunction->None], "(", "y", ")"}]}], "+", RowBox[{"\[ImaginaryI]", " ", "m", " ", RowBox[{ SubsuperscriptBox["\[Psi]", "1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "y", ")"}]}]}], ")"}]}], "m"]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["bc1O1", "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Phi]", "1", "\[Prime]", MultilineFunction->None], "(", "1", ")"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["bc2O1", "Input"], Cell[BoxData[ \(TraditionalForm\`\(\[Phi]\_1\)(1)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[" bc3O1 ", "Input"], Cell[BoxData[ FormBox[ RowBox[{ \(-\(\(a1\ \(( c1\ \(\(\[Phi]\_0\)(0)\) - \((c0 - 1)\)\ \(\(\[Phi]\_1\)(0)\))\)\)\/\((c0 - 1)\)\^2 \)\), "+", \(\(a2\ \(( c1\ \(\(\[Phi]\_0\)(0)\) - \((c0 - 1)\)\ \(\(\[Phi]\_1\)(0)\)) \)\)\/\((c0 - 1)\)\^2\), "+", RowBox[{ SubsuperscriptBox["\[Phi]", "1", "\[Prime]", MultilineFunction->None], "(", "0", ")"}], "-", RowBox[{ SubsuperscriptBox["\[Psi]", "1", "\[Prime]", MultilineFunction->None], "(", "0", ")"}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["bc4O1", "Input"], Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\_1\)(0) - \(\[Phi]\_1\)(0)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["bc5O1", "Input"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubsuperscriptBox["\[Phi]", "1", "\[DoublePrime]", MultilineFunction->None], "(", "0", ")"}], "-", RowBox[{"m", " ", RowBox[{ SubsuperscriptBox["\[Psi]", "1", "\[DoublePrime]", MultilineFunction->None], "(", "0", ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["bc6O1", "Input"], Cell[BoxData[ FormBox[ RowBox[{ \(\(-\[ImaginaryI]\)\ a1\ R\ \(\(\[Phi]\_0\)(0)\)\), "-", \(\(\[ImaginaryI]\ F\ R\ \(\(\[Phi]\_0\)(0)\)\)\/\(c0 - 1\)\), "+", \(\[ImaginaryI]\ a2\ r\ R\ \(\(\[Psi]\_0\)(0)\)\), "-", RowBox[{"\[ImaginaryI]", " ", "c0", " ", "R", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "0", "\[Prime]", MultilineFunction->None], "(", "0", ")"}]}], "+", RowBox[{"\[ImaginaryI]", " ", "R", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "0", "\[Prime]", MultilineFunction->None], "(", "0", ")"}]}], "+", RowBox[{"\[ImaginaryI]", " ", "c0", " ", "r", " ", "R", " ", RowBox[{ SubsuperscriptBox["\[Psi]", "0", "\[Prime]", MultilineFunction->None], "(", "0", ")"}]}], "-", RowBox[{"\[ImaginaryI]", " ", "r", " ", "R", " ", RowBox[{ SubsuperscriptBox["\[Psi]", "0", "\[Prime]", MultilineFunction->None], "(", "0", ")"}]}], "-", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((3)\), Derivative], MultilineFunction->None], "(", "0", ")"}], "+", RowBox[{"m", " ", RowBox[{ SubsuperscriptBox["\[Psi]", "1", TagBox[\((3)\), Derivative], MultilineFunction->None], "(", "0", ")"}]}]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["bc7O1", "Input"], Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox["\[Psi]", "1", "\[Prime]", MultilineFunction->None], "(", \(-n\), ")"}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["bc8O1", "Input"], Cell[BoxData[ \(TraditionalForm\`\(\[Psi]\_1\)(\(-n\))\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["The solution to the first order problem", "Subsubsection"], Cell["\<\ We are fortunate again that c1 appears only in bc3. This condition \ will be saved for last. We will solve the two OS equations and boundary conditions in three sets \ because they are so big. \ \>", "Text"], Cell[CellGroupData[{ Cell["Start with the upper phase OS eq, + top wall BC's ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eqs12 = Simplify[{os1O1 == 0, \ bc1O1 == 0, bc2O1 == 0} /. { \[Phi]\_0[0] -> \((phi0 /. y -> 0)\)\ , \ \ \ \ \(\[Phi]\_0'\)[0] -> \((D[phi0, y] /. y -> 0)\), \n\t\t\t \(\(\[Phi]\_0'\)'\)[0] -> \((D[phi0, {y, 2}] /. y -> 0)\), \n \t\t\t\(\(\(\[Phi]\_0'\)'\)'\)[0] -> \((D[phi0, {y, 3}] /. y -> 0)\), \n\t\t\t\[Phi]\_0[y] -> phi0, D[\ \[Phi]\_0[y], {y, a1_}] :> D[phi0, {y, a1}], \n\t\t\ \ \ \[Psi]\_0[0] -> \((psi0 /. y -> 0)\)\ , \ \(\[Psi]\_0'\)[0] -> \((D[psi0, y] /. y -> 0)\), \n\t\t\t \(\(\[Psi]\_0'\)'\)[0] -> \((D[psi0, {y, 2}] /. y -> 0)\), \n \t\t\t\(\(\(\[Psi]\_0'\)'\)'\)[0] -> \((D[psi0, {y, 3}] /. y -> 0)\), \n\t\t\t\ \[Psi]\_0[y] -> psi0\ , D[\ \[Psi]\_0[y], {y, a2_}] :> D[psi0, {y, a2}], \n\t\t\t c0 -> c0eig}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(-\[ImaginaryI]\), " ", RowBox[{"(", RowBox[{ \(-\(\(b1\ R\ \((2\ n\^3 + \((y + 2)\)\ n\^2 - m\ y)\)\ \((y - 1)\)\^2\)\/\(n\^2\ \((n + 1)\)\)\)\), "+", \(\(R\ \(( b1\ y\^2 + a1\ y - \(2\ \((m - 1)\)\ n\ \((m - n\^2)\)\)\/\(n\^4 + 2\ m\ \((n\ \((2\ n + 3)\) + 2)\)\ n + m\^2\)) \)\ \((\((2\ n + 3\ y)\)\ n\^2 + m\ \((2 - 3\ y)\)) \)\)\/\(n\^2\ \((n + 