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McCready\nProfessor and Chair of Chemical Engineering\nUniversity \ of Notre Dame\nNotre Dame, IN 46556\nUSA", FontSize->14], "\n\n", ButtonBox["mjm@nd.edu", ButtonStyle->None], "\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by which ever laws pertain \ to the World Wide Web and the Internet.\n\nI would hope that as a \ professional courtesy, this notice remain visible to other users. \nThere is \ no charge for copying and dissemination \n\nVersion: 10/15/99\nMore recent \ versions of this notebook may be available at:\n", Cell[BoxData[ FormBox[ ButtonBox[ FormBox[ ButtonBox[ \(\(http : \) // \(www . nd . edu/\(~mjm\)\)/lubricatedflow . nb\), ButtonData:>{"lubricatedflow.nb", " "}, ButtonStyle->"Hyperlink"], "TraditionalForm"], ButtonData:>{ URL[ "http://www.nd.edu/~mjm/lubricatedflow.nb"], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]], "\n\nThis notebook uses the notation of and is intended as a companion to:\n\ ", StyleBox["An Introduction to Fluid Dynamics", FontSlant->"Italic"], " by Stanley Middleman, John Wiley, 1999. " }], "Text"], Cell[CellGroupData[{ Cell["Keywords", "Subtitle"], Cell["\<\ lubricated transport, two-layer flows, multifluid flow, two-phase \ flow, laminar flow \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Motivation", "Subtitle"], Cell[TextData[{ "We have found that pressure drop is proportional to viscosity and diameter \ to the fourth power for laminar flows. Thus as the viscosity of a liquid \ gets much larger than water, it will require either very high pressure drop \ (high operating costs) or large pipes (high capital costs) to pump. \ Obviously, if the liquid is valuable enough then it will be worth it and \ pipeline built. However for a less valuable material, pumping as a single \ phase liquid may not be economically viable. Such is the case for very heavy \ crudes. Pumping costs can be so large that the oil could never be sold for \ profit.\n\nHowever, what if your company or country has access to only this \ kind of heavy crude oil. It is in your interests to solve this problem and \ produce a new technology that transport the crude efficiently enough so that \ it can be sold for profit.\n\nThe idea that could be used is \"lubricated\" \ transport where the viscous oil is confined to a \"core\" region in the \ center of the pipe and a much less viscous liquid (e.g. cheap also, water) \ flows in the annular region. The pressure drop is reduced because while the \ water is sheared highly, the viscous oil core is not subject to much \ deformation thus does not contribute to pressure drop.\n\nFor more about the \ real problem you will need to read elsewhere. Real core-annular flows have \ waves on the interface which usually act to stabilize the core flow, although \ could sometimes break up the core. \n\nA general reference for this topic is \ ", StyleBox["Fundamentals of two-fluid dynamics", FontSlant->"Italic"], " by D. D. Joseph and Y. Y. Renardy, Springer Verlag, 1993. \n\nThere is \ also a CNN science/technology video on this topic \"staring\" Dan Joseph and \ his lab. ", " " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Preview of major points", "Subtitle", PageBreakAbove->True], Cell[TextData[{ "This notebook solves the \"lubricated\" flow in a pipe problem and in \ doing so demonstrates several major fundamental points in Transport \ Phenomena. \n\n1. For flows where more than one fluid is flowing together, \ there is a ", ButtonBox["separate equation", ButtonData:>"twoequations", ButtonStyle->"Hyperlink"], " for the velocity profile for each fluid. These equations differ just by \ the fluid properties. \n\n2. To solve these two-fluid problems you need to \ recognize:\n\t\[Bullet] The ", Cell[BoxData[ FormBox[ ButtonBox[\(pressure\ drop\ is\ the\ same\ in\ each\ phase\), ButtonData:>"pressuresame", ButtonStyle->"Hyperlink"], TraditionalForm]]], " if gravity can be ignored.\n\t\[Bullet] The ", ButtonBox["velocity of the two fluids matches", ButtonData:>"vmatch", ButtonStyle->"Hyperlink"], " at the interface.