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Notebook[{
Cell["\<\
Demonstration of the effect of flow regime on pressure drop in \
multifluid flows\
\>", "Title"],

Cell[TextData[{
  "This notebook has been written in  ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "by \n\n",
  StyleBox["Mark J. McCready\nProfessor and Chair of Chemical Engineering\n\
University 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: 12/22/99\nMore recent \
versions of this notebook may be available at:\n",
  ButtonBox["http://www.nd.edu/~mjm/Effect.of.Flow.Regime.nb",
    ButtonData:>{
      URL[ "http://www.nd.edu/~mjm/Effect.of.Flow.Regime.nb"], None},
    ButtonStyle->"Hyperlink"]
}], "Text"],

Cell[CellGroupData[{

Cell["Summary", "Subtitle"],

Cell[TextData[{
  "This notebook is intended to give a first introduction to multifluid flows \
through the use of \"model\"  ",
  StyleBox["flow",
    FontSlant->"Italic"],
  " ",
  StyleBox["regimes ",
    FontSlant->"Italic"],
  "calculated from exact solutions for laminar flow in different \
configurations.  By comparing pressure drop over a range of flow rates for \
these different configurations, that ",
  StyleBox["show differences of factors of up to 30",
    FontWeight->"Bold"],
  ", the importance of knowing the flow regime is demonstrated.  Insight into \
the physical reasons for the variation in pressure drop with flow rates and \
physical properties is given.\n "
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True]
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Cell[CellGroupData[{

Cell["Keywords", "Subtitle"],

Cell["\<\
Flow Regimes, Flow Regime Transitions, Gas-Liquid Flow, Multiphase \
Flow, Two-phase Flow, Stratified Flow, Bubbly Flow, Slug Flow, Multifluid \
Flows, \
\>", "Text"]
}, Closed]],

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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 would like to view the ",
  ButtonBox["movies of flow regimes",
    ButtonData:>"movies",
    ButtonStyle->"Hyperlink"],
  " you will need web access and a web browser (which you might need to have \
turned on first).  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\nIf you have to, you might be able to get some of the message \
from reading the .pdf file.   \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"],
  ".  \n      "
}], "Text",
  CellTags->"howto"]
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Cell[CellGroupData[{

Cell["Motivation: Importance of multifluid flows", "Subtitle",
  Background->RGBColor[0.8, 0.919997, 0.919997]],

Cell[TextData[{
  "There are several reasons that multifluid flows are of interest.  In the \
process industries, operations such as extraction, distillation and stripping \
involve contacting of immiscible fluids and thus multifluid flows.  Key \
questions are the amount of interfacial area produced by various mixing \
schemes, the interfacial transport rates and possibly pressure drop for \
pumping.  In the petroleum industry, long distance transportation by pipe of \
gas-liquid mixtures is quite common.  Prediction of flow regime and pressure \
drop ",
  StyleBox["\[LongDash]",
    FontSize->11],
  " flowrate relations are crucial to success of design and operation ",
  StyleBox["--",
    FontSize->10],
  " particularly of facilities to produce oil from deep water.  \n\nThe \
academic interest of multifluid flows stems from the inherent complexity of \
these flows.  For a horizontal two-fluid flow in a simple geometry (circular \
or 2-D), there are six dimensionless groups.  Thus a general predictive \
relation for say, flowrate ",
  StyleBox["\[LongDash]",
    FontSize->10],
  StyleBox[" ",
    FontSize->11],
  "pressure drop, cannot be obtained in a simple a form such as the ",
  StyleBox["f",
    FontSlant->"Italic"],
  " - ",
  StyleBox["Re",
    FontSlant->"Italic"],
  " relation for single phase flow.  With two or more fluids, the new \
problems that arise and really make the problem more interesting (and hard) \
are the presence of ",
  StyleBox["interface",
    FontSlant->"Italic"],
  " with dynamics different from either single phase and the issue of the \
location of the phases.         "
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Cell["Importance of knowing the flow regime", "Subtitle"],

Cell["\<\
Prediction of flow regime for process and pipeline flows is still \
considered one of the major unsolved problems of multifluid flows because \
prediction procedures typically have limited (and sometimes unknown) ranges \
of validity and the regime exerts a zero order effect on the important \
macroscopic properties of the flow.  
\
\>", "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[
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  StyleBox[" = ",
    FontSize->15],
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  " = .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},
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  "\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, 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",
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  "\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"],
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  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\
"
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Cell[CellGroupData[{

Cell["Intent of this notebook", "SectionFirst"],

Cell[TextData[{
  "\nThis notebook is intended to serve as a basic introduction into one \
aspect of multifluid flows \[Dash] the importance of knowing the flow \
geometry, that is the ",
  StyleBox["flow regime, ",
    FontSlant->"Italic"],
  "when a flowrate \[Dash] pressure drop relation is needed.  To do this we \
use three idealized flows that have (more or less) exact solutions of the \
governing equations.  It should be noted that for various parameter ranges it \
may not be possible to physically construct the flows that are being \
computed.  However, it should also be realized that this demonstration \
exercise, ",
  StyleBox["does not",
    FontWeight->"Bold"],
  " in any way, ",
  StyleBox["overemphasize",
    FontWeight->"Bold"],
  " the importance of correct knowledge that the ",
  StyleBox["regime",
    FontSlant->"Italic"],
  " has on the design of a process flow!  \n\nThere are very few exact \
solutions of the governing equations for multifluid flows.  Here we use one \
exact solution, a two-layer laminar, ",
  StyleBox["stratified",
    FontSlant->"Italic"],
  " flow and two idealized solutions, an alternating fluid1-fluid2 laminar \
flow (which we term a ",
  StyleBox["slug",
    FontSlant->"Italic"],
  " flow) and a ",
  StyleBox["dispersed",
    FontSlant->"Italic"],
  " fluid1-fluid 2 flow that is laminar with fluid properties described by an \
average viscosity and density.\n\nOur goal is to first show the extent of the \
predicted difference in the pressure drop depending upon the arrangement of \
the phases.  Second we would like to gain some physical understanding of  the \
differences between multifluid flows and single phase flows.  Finally, some \
effort is given to demonstrating various mathematical manipulations that are \
useful for studying the equations of multifluid flows.        "
}], "Text"]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Preview of Major points", "Subtitle",
  Background->RGBColor[0.6, 0.6, 1],
  CellTags->{"dp30ret", "retroreturn"}],