1)\)\)\), "+", RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubsuperscriptBox["\[Phi]", "1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "y", ")"}]}]}], ")"}]}], "==", "0"}], ",", RowBox[{ RowBox[{ SubsuperscriptBox["\[Phi]", "1", "\[Prime]", MultilineFunction->None], "(", "1", ")"}], "==", "0"}], ",", \(\(\[Phi]\_1\)(1) == 0\)}], "}"}], TraditionalForm]], "Output"], Cell[BoxData[ \(\(ans12phi = DSolve[eqs12, \[Phi]\_1[y], y]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(phi1temp1 = ans12phi /. {C[3] -> c3f, C[4] -> c4f}; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(phi1temp2 = \[Phi]\_1[y] /. phi1temp1[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\(\[ImaginaryI]\ b1\ \((n\^2 - m)\)\ R\ y\^7\)\/\(420\ n\^2\ \((n + 1)\)\) + \(\[ImaginaryI]\ a1\ \((n\^2 - m)\)\ R\ y\^6\)\/\(120\ n\^2\ \((n + 1)\)\) + \((\[ImaginaryI]\ \((2\ a1\ n\^7 + 4\ b1\ n\^7 + 3\ b1\ n\^6 + 8\ a1\ m\ n\^6 + 16\ b1\ m\ n\^6 + 12\ a1\ m\ n\^5 + 36\ b1\ m\ n\^5 + 6\ m\ n\^5 - 6\ n\^5 + 10\ a1\ m\ n\^4 + 35\ b1\ m\ n\^4 + 10\ a1\ m\^2\ n\^3 + 8\ b1\ m\^2\ n\^3 - 12\ m\^2\ n\^3 + 12\ b1\ m\ n\^3 + 12\ m\ n\^3 + 12\ a1\ m\^2\ n\^2 + 9\ b1\ m\^2\ n\^2 + 6\ m\^3\ n + 8\ a1\ m\^2\ n + 4\ b1\ m\^2\ n - 6\ m\^2\ n + 2\ a1\ m\^3 + b1\ m\^3)\)\ R\ y\^5)\)/\(( 120\ n\^2\ \((n + 1)\)\ \((n\^4 + 4\ m\ n\^3 + 6\ m\ n\^2 + 4\ m\ n + m\^2)\))\) - \((\[ImaginaryI]\ \((b1\ n\^6 + b1\ n\^5 + 4\ b1\ m\ n\^5 - 2\ m\ n\^5 + 2\ n\^5 + 10\ b1\ m\ n\^4 + 2\ m\^2\ n\^3 + 10\ b1\ m\ n\^3 - 2\ m\ n\^3 + b1\ m\^2\ n\^2 - 2\ m\^2\ n\^2 + 4\ b1\ m\ n\^2 + 2\ m\ n\^2 + b1\ m\^2\ n + 2\ m\^3 - 2\ m\^2) \)\ R\ y\^4)\)/ \((12\ n\ \((n + 1)\)\ \((n\^4 + 4\ m\ n\^3 + 6\ m\ n\^2 + 4\ m\ n + m\^2)\))\) + c4f\ y\^3 + c3f\ y\^2 + \((\(-2\)\ c3f - 3\ c4f - \(\[ImaginaryI]\ a1\ \((n\^2 - m)\)\ R\)\/\(20\ n\^2\ \((n + 1)\)\) - \(\[ImaginaryI]\ b1\ \((n\^2 - m)\)\ R\)\/\(60\ n\^2\ \((n + 1)\)\) + \((\[ImaginaryI]\ \((b1\ n\^6 + b1\ n\^5 + 4\ b1\ m\ n\^5 - 2\ m\ n\^5 + 2\ n\^5 + 10\ b1\ m\ n\^4 + 2\ m\^2\ n\^3 + 10\ b1\ m\ n\^3 - 2\ m\ n\^3 + b1\ m\^2\ n\^2 - 2\ m\^2\ n\^2 + 4\ b1\ m\ n\^2 + 2\ m\ n\^2 + b1\ m\^2\ n + 2\ m\^3 - 2\ m\^2)\)\ R)\)/ \((3\ n\ \((n + 1)\)\ \((n\^4 + 4\ m\ n\^3 + 6\ m\ n\^2 + 4\ m\ n + m\^2)\))\) - \((\[ImaginaryI]\ \((2\ a1\ n\^7 + 4\ b1\ n\^7 + 3\ b1\ n\^6 + 8\ a1\ m\ n\^6 + 16\ b1\ m\ n\^6 + 12\ a1\ m\ n\^5 + 36\ b1\ m\ n\^5 + 6\ m\ n\^5 - 6\ n\^5 + 10\ a1\ m\ n\^4 + 35\ b1\ m\ n\^4 + 10\ a1\ m\^2\ n\^3 + 8\ b1\ m\^2\ n\^3 - 12\ m\^2\ n\^3 + 12\ b1\ m\ n\^3 + 12\ m\ n\^3 + 12\ a1\ m\^2\ n\^2 + 9\ b1\ m\^2\ n\^2 + 6\ m\^3\ n + 8\ a1\ m\^2\ n + 4\ b1\ m\^2\ n - 6\ m\^2\ n + 2\ a1\ m\^3 + b1\ m\^3)\)\ R)\)/ \((24\ n\^2\ \((n + 1)\)\ \((n\^4 + 4\ m\ n\^3 + 6\ m\ n\^2 + 4\ m\ n + m\^2)\))\)) \)\ y + c3f + 2\ c4f + \(\[ImaginaryI]\ a1\ \((n\^2 - m)\)\ R\)\/\(24\ n\^2\ \((n + 1)\)\) + \(\[ImaginaryI]\ b1\ \((n\^2 - m)\)\ R\)\/\(70\ n\^2\ \((n + 1)\)\) - \((\[ImaginaryI]\ \((b1\ n\^6 + b1\ n\^5 + 4\ b1\ m\ n\^5 - 2\ m\ n\^5 + 2\ n\^5 + 10\ b1\ m\ n\^4 + 2\ m\^2\ n\^3 + 10\ b1\ m\ n\^3 - 2\ m\ n\^3 + b1\ m\^2\ n\^2 - 2\ m\^2\ n\^2 + 4\ b1\ m\ n\^2 + 2\ m\ n\^2 + b1\ m\^2\ n + 2\ m\^3 - 2\ m\^2) \)\ R)\)/ \((4\ n\ \((n + 1)\)\ \((n\^4 + 4\ m\ n\^3 + 6\ m\ n\^2 + 4\ m\ n + m\^2)\))\) + \((\[ImaginaryI]\ \((2\ a1\ n\^7 + 4\ b1\ n\^7 + 3\ b1\ n\^6 + 8\ a1\ m\ n\^6 + 16\ b1\ m\ n\^6 + 12\ a1\ m\ n\^5 + 36\ b1\ m\ n\^5 + 6\ m\ n\^5 - 6\ n\^5 + 10\ a1\ m\ n\^4 + 35\ b1\ m\ n\^4 + 10\ a1\ m\^2\ n\^3 + 8\ b1\ m\^2\ n\^3 - 12\ m\^2\ n\^3 + 12\ b1\ m\ n\^3 + 12\ m\ n\^3 + 12\ a1\ m\^2\ n\^2 + 9\ b1\ m\^2\ n\^2 + 6\ m\^3\ n + 8\ a1\ m\^2\ n + 4\ b1\ m\^2\ n - 6\ m\^2\ n + 2\ a1\ m\^3 + b1\ m\^3)\)\ R)\)/ \((30\ n\^2\ \((n + 1)\)\ \((n\^4 + 4\ m\ n\^3 + 6\ m\ n\^2 + 4\ m\ n + m\^2)\))\)\)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Here is the lower phase OS equation + bottom wall BC's", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eqs12a = Simplify[{os2O1 == 0, \ bc7O1 == 0, bc8O1 == 0} /. { \[Phi]\_0[0] -> \((phi0 /. y -> 0)\)\ , \ \ \ \ \(\[Phi]\_0'\)[0] -> \((D[phi0, y] /. y -> 0)\), \n\t\t\t \(\(\[Phi]\_0'\)'\)[0] -> \((D[phi0, {y, 2}] /. y -> 0)\), \n \t\t\t\(\(\(\[Phi]\_0'\)'\)'\)[0] -> \((D[phi0, {y, 3}] /. y -> 0)\), \n\t\t\t\[Phi]\_0[y] -> phi0, D[\ \[Phi]\_0[y], {y, a1_}] :> D[phi0, {y, a1}], \n\t\t\ \ \ \[Psi]\_0[0] -> \((psi0 /. y -> 0)\)\ , \ \(\[Psi]\_0'\)[0] -> \((D[psi0, y] /. y -> 0)\), \n\t\t\t \(\(\[Psi]\_0'\)'\)[0] -> \((D[psi0, {y, 2}] /. y -> 0)\), \n \t\t\t\(\(\(\[Psi]\_0'\)'\)'\)[0] -> \((D[psi0, {y, 3}] /. y -> 0)\), \n\t\t\t\ \[Psi]\_0[y] -> psi0\ , D[\ \[Psi]\_0[y], {y, a2_}] :> D[psi0, {y, a2}], \n\t\t\t c0 -> c0eig}]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", RowBox[{\(1\/m\^2\), RowBox[{"(", RowBox[{"\[ImaginaryI]", " ", RowBox[{"(", RowBox[{ RowBox[{"\[ImaginaryI]", " ", RowBox[{ SubsuperscriptBox["\[Psi]", "1", TagBox[\((4)\), Derivative], MultilineFunction->None], "(", "y", ")"}], " ", \(m\^2\)}], "-", \(\(b2\ r\ R\ \((n + y)\)\^2\ \((y\ n\^2 + m\ \((2\ n - y + 2)\))\)\)\/\(n\^2 \ \((n + 1)\)\)\), "+", \(\(r\ R\ \((b2\ y\^2 + a2\ y - \(2\ \((m - 1)\)\ n\ \((m - n\^2)\)\)\/\(n\^4 + 2\ m\ \((n\ \((2\ n + 3)\) + 2)\)\ n + m\^2\))\)\ \((\((2\ n + 3\ y)\)\ n\^2 + m\ \((2 - 3\ y)\)) \)\)\/\(n\^2\ \((n + 1)\)\)\)}], ")"}]}], ")"}]}]}], "==", "0"}], ",", RowBox[{ RowBox[{ SubsuperscriptBox["\[Psi]", "1", "\[Prime]", MultilineFunction->None], "(", \(-n\), ")"}], "==", "0"}], ",", \(\(\[Psi]\_1\)(\(-n\)) == 0\)}], "}"}], TraditionalForm]], "Output"], Cell[BoxData[ \(\(ans12psi = DSolve[eqs12a, \[Psi]\_1[y], y]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(psi1temp2 = \[Psi]\_1[y] /. ans12psi[\([1]\)]; \)\)], "Input"], Cell[BoxData[ \(Short[psi1temp2]\)], "Input"], Cell[BoxData[ FormBox[ TagBox[\(\[LeftSkeleton]1\[RightSkeleton]\), Short], TraditionalForm]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Here are the remaining boundary conditions, except for the one that \ has c1 in it\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(eqsO1a = \(({bc4O1 == 0, bc5O1 == 0, bc6O1 == 0})\) /. { \[Phi]\_0[0] -> \((phi0 /. y -> 0)\)\ , \ \ \ \ \(\[Phi]\_0'\)[0] -> \((D[phi0, y] /. y -> 0)\), \n\t\t\t \(\(\[Phi]\_0'\)'\)[0] -> \((D[phi0, {y, 2}] /. y -> 0)\), \n\t\t\t \(\(\(\[Phi]\_0'\)'\)'\)[0] -> \((D[phi0, {y, 3}] /. y -> 0)\), \n \t\t\t\[Phi]\_0[y] -> phi0, D[\ \[Phi]\_0[y], {y, a1_}] :> D[phi0, {y, a1}], \n\t\t\ \ \ \[Psi]\_0[0] -> \((psi0 /. y -> 0)\)\ , \ \(\[Psi]\_0'\)[0] -> \((D[psi0, y] /. y -> 0)\), \n\t\t\t \(\(\[Psi]\_0'\)'\)[0] -> \((D[psi0, {y, 2}] /. y -> 0)\), \n\t\t\t \(\(\(\[Psi]\_0'\)'\)'\)[0] -> \((D[psi0, {y, 3}] /. y -> 0)\), \n \t\t\t\ \[Psi]\_0[y] -> psi0\ , D[\ \[Psi]\_0[y], {y, a2_}] :> D[psi0, {y, a2}], \[Phi]\_1[0] -> \((phi1temp2 /. y -> 0)\)\ , \ \ \ \ \(\[Phi]\_1'\)[0] -> \((D[phi1temp2, y] /. y -> 0)\), \n\t\t\t \(\(\[Phi]\_1'\)'\)[0] -> \((D[phi1temp2, {y, 2}] /. y -> 0)\), \n \t\t\t\(\(\(\[Phi]\_1'\)'\)'\)[0] -> \((D[phi1temp2, {y, 3}] /. y -> 0)\), \n\t\t\t \[Phi]\_1[y] -> phi1temp2, D[\[Phi]\_1[y], {y, a1_}] :> D[phi1temp2, {y, a1}], \n\t\t\t \[Psi]\_1[0] -> \((psi1temp2 /. y -> 0)\)\ , \ \ \ \ \(\[Psi]\_1'\)[0] -> \((D[psi1temp2, y] /. y -> 0)\), \n\t\t\t \(\(\[Psi]\_1'\)'\)[0] -> \((D[psi1temp2, {y, 2}] /. y -> 0)\), \n \t\t\t\(\(\(\[Psi]\_1'\)'\)'\)[0] -> \((D[psi1temp2, {y, 3}] /. y -> 0)\), \n\t\t\t \[Psi]\_1[y] -> psi1temp2, D[\[Psi]\_1[y], {y, a1_}] :> D[psi1temp2, {y, a1}], \n\t\t\t c0 -> c0eig}; \)\)], "Input"], Cell[BoxData[ \(Short[eqsO1a]\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ \({\(-\(\(\[ImaginaryI]\ b2\ \((m - n\^2)\)\ r\ R\ n\^5\)\/\(420\ m\^2\ \((n + 1)\)\)\)\) + \[LeftSkeleton]1\[RightSkeleton]\/\[LeftSkeleton]1 \[RightSkeleton] + \[LeftSkeleton]19\[RightSkeleton] == 0, \[LeftSkeleton]2\[RightSkeleton]}\), Short], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(O1bcstemp = Solve[eqsO1a, {C[4], c3f, c4f}]; \)\)], "Input"], Cell[BoxData[ \(Short[O1bcstemp]\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{ SubscriptBox[ TagBox["c", C], "4"], "\[Rule]", \(\(-\(\(\(-\[LeftSkeleton]1\[RightSkeleton]\) + \[LeftSkeleton]9\[RightSkeleton]\)\/\(6\ m\)\)\) - \(\[LeftSkeleton]1\[RightSkeleton]\/\[LeftSkeleton]1 \[RightSkeleton] - \(6\ \[LeftSkeleton]2 \[RightSkeleton]\)\/\[LeftSkeleton]1 \[RightSkeleton]\)\/m\)}], ",", \(\[LeftSkeleton]1\[RightSkeleton]\), ",", RowBox[{"c3f", "\[Rule]", RowBox[{"m", " ", SubscriptBox[ TagBox["c", C], "3"]}]}]}], "}"}], "}"}], Short], TraditionalForm]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Finally we work on bc3 which contains c1", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(O1bc3 = bc3O1 /. {\[Phi]\_0[0] -> \((phi0 /. y -> 0)\)\ , \ \ \ \ \(\[Phi]\_0'\)[0] -> \((D[phi0, y] /. y -> 0)\), \n\t\t\t \(\(\[Phi]\_0'\)'\)[0] -> \((D[phi0, {y, 2}] /. y -> 0)\), \n\t\t\t \(\(\(\[Phi]\_0'\)'\)'\)[0] -> \((D[phi0, {y, 3}] /. y -> 0)\), \n \t\t\t\[Phi]\_0[y] -> phi0, D[\ \[Phi]\_0[y], {y, a1_}] :> D[phi0, {y, a1}], \n\t\t\ \ \ \[Psi]\_0[0] -> \((psi0 /. y -> 0)\)\ , \ \(\[Psi]\_0'\)[0] -> \((D[psi0, y] /. y -> 0)\), \n\t\t\t \(\(\[Psi]\_0'\)'\)[0] -> \((D[psi0, {y, 2}] /. y -> 0)\), \n\t\t\t \(\(\(\[Psi]\_0'\)'\)'\)[0] -> \((D[psi0, {y, 3}] /. y -> 0)\), \n \t\t\t\ \[Psi]\_0[y] -> psi0\ , D[\ \[Psi]\_0[y], {y, a2_}] :> D[psi0, {y, a2}], \[Phi]\_1[0] -> \((phi1temp2 /. y -> 0)\)\ , \ \ \ \ \(\[Phi]\_1'\)[0] -> \((D[phi1temp2, y] /. y -> 0)\), \n\t\t\t \(\(\[Phi]\_1'\)'\)[0] -> \((D[phi1temp2, {y, 2}] /. y -> 0)\), \n \t\t\t\(\(\(\[Phi]\_1'\)'\)'\)[0] -> \((D[phi1temp2, {y, 3}] /. y -> 0)\), \n\t\t\t \[Phi]\_1[y] -> phi1temp2, D[\[Phi]\_1[y], {y, a1_}] :> D[phi1temp2, {y, a1}], \n\t\t\t \[Psi]\_1[0] -> \((psi1temp2 /. y -> 0)\)\ , \ \ \ \ \(\[Psi]\_1'\)[0] -> \((D[psi1temp2, y] /. y -> 0)\), \n\t\t\t \(\(\[Psi]\_1'\)'\)[0] -> \((D[psi1temp2, {y, 2}] /. y -> 0)\), \n \t\t\t\(\(\(\[Psi]\_1'\)'\)'\)[0] -> \((D[psi1temp2, {y, 3}] /. y -> 0)\), \n\t\t\t \[Psi]\_1[y] -> psi1temp2, D[\[Psi]\_1[y], {y, a1_}] :> D[psi1temp2, {y, a1}], \n\t\t\t c0 -> c0eig}; \)\)], "Input"], Cell[BoxData[ \(Short[O1bc3]\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ \(\(-\(\(\[ImaginaryI]\ b2\ \((m - n\^2)\)\ r\ R\ n\^4\)\/\(60\ m\^2\ \((n + 1)\)\)\)\) + \[LeftSkeleton]1\[RightSkeleton]\/\[LeftSkeleton]1 \[RightSkeleton] + \[LeftSkeleton]19\[RightSkeleton] + \(a2\ \[LeftSkeleton]2\[RightSkeleton]\)\/\(4\ \((\[LeftSkeleton]1\[RightSkeleton])\)\^2\ \[LeftSkeleton]1\[RightSkeleton]\ n\^2\)\), Short], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(bc3O1t2 = Solve[O1bc3 == 0, c1]; \)\)], "Input"], Cell[BoxData[ \(Short[bc3O1t2]\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ \({{c1 \[Rule] \(-\(\(\(-\[LeftSkeleton]1\[RightSkeleton]\) + \[LeftSkeleton]21 \[RightSkeleton]\)\/\(\(a2\ \[LeftSkeleton]1\[RightSkeleton]\ \[LeftSkeleton]1\[RightSkeleton]\^2\)\/\(8\ \[LeftSkeleton]3\[RightSkeleton]\ \((\[LeftSkeleton]1\[RightSkeleton])\)\^2\) - \[LeftSkeleton]1\[RightSkeleton]\)\)\)}}\), Short], TraditionalForm]], "Output"] }, Open ]], Cell[BoxData[ \(\(bc3O1t3 = bc3O1t2 /. O1bcstemp[\([1]\)]; \)\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Here is the answer for c1", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(c1eig1 = bc3O1t3 //. {a1 -> \((m - n^2)\)/\((n^2 + n)\), \n\t\t\t\ta2 -> a1/m, \ b1 -> \(-\((m + n)\)\)/\((n^2 + n)\), \n\t\t\t\tb2 -> b1/m}; \)\)], "Input"], Cell[BoxData[ \(Short[c1eig1]\)], "Input"], Cell[BoxData[ FormBox[ TagBox[ \({{c1 \[Rule] \(-\(\(\(-\(\(\[ImaginaryI]\ \[LeftSkeleton]4\[RightSkeleton]\ n\^4\)\/\(60\ \[LeftSkeleton]2\[RightSkeleton]\ \((n\^2 + n)\)\)\)\) + \[LeftSkeleton]21 \[RightSkeleton]\)\/\(\(\(( \[LeftSkeleton]1\[RightSkeleton])\)\ \[LeftSkeleton]1\[RightSkeleton]\)\/\(8\ \[LeftSkeleton]5\[RightSkeleton]\ \((n\^2 + n)\)\) - \[LeftSkeleton]1\[RightSkeleton]\)\)\)}}\), Short], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[{ "Finally if we get the expression by itself and simplify it, the remaining \ constant, ", Cell[BoxData[ \(c\_3\)]], " drops out." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(see1 = Simplify[c1 /. c1eig1[\([1]\)]]\)], "Input"], Cell[BoxData[ \(TraditionalForm \`\((\[ImaginaryI]\ \((\((n + 6)\)\ r\ n\^15 - m\ \((n\^4 + 8\ n\^3 + 10\ n\^2 + 126\ n + 88)\)\ r\ n\^12 + m\^2\ \(( 2\ r\ n\^6 + \((42\ r - 140\ F)\)\ n\^5 + \((\(-280\)\ F + 248\ r + 224)\)\ n\^4 + \((\(-140\)\ F - 125\ r + 1043)\)\ n\^3 + \((60\ r + 1486)\)\ n\^2 + 32\ \((16\ r + 21)\)\ n + 224\ r)\)\ n\^9 - m\^3\ \(( 32\ r\ n\^6 + 2\ \((630\ F + 61\ r + 112)\)\ n\^5 + \((4200\ F - 197\ r + 651)\)\ n\^4 + 4\ \((1435\ F + 156\ r - 94)\)\ n\^3 + \((3920\ F - 356\ r - 1306)\)\ n\^2 + \((1120\ F - 2180\ r + 406)\)\ n - 1088\ r + 944)\)\ n\^8 - m\^4\ \((8\ \((420\ F - 2\ r + 49)\)\ n\^7 + 6\ \((2520\ F - 94\ r + 273)\)\ n\^6 + 2\ \((15540\ F + 578\ r + 429)\)\ n\^5 + 8\ \((4620\ F + 110\ r - 83)\)\ n\^4 + \((26600\ F - 1287\ r + 1431)\)\ n\^3 + \((11200\ F + 258\ r + 1524)\)\ n\^2 + 32\ \((70\ F + 35\ r - 9)\)\ n + 224\ r)\)\ n\^5 - m\^5\ \(( 224\ \((10\ F + 1)\)\ n\^7 + 32\ \((350\ F - 9\ r + 35)\)\ n\^6 + \((26600\ F + 1524\ r + 258)\)\ n\^5 + 3\ \((12320\ F + 477\ r - 429)\)\ n\^4 + 8\ \((3885\ F - 83\ r + 110)\)\ n\^3 + 2\ \((7560\ F + 429\ r + 578)\)\ n\^2 + 6\ \((560\ F + 273\ r - 94)\)\ n + 392\ r - 16)\)\ n\^4 - m\^6\ \(( 16\ \((70\ F + 59\ r - 68)\)\ n\^6 + \((3920\ F + 406\ r - 2180)\)\ n\^5 + 2\ \((2870\ F - 653\ r - 178)\)\ n\^4 + 8\ \((525\ F - 47\ r + 78)\)\ n\^3 + \((1260\ F + 651\ r - 197)\)\ n\^2 + 2\ \((112\ r + 61)\)\ n + 32)\)\ n\^2 + m\^7\ \(( 224\ n\^6 + 32\ \((21\ r + 16)\)\ n\^5 + \((\(-140\)\ F + 1486\ r + 60)\)\ n\^4 + \((\(-280\)\ F + 1043\ r - 125)\)\ n\^3 - 4\ \((35\ F - 56\ r - 62)\)\ n\^2 + 42\ n + 2)\)\ n + m\^9\ \((6\ n + 1)\) - m\^8\ \((88\ n\^4 + 126\ n\^3 + 10\ n\^2 + 8\ n + 1)\))\)\ R)\)/ \((420\ m\^2\ \((n + 1)\)\^2\ \((n\^4 + 2\ m\ \((2\ n\^2 + 3\ n + 2)\)\ n + m\^2)\)\^3)\)\)], "Output"] }, Open ]] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Here is a check against the answers of Yiantsios and Higgins", "Subtitle"], Cell[CellGroupData[{ Cell["We need can see that the real part of c1 is 0. 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