\n\t\[Bullet] The ", ButtonBox["shear stress", ButtonData:>"stressmatch", ButtonStyle->"Hyperlink"], " in each fluid matches at the interface. \nThe last two of these are \ common boundary conditions. \n\nFor this specific situation the major \ results are:\n\nLubricated transport can cause a flow rate ", Cell[BoxData[ FormBox[ ButtonBox[\(enhancement\ of\ the\ viscous\ fluid\), ButtonData:>"enhancement", ButtonStyle->"Hyperlink"], TraditionalForm]]], " (compared to this fluid flowing alone in the pipe) on the ", ButtonBox["order of the viscosity ratio", ButtonData:>"scaledenhancemnt", ButtonStyle->"Hyperlink"], " of two fluids and the maximum enhancement occurs when the less viscous \ fluid occupies the outer 30% of the radius. " }], "Text", CellTags->"preview"] }, Open ]], Cell[CellGroupData[{ Cell["Flow geometry", "Subtitle"], Cell[TextData[{ "The problem we are considering is the flow of a very viscous liquid (Fluid \ I) confined to the center of the pipe, with a much less viscous liquid (Fluid \ II) flowing around Fluid I and contacting the walls. The pipe diameter is ", StyleBox["D", FontSlant->"Italic"], " and the thickness of the lubricating layer is ", StyleBox["h", FontSlant->"Italic"], ". 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0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Open ]], Cell[CellGroupData[{ Cell["Solution of the example", "Subtitle"], Cell[CellGroupData[{ Cell["Basic equations", "Section"], Cell[CellGroupData[{ Cell["The momentum equations for steady, unidirectional flow", "Subsection"], Cell["\<\ We have derived the equation for steady laminar flow in a tube. \ When we did this, we did not make restrictions on the type of fluid or even \ if there is more than one fluid. The equation is\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq1 = \(1\/r\) D[r\ \[Tau]\_zr[r], r] - \ D[P[z], z]\ + \ \[Rho]\ g\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{\(g\ \[Rho]\), "-", RowBox[{ SuperscriptBox["P", "\[Prime]", MultilineFunction->None], "(", "z", ")"}], "+", FractionBox[ RowBox[{\(\(\[Tau]\_zr\)(r)\), "+", RowBox[{"r", " ", RowBox[{ SubsuperscriptBox["\[Tau]", "zr", "\[Prime]", MultilineFunction->None], "(", "r", ")"}]}]}], "r"]}], TraditionalForm]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["The Newtonian stress constitutive equation", "Subsubsection"], Cell[TextData[{ "From the previous example in Middleman (3.2.1, page 68-74) you may recall \ two points. First, that P[z] was a linear function and thus P'[z] = const. \ We will call this constant, ", StyleBox["dpdz.", FontSlant->"Italic"], " Second, we need a constitutive equation for the stress component, ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_zr\)]], ", in terms of the velocity gradient. Since we cannot easily derive this, \ we just stated that it is: \n\t\t", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_zr = \(-\[Mu]\)\ \ \[PartialD]v\_z\/\[PartialD]r\)]], ".\n\t\t\nNow we need to recognize that there are two fluids and thus two \ values of viscosity for the fluids in different regions. Thus we will have \ two versions of eq1, one for fluid I and one for fluid II. Since we know the \ constitutive equations and we know that P'[z] is a constant, we might as well \ substitute and start to solve it. We also make the restriction that gravity \ is not important and thus ignore its possible effects by making g=0. \n\nThe \ ", StyleBox["z", FontSlant->"Italic"], " momentum equations are:" }], "Text", CellTags->"twoequations"] }, Open ]], Cell[TextData[Cell[BoxData[ FormBox[ ButtonBox[\(Back\ to\ preview\), ButtonData:>"preview", ButtonStyle->"Hyperlink"], TraditionalForm]]]], "Text"], Cell[TextData[ButtonBox["Back to conclusions", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"]], "Text"], Cell[CellGroupData[{ Cell["fluid I equation", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(eq11 = eq1 /. {\(\(Derivative[i_]\)[\[Tau]\_zr]\)[r] \[RuleDelayed] \ D[\(-\[Mu]\_1\) D[v\_z\%I[r], r], {r, i}], \[IndentingNewLine]\[Tau]\_zr[r] -> \(-\[Mu]\_1\) D[v\_z\%I[r], r], \(P'\)[z] \[Rule] dpdz1, g \[Rule] 0}\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ RowBox[{\(-\[Mu]\_1\), " ", RowBox[{ SuperscriptBox[\((v\_z\%\[ImaginaryI])\), "\[Prime]", MultilineFunction->None], "\[InvisibleApplication]", \((r)\)}]}], "-", RowBox[{"r", " ", \(\[Mu]\_1\), " ", RowBox[{ SuperscriptBox[\((v\_z\%\[ImaginaryI])\), "\[Prime]\[Prime]", MultilineFunction->None], "\[InvisibleApplication]", \((r)\)}]}]}], "r"], "-", "dpdz1"}], TraditionalForm]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["fluid II equation", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(eq12 = eq1 /. {\(\(Derivative[i_]\)[\[Tau]\_zr]\)[r] \[RuleDelayed] \ D[\(-\[Mu]\_2\) D[v\_z\%II[r], r], {r, i}], \[IndentingNewLine]\[Tau]\_zr[r] -> \(-\[Mu]\_2\) D[v\_z\%II[r], r], \(P'\)[z] \[Rule] dpdz2, g \[Rule] 0}\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{ RowBox[{\(-\[Mu]\_2\), " ", RowBox[{ SuperscriptBox[\((v\_z\%II)\), "\[Prime]", MultilineFunction->None], "\[InvisibleApplication]", \((r)\)}]}], "-", RowBox[{"r", " ", \(\[Mu]\_2\), " ", RowBox[{ SuperscriptBox[\((v\_z\%II)\), "\[Prime]\[Prime]", MultilineFunction->None], "\[InvisibleApplication]", \((r)\)}]}]}], "r"], "-", "dpdz2"}], TraditionalForm]], "Output"] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["A note on Mathematica", "Subsubsection"], Cell[TextData[{ "The\"/.\" command is a substitution command, which is used frequently when \ doing computer algebra. On the left side of /. is the term that is receiving \ the substitution. On the right side of the /. is the designation for what we \ are replacing, either any derivative of ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_zr\)]], "or just ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_zr\)]], "(we can't do the 0 derivative substitution with this command), with the \ velocity gradient substituted in. " }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Solution of the governing equations", "Section"], Cell[CellGroupData[{ Cell["General solutions to these differential equations", "Subsection"], Cell["\<\ Let's solve what we can. Start with the middle of the fluid. We can first get a general solution to \ this equation. \ \>", "Text"], Cell["Equation 1 solution", "Subsubsection", CellTags->"diffsolution"], Cell[CellGroupData[{ Cell[BoxData[ \(eq1soln = DSolve[{eq11 \[Equal] 0}, v\_z\%I[r], r]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{\(v\_z\%\[ImaginaryI][r]\), "\[Rule]", RowBox[{\(-\(\(dpdz1\ r\^2\)\/\(4\ \[Mu]\_1\)\)\), "+", SubscriptBox[ TagBox["c", C], "2"], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "1"], " ", \(log(r)\)}]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell[TextData[ButtonBox["Back to conclusions", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"]], "Text"], Cell[TextData[{ "It is convenient to examine this solution to see if an obvious \ simplification arises. It does. At r=0, we don't anticipate any reason for \ the velocity to be infinite. Since there is no other term to cancel log(r), \ we must make ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], "=0. We can't do anything with the other constant at this moment. Recall \ however that in the single phase example, ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], " was adjusted to make the velocity = 0 at the fixed wall. \n\nThis real \ ugly command sets ", Cell[BoxData[ \(TraditionalForm\`c\_1\)]], "=0, and changes the name of ", Cell[BoxData[ \(TraditionalForm\`c\_2\)]], "to f1 (to sort of resemble Middleman's notation.). It also picks out just \ the equation without the braces. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq1a\ = \ v\_z\%\[ImaginaryI][ r] /. \((eq1soln /. {C[1] \[Rule] 0, C[2] \[Rule] f1})\)[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`f1 - \(dpdz1\ r\^2\)\/\(4\ \[Mu]\_1\)\)], "Output"] }, Open ]], Cell["Equation 2 solution", "Subsubsection"], Cell["\<\ We can solve the second equation the same way. This time the \ domain of r does not go to 0 and there is no need (or reason) to exclude the \ log(r) term -- so it stays. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq2soln = DSolve[{eq12 \[Equal] 0}, v\_z\%II[r], r]\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"{", RowBox[{\(v\_z\%II[r]\), "\[Rule]", RowBox[{\(-\(\(dpdz2\ r\^2\)\/\(4\ \[Mu]\_2\)\)\), "+", SubscriptBox[ TagBox["c", C], "2"], "+", RowBox[{ SubscriptBox[ TagBox["c", C], "1"], " ", \(log(r)\)}]}]}], "}"}], "}"}], TraditionalForm]], "Output"] }, Open ]], Cell["\<\ Change the name of the constants to e2 and f2 and extract the \ equation as \"eq2a\"\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq2a\ = \ v\_z\%II[r] /. \((eq2soln /. {C[1] \[Rule] e2, C[2] \[Rule] f2})\)[\([1]\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-\(\(dpdz2\ r\^2\)\/\(4\ \[Mu]\_2\)\)\) + f2 + e2\ \(log(r)\)\)], "Output"] }, Open ]], Cell["\<\ We can take stock at this point. We don't know the three \ constants, f1, e2, f2 and we expect to need three boundary conditions. We \ also worry about and will need to resolve if dpdx1 = dpdx2. (They do, but we \ should discuss this -- see page 80). \ \>", "Text", CellTags->"pressurematch"] }, Closed]], Cell[CellGroupData[{ Cell["Boundary conditions", "Subsection", CellTags->"boundaryconditions"], Cell[TextData[ButtonBox["Back to conclusions", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"]], "Text"], Cell[CellGroupData[{ Cell["Velocity match at the interface", "Subsubsection", CellTags->"vmatch"], Cell[TextData[Cell[BoxData[ FormBox[ ButtonBox[\(Back\ to\ preview\), ButtonData:>"preview", ButtonStyle->"Hyperlink"], TraditionalForm]]]], "Text"], Cell[TextData[{ "Just as fluids stick to walls, they also stick to each other so we require \ that at the interface, ", Cell[BoxData[ \(TraditionalForm\`r = \ D/2 - h\)]], ", the velocities in the two fluids be the same. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(vmatcheq = \((eq1a /. r \[Rule] \((D/2 - h)\))\)\ - \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((eq2a /. r \[Rule] \((D/2 - h)\))\) \[Equal] 0\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-\(\(dpdz1\ \((D\/2 - h)\)\^2\)\/\(4\ \[Mu]\_1\)\)\) \ + \(dpdz2\ \((D\/2 - h)\)\^2\)\/\(4\ \[Mu]\_2\) + f1 - f2 - e2\ \(log(D\/2 - h)\) == 0\)], "Output"] }, Open ]] }, Open ]], Cell[TextData[ButtonBox["Back to conclusions", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"]], "Text"], Cell[CellGroupData[{ Cell["No slip at the wall", "Subsubsection", CellTags->"noslip"], Cell["\<\ The outer fluid, phase II, is touching the pipe wall and thus the \ velocity there is 0. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(noslipeq = \((eq2a /. r \[Rule] \((D/2)\))\) \[Equal] 0\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-\(\(dpdz2\ D\^2\)\/\(16\ \[Mu]\_2\)\)\) + f2 + e2\ \(log(D\/2)\) == 0\)], "Output"] }, Open ]] }, Open ]], Cell[TextData[ButtonBox["Back to conclusions", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"]], "Text"], Cell[CellGroupData[{ Cell["Stress match at the interface", "Subsubsection", CellTags->"stressmatch"], Cell[TextData[Cell[BoxData[ FormBox[ ButtonBox[\(Back\ to\ preview\), ButtonData:>"preview", ButtonStyle->"Hyperlink"], TraditionalForm]]]], "Text"], Cell["\<\ We have used up conditions on the velocity at locations that we can \ think of (center of pipe, wall, interface) thus we think of possible boundary \ conditions on the first derivative. This, of course, relates to the shear \ stress in the fluid and we have condition that arises from the equation that \ was derived originally. The original equation (eq1), tells us that the \ stress profile is linear across the pipe with a value of 0 in the middle and \ a largest (in magnitude) value at the wall. This result is exclusive of the \ values of viscosity of the fluid or if there are different fluids in the \ pipe. Thus a condition on the first derivative is that the shear stress is \ continuous across the pipe. This means that the shear stress in each fluid \ is equal at the interface. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \ \)\(stresseq = \((\[Mu]\_1\ \ D[eq1a, r] /. r \[Rule] \((D/2 - h)\))\)\ - \ \ \ \ \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \((\(\[Mu]\_2\) D[eq2a, r] /. r \[Rule] \((D/2 - h)\))\) \[Equal] 0\)\)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-\(1\/2\)\)\ dpdz1\ \((D\/2 - h)\) - \((e2\/\(D\/2 - h\) - \(dpdz2\ \((D\/2 - h)\)\)\/\(2\ \ \[Mu]\_2\))\)\ \[Mu]\_2 == 0\)], "Output"] }, Open ]] }, Open ]], Cell[TextData[ButtonBox["Back to conclusions", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"]], "Text"], Cell[CellGroupData[{ Cell["\<\ Solve for the constants using the three boundary conditions. \ \>", \ "Subsubsection"], Cell["\<\ We can solve for the three constants using the 3 boundary condition \ equations. The answer is in the form of a substitution rule telling what, \ f1, f2 and e2 should be replaced by. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bceqs1 = \ {vmatcheq, noslipeq, stresseq}\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{\(-\(\(dpdz1\ \((D\/2 - h)\)\^2\)\/\(4\ \ \[Mu]\_1\)\)\) + \(dpdz2\ \((D\/2 - h)\)\^2\)\/\(4\ \[Mu]\_2\) + f1 - f2 - e2\ \(log(D\/2 - h)\) == 0, \(-\(\(dpdz2\ D\^2\)\/\(16\ \[Mu]\_2\)\)\) + f2 + e2\ \(log(D\/2)\) == 0, \(-\(1\/2\)\)\ dpdz1\ \((D\/2 - h)\) - \((e2\/\(D\/2 - h\) - \(dpdz2\ \((D\/2 - h)\)\)\/\(2\ \ \[Mu]\_2\))\)\ \[Mu]\_2 == 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(bceqs2 = \ Simplify[bceqs1]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{\(dpdz2\ \((D - 2\ h)\)\^2\)\/\(16\ \[Mu]\_2\) + f1 == \(dpdz1\ \((D - 2\ h)\)\^2\)\/\(16\ \[Mu]\_1\) + f2 + e2\ \(log(1\/2\ \((D - 2\ h)\))\), f2 + e2\ \(log(D\/2)\) == \(D\^2\ dpdz2\)\/\(16\ \[Mu]\_2\), 0 == \(\((dpdz1 - dpdz2)\)\ \((D - 2\ h)\)\^2 + 8\ e2\ \[Mu]\_2\)\/\(4\ \ \((D - 2\ h)\)\)}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(bcans = Solve[bceqs2, {e2, f2, \ f1}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{f1 \[Rule] \(dpdz2\ D\^2\)\/\(16\ \[Mu]\_2\) + \ \(dpdz1\ \((D - 2\ h)\)\^2\)\/\(16\ \[Mu]\_1\) - \(dpdz2\ \((D - 2\ h)\)\^2\)\ \/\(16\ \[Mu]\_2\) + \(\((dpdz1 - dpdz2)\)\ \((D - 2\ h)\)\^2\ \ \(log(D\/2)\)\)\/\(8\ \[Mu]\_2\) - \(\((dpdz1 - dpdz2)\)\ \((D - 2\ h)\)\^2\ \ \(log(1\/2\ \((D - 2\ h)\))\)\)\/\(8\ \[Mu]\_2\), f2 \[Rule] \(dpdz2\ D\^2\)\/\(16\ \[Mu]\_2\) + \(\((dpdz1 - dpdz2)\)\ \ \((D - 2\ h)\)\^2\ \(log(D\/2)\)\)\/\(8\ \[Mu]\_2\), e2 \[Rule] \(-\(\(\((dpdz1 - dpdz2)\)\ \((D - 2\ h)\)\^2\)\/\(8\ \ \[Mu]\_2\)\)\)}}\)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Pressure drop is same in both phases", "Subsection"], Cell[TextData[Cell[BoxData[ FormBox[ ButtonBox[\(Back\ to\ preview\), ButtonData:>"preview", ButtonStyle->"Hyperlink"], TraditionalForm]]]], "Text"], Cell[TextData[{ "We can make an argument that the pressure across a cross-section must be \ constant, (or just increase linearly downward because of the gravitational \ hydrostatic force) otherwise there will be flow in the radial direction to \ relax the pressure. Since we are requiring only one dimensional flow, there \ can be no pressure change in the ", StyleBox["r", FontSlant->"Italic"], " direction. Now that the pressure is constant across a cross section, we \ realize that the pressure change in the axial (", StyleBox["z", FontSlant->"Italic"], ") direction must be the same in each phase. " }], "Text", CellTags->"pressuresame"] }, Closed]], Cell[CellGroupData[{ Cell["Velocity profiles", "Subsection"], Cell["\<\ Let's get the answers for the two velocity profiles by substituting \ in the answers from the boundary conditions. Note that \"[[1]]\" takes the \ expression that is inside one set of braces, { $$$ }. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(v1ans = FullSimplify[\((eq1a /. bcans[\([1]\)])\) /. {dpdz2 \[Rule] dpdz, dpdz1 \[Rule] dpdz}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(4\ dpdz\ \((D - h)\)\ h\ \[Mu]\_1 + dpdz\ \((\((D - \ 2\ h)\)\^2 - 4\ r\^2)\)\ \[Mu]\_2\)\/\(16\ \[Mu]\_1\ \[Mu]\_2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(v2ans = FullSimplify[\((eq2a /. bcans[\([1]\)])\) /. {dpdz2 \[Rule] dpdz, dpdz1 \[Rule] dpdz}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(dpdz\ \((D\^2 - 4\ r\^2)\)\)\/\(16\ \[Mu]\_2\)\)], \ "Output"] }, Open ]], Cell["Dimensionless profiles", "Subsubsection"], Cell["\<\ Following Middleman, we can simplify the equations and get a better \ feel for how they behave if we nondimensionalize these. Note how we do \ substitutions to get these forms. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(v1profile = FullSimplify[\((v1ans /. {D \[Rule] 2\ R, \ r \[Rule] s\ R, \ \[Mu]\_2 \[Rule] \[Mu]\_1/M, h \[Rule] \[Eta]\ R})\)/\((dpdz\ \(R^2/ 4\)/\[Mu]\_1)\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-s\^2\) - \((M - 1)\)\ \((\[Eta] - 2)\)\ \[Eta] + 1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(v2profile = FullSimplify[\((v2ans /. {D \[Rule] 2\ R, \ r \[Rule] s\ R, \ \[Mu]\_2 \[Rule] \[Mu]\_1/M})\)/\((dpdz\ \(R^2/ 4\)/\[Mu]\_1)\)]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`M - M\ s\^2\)], "Output"] }, Open ]], Cell["\<\ These match equations 3.2.53 and 3.2.54, although you have to re \ arrange the first one a little. \ \>", "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Investigation of the solution", "Section"], Cell[CellGroupData[{ Cell["Plots of the velocity profiles", "Subsection"], Cell[CellGroupData[{ Cell["\<\ Let's make some plots to show how it behaves. 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We realize \ that while adding more outside fluid reduces shear, and thus increases flow, \ it also reduces the area available for flow and thus could ultimately reduce \ the flow of the inner flow. Thus a fair measure of the enhancement is the ratio of the volumetric flow of \ the inner fluid for the lubricated case to its volumetric flow when \[Eta]=0, \ that is when there is no outside fluid. \ \>", "Text"], Cell[CellGroupData[{ Cell["Expression for the flow rate enhancement", "Subsubsection"], Cell["\<\ To get an expression for the volumetric flow rate, we integrate the \ velocity of the inner fluid in the region that there flow of the inner flow. \ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(q1\ = \ Simplify[Integrate[ 2\ \[Pi]\ s\ v1profile, {s, 0, 1 - \[Eta]}]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-\(1\/2\)\)\ \[Pi]\ \((\[Eta] - 1)\)\^2\ \((\((2\ M \ - 1)\)\ \[Eta]\^2 + \((2 - 4\ M)\)\ \[Eta] - 1)\)\)], "Output"] }, Open ]], Cell["\<\ If we see \[Eta] =0 we can get the dimensionless flow rate with no \ outside phase.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(q1\[Eta]0 = \ q1 /. \[Eta] \[Rule] 0\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\[Pi]\/2\)], "Output"] }, Open ]], Cell["Now the enhancement is this ratio:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\[CapitalPsi]\ = \ FullSimplify[q1\ /q1\[Eta]0]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-\((\[Eta] - 1)\)\^2\)\ \((\((2\ M - 1)\)\ \((\[Eta] - 2)\)\ \[Eta] - 1)\)\)], "Output"] }, Open ]], Cell["\<\ Which matches equation 3.2.58 and gives the relative enhancement \ due to lubrication. \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Plots of the enhancement ratio.", "Subsubsection"], Cell["\<\ We would like to make some log-log plots and we need to load the \ Graphics package.\ \>", "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input", AspectRatioFixed->True], Cell[TextData[{ "Note for ", StyleBox["Mathematica", FontSlant->"Italic"], " 3 users: the Log-Log plotting package may give trouble and not show \ correct axis numbering. 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(How much of the area is this??) Below this \ value, there is an advantage to increasing the less viscous flow to reduce \ shear. Above this point, there is not enough flow area available for the \ desired fluid and the enhancement drops. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Velocity profiles for different values of \[Eta] at constant ", StyleBox["M", FontSlant->"Italic"], " to show how the enhancement occurs. ", StyleBox[" ", FontSlant->"Italic"] }], "Subsubsection"], Cell[CellGroupData[{ Cell["\<\ It is now worth taking a look at the velocity profiles along one of \ these curves. We make a series of plots but don't show them until the end \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(vp2hp1 = Plot[v2profile /. M \[Rule] 10, {s, 1, .9}, \ DisplayFunction \[Rule] Identity];\)\)], "Input", PageBreakAbove->True], Cell[BoxData[ \(\(vp2hp2 = Plot[v2profile /. 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1P00HkmS_`00Hkl000=S_`03001S_f>o0==S_`000f>o00D006>o001S_`0000=S_`04001S_f>o0003 Hkl00`00HkmS_`0=Hkl00`00HkmS_`03Hkl01`00HkmS_f>o001S_`00009S_`03001S_f>o009S_`03 001S_f>o00=S_`03001S_f>o009S_`06001S_f>o001S_`000f>o00<006>oHkl0df>o0005Hkl00`00 HkmS_`03Hkl02000HkmS_`00Hkl006>o000@Hkl40002Hkl30003Hkl20005Hkl01000HkmS_f>o0`00 0f>o10000f>o1@00e6>o0003Hkl50004Hkl01@00Hkl00000Hkl00P008f>o00<006>oHkl0if>o000> Hkl01000HkmS_f>o0P0086>o0`001F>o00<006>oHkl0hF>o003oHklQHkl00?mS_b5S_`00of>o8F>o 003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-2.39807, -1.85922, \ 0.00885373, 0.0124864}}] }, Open ]] }, Open ]], Cell["\<\ We see that the scaling is correct and that the maximum enhancement \ is about 50% of the viscosity ratio. Now you know enough to tell if this \ process for transporting oil will be worthwhile! \ \>", "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Where is the flowrate maximum located?", "Subsection"], Cell["\<\ I can't resist one more calculation. We can find the maximum \ enhancement readily. First take the derivative. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(max1 = FullSimplify[D[\[CapitalPsi], \[Eta]]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(-4\)\ \((\[Eta] - 1)\)\ \((M\ \((2\ \((\[Eta] - 2)\)\ \[Eta] + 1)\) - \((\[Eta] - 1)\)\^2)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Set it to 0 and find the value of \[Eta]. Isn't ", StyleBox["Mathematica", FontSlant->"Italic"], " great!!. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(max2 = Solve[max1 \[Equal] 0, \[Eta]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{\[Eta] \[Rule] 1}, {\[Eta] \[Rule] \(2\ M - \@\(2\ M - 1\)\ \@M - 1\)\/\(2\ M - \ 1\)}, {\[Eta] \[Rule] \(2\ M + \@\(2\ M - 1\)\ \@M - 1\)\/\(2\ M - 1\)}}\)], \ "Output"] }, Open ]], Cell[TextData[{ "We have observed that it does not change much with M. Lets see why by \ looking at the solution for large ", StyleBox["M.", FontSlant->"Italic"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(max3 = Limit[max2, M \[Rule] Infinity]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{\[Eta] \[Rule] 1}, {\[Eta] \[Rule] 1 - 1\/\@2}, {\[Eta] \[Rule] 1 + 1\/\@2}}\)], "Output"] }, Open ]], Cell["\<\ The value of the maximum is about 2.9. You now know even more \ about this problem. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[max3]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{\[Eta] \[Rule] 1.`}, {\[Eta] \[Rule] 0.29289321881345254`}, {\[Eta] \[Rule] 1.7071067811865475`}}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Does it make sense that ", StyleBox["increasing", FontVariations->{"Outline"->True}], " one of the flow rates will ", StyleBox["decrease", FontSlant->"Italic"], " the pressure drop?" }], "Subsection"], Cell[TextData[{ "There is one more interesting point to make about this calculation. We \ can examine how the pressure drop changes as the flowrate of the less viscous \ fluid is increased. If we increase the flowrate of a fluid in a single phase \ flow the pressure drop will increase. This will be linearly for a laminar \ flow. However, for a two-fluid flow will this always be the case? We will \ find that it is not the case. For a range of flow rates, increasing the less \ viscous fluid flowrate will ", StyleBox["decrease", FontSlant->"Italic"], " the pressure drop. \n\nCalculate the ", StyleBox["dimensional", FontSlant->"Italic", FontVariations->{"Underline"->True}], " flowrate of the inside fluid starting with the dimensional velocity \ profile. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(v1prof = \(4\ dpdz\ \((D - h)\)\ h\ \[Mu]\_1 + dpdz\ \((\((D - 2\ \ h)\)\^2 - 4\ r\^2)\)\ \[Mu]\_2\)\/\(16\ \[Mu]\_1\ \[Mu]\_2\)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(4\ dpdz\ \((D - h)\)\ h\ \[Mu]\_1 + dpdz\ \((\((D - \ 2\ h)\)\^2 - 4\ r\^2)\)\ \[Mu]\_2\)\/\(16\ \[Mu]\_1\ \[Mu]\_2\)\)], "Output"] }, Open ]], Cell["The flowrate is,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(volflow1 = Simplify[Integrate[2\ \[Pi]\ r\ v1prof, {r, 0, D/2 - h}]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(dpdz\ \((D - 2\ h)\)\^2\ \[Pi]\ \((\[Mu]\_2\ \((D - \ 2\ h)\)\^2 + 8\ \((D - h)\)\ h\ \[Mu]\_1)\)\)\/\(128\ \[Mu]\_1\ \ \[Mu]\_2\)\)], "Output"] }, Open ]], Cell["Now do the same for the outside fluid. ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(v2prof = \(dpdz\ \((D\^2 - 4\ r\^2)\)\)\/\(16\ \[Mu]\_2\)\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(dpdz\ \((D\^2 - 4\ r\^2)\)\)\/\(16\ \[Mu]\_2\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(volflow2 = Simplify[Integrate[2\ \[Pi]\ r\ v2prof, {r, D/2 - h, D/2}]]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`\(dpdz\ \((D - h)\)\^2\ h\^2\ \[Pi]\)\/\(8\ \[Mu]\_2\)\ \)], "Output"] }, Open ]], Cell[TextData[{ "To make the comparison we will have to solve for the pressure drop, dpdz \ and ", StyleBox["h", FontSlant->"Italic"], ". We will then plot how pressure drop changes with changes in the \ volumetric flow rate of the outside fluid. We will use numbers to show this.\ \n\nHere is what is solved to get the plot. Note the order of ", StyleBox["h", FontSlant->"Italic"], ". " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(testeqs = {volflow1 - qq1, volflow2 - qq2}\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{\(dpdz\ \((D - 2\ h)\)\^2\ \[Pi]\ \((\[Mu]\_2\ \((D - \ 2\ h)\)\^2 + 8\ \((D - h)\)\ h\ \[Mu]\_1)\)\)\/\(128\ \[Mu]\_1\ \[Mu]\_2\) - qq1, \(dpdz\ \((D - h)\)\^2\ h\^2\ \[Pi]\)\/\(8\ \[Mu]\_2\) - qq2}\)], "Output"] }, Open ]], Cell[TextData[{ "First find the pressure drop for a single phase flow. We can do this if \ ", StyleBox["h", FontSlant->"Italic"], "=0. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ansx1 = Solve[\((testeqs /. h \[Rule] 0)\) \[Equal] 0, dpdz]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{dpdz \[Rule] \(128\ qq1\ \[Mu]\_1\)\/\(D\^4\ \ \[Pi]\)}}\)], "Output"] }, Open ]], Cell["\<\ This looks correct and if we pick some numbers in cgs units for the \ variables we might find, \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[ansx1 /. {D \[Rule] 10, qq1 \[Rule] 1000, qq2 \[Rule] que2, \[Mu]\_2 \[Rule] .01, \[Mu]\_1 \[Rule] 10}]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{{dpdz \[Rule] 40.74366543152521`}}\)], "Output"] }, Open ]], Cell[TextData[{ "So this is the pressure drop with no lubricating flow. Adding outside \ fluid will decrease dpdz from this value. \n\nNow find the pressure drop as \ a function of qq2. We need to use a nonlinear solution technique to get the \ answer since there are at least 6 solutions for different values of ", StyleBox["h", FontSlant->"Italic"], ". You can explore this more if you wish, or just trust me. Note that ", StyleBox["h", FontSlant->"Italic"], " must be positive and less than D. \n\n(To make this work in ", StyleBox["Mathematica", FontSlant->"Italic"], " 4, I needed to pick out root 5 \"[[5]]\" of the 6 roots. I also chose \ the range of the entire calculation \"i,1,20\" and the scaling of que2, \" \ 3+(i-1)*Exp[i/3.3]\" to show the particular region that I wanted. This type \ of behavior will exist for almost any \"qq1\" but you might imagine that it \ is difficult to show because it involves a six-order equation for ", StyleBox["h.)\n\n", FontSlant->"Italic"], "(To make this work in Mathematica 3, you need to pick the first root, \ [[1]]. )" }], "Text"], Cell["\<\ Here is what the set of two equations look like for one set of \ numbers:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(testeqs /. {D \[Rule] 10, qq1 \[Rule] 1000, qq2 \[Rule] que2, \[Mu]\_2 \[Rule] .01, \[Mu]\_1 \[Rule] 10}\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{0.2454369260617026`\ dpdz\ \((10 - 2\ h)\)\^2\ \ \((0.01`\ \((10 - 2\ h)\)\^2 + 80\ \((10 - h)\)\ h)\) - 1000, 39.269908169872416`\ dpdz\ \((10 - h)\)\^2\ h\^2 - 8147.0676034796`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ExpandAll[%]\)], "Input"], Cell[BoxData[ \(TraditionalForm\`{\(-78.50054643157495`\)\ dpdz\ h\^4 + 1570.0109286314992`\ dpdz\ h\^3 - 9811.586556242622`\ dpdz\ h\^2 + 19615.319130851272`\ dpdz\ h + 24.543692606170257`\ dpdz - 1000, 39.269908169872416`\ dpdz\ h\^4 - 785.3981633974483`\ dpdz\ h\^3 + 3926.9908169872415`\ dpdz\ h\^2 - 8147.0676034796`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["Now get the answers for the pressure drop decrease", "Subsubsection"], Cell[TextData[{ "Note for ", StyleBox["Mathematica", FontSlant->"Italic"], " 3 users: You need to change the [[5]] to [[1]] to pick out the first \ root. 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"0.12671985464476165`"}, {"189.331548855916`", "0.10206182019101107`"}, {"283.3162489452614`", "0.08814026990148704`"}, {"420.48929747086396`", "0.08204694476444274`"}, {"619.6493754386614`", "0.08239897011662833`"}, {"907.4921264073827`", "0.08892507062410004`"}, {"1321.8453198533732`", "0.1022957033419516`"}, {"1916.204855288611`", "0.12412584602134302`"}, {"2766.0871707965402`", "0.15712227625772357`"}, {"3977.9156176607826`", "0.20537626289296895`"}, {"5701.438765251701`", "0.27482578775379807`"}, {"8147.0676034796`", "0.37393399071879685`"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Plot of pressure drop decrease", "Subsubsection"], Cell[TextData[ButtonBox["Back to conclusions", ButtonData:>"conclusions", ButtonStyle->"Hyperlink"]], "Text"], Cell[TextData[{ "Note for ", StyleBox["Mathematica", FontSlant->"Italic"], " 3 users: The log-log plot labeling problem mare arise here." }], "Text", 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This makes sense because ", StyleBox["h", FontSlant->"Italic"], " is supposed to be about 30% of D/2 (= 1.5 for this calculation) for the \ most efficient flow. ", " " }], "Text"] }, Closed]] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Conclusions", "Subtitle"], Cell[TextData[{ "1. There is a ", ButtonBox["separate equation", ButtonData:>"twoequations", ButtonStyle->"Hyperlink"], " for the velocity profile of each fluid, but only one equation for the \ stress profile. \n\n2. The form of the differential operator (the \ differential equation) for a circular geometry leads to a ", ButtonBox["logarithmic term", ButtonData:>"diffsolution", ButtonStyle->"Hyperlink"], " in the solution. This term does not give a physically realizable profile \ for the middle of the pipe and must be neglected. (It is kept for fluids \ confined away from the middle.\n\n3. 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