Cell[TextData[{
  "We have shown, using simple models for flow regimes,  ",
  ButtonBox["stratified",
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    ButtonStyle->"Hyperlink"],
  ", ",
  ButtonBox["slug",
    ButtonData:>"slugpic",
    ButtonStyle->"Hyperlink"],
  " and ",
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    ButtonStyle->"Hyperlink"],
  ", that\n\n1.\tThe ",
  StyleBox["qualitative",
    FontSlant->"Italic"],
  " as well as the ",
  StyleBox["quantitative",
    FontSlant->"Italic"],
  " behavior of multiphase flows will change as the ratios of flow rates and \
physical properties change.\n\n2.\tThe pressure drop predictions differ \
substantially with flow configuration.  The pressure drop for dispersed flow \
was predicted to be a ",
  ButtonBox["factor of 35",
    ButtonData:>"dp30",
    ButtonStyle->"Hyperlink"],
  " higher than for slug flow in one case and a",
  ButtonBox[" factor of 20",
    ButtonData:>"dp20a",
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  " greater than stratified flow for another case  This key result is true \
for process flows and makes correct prediction of the flow regime crucial to \
successful design of multifluid systems.  Most engineering designs cannot \
stand an uncertainty of a factor of 2 in the main design variable, let alone \
30. \n\n3.\t",
  ButtonBox["Stratified flow",
    ButtonData:>"stratpic",
    ButtonStyle->"Hyperlink"],
  "  is the most efficient configuration, of the three tested here (compare \
stratified/slug, dispersed/stratified), for fluid transport when the more \
viscous fluid has a higher flow rate.  This is due to the lubricating effect \
of the less viscous fluid that reduces shear in the more viscous fluid.  This \
is the basis of ",
  ButtonBox["lubricated",
    ButtonData:>"lubricating",
    ButtonStyle->"Hyperlink"],
  " pipeline transport of heavy oil (See D. D. Joseph and Y. Y. Renardy, ",
  StyleBox["Fundamentals of Two-Fluid Dynamics",
    FontSlant->"Italic"],
  ", Springer-Verlag, 1993, Vol. 2.)  If the more viscous fluid is present in \
lessor amounts the advantage is lost because it is subjected to high shear \
and acts to reduce the available flow area for the less viscous fluid.  \n\n\
4.\tThe loss of lubricating effect of a less viscous fluid in stratified flow \
can cause a region where ",
  StyleBox["decreasing",
    FontSlant->"Italic"],
  " the flow rate of the less viscous fluid, ",
  StyleBox["increases",
    FontSlant->"Italic"],
  " the pressure drop (",
  ButtonBox["click",
    ButtonData:>"retrograde",
    ButtonStyle->"Hyperlink"],
  " for specifics about ",
  StyleBox["retrograde",
    FontSlant->"Italic"],
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flow situations.  \n\n5.\tThe specific conclusions for  ",
  ButtonBox["dispersed/slug,",
    ButtonData:>"dispersedslugconclusions",
    ButtonStyle->"Hyperlink"],
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  ButtonBox["dispersed/stratified",
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  ButtonBox["stratified/slug",
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pressure drop with configuration for the examples in this notebook is that \
the dissipation is altered.  Differences in dissipation arise primarily when \
fluids of different viscosities are located in regions of different stress.  \
We also find that changing the effective flow area (i.e., by having a \
stratified region of more viscous fluid) for the fast moving fluid changes \
the dissipation significantly.  These general observations should hold for \
either laminar flow (as shown here) or turbulent flow.  However, if the \
primary contributions to pressure drop are from fluid acceleration,  or \
gravity, then the pressure drop differences  caused by the flow regime could \
be less than shown here.   Examples are unsteady or transient flows, \
developing flows or vertical flows.    \n\n",
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Cell[TextData[{
  "As shown in the picture, we consider a two-layer, steady, laminar flow \
with a flat interface.  The two fluids are presumed to have different \
viscosities and densities (and be immiscible) so that the separate regions \
are governed by separate flow equations.  The no-slip condition will be \
enforced at the lower (",
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boundary conditions.\n\nExperimental and process flows are often stratified \
although the usual case would be that waves form.  (Consider also the case of \
wind blowing over a body of water.  There will be a current caused by the \
wind and waves may form.)  If you would like to see ",
  ButtonBox["still pictures of waves",
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  ", 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 ."
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I have used a mix of input notation to show how it can be done. \
\
\>", "Text"],

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I would enter the dynamic boundary condition from a key pad as\
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Note that you can convert any \"cryptic\" input expression by \
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We can solve all of the above equations together and simplify the \
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\>", "Text"],

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We should check the profiles to see if they make sense.  Here we \
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\>", "Text"],

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\>", "Text"],

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Cell[CellGroupData[{

Cell["Average velocities", "Subsubsection"],

Cell["\<\
We need the average velocities to make comparisons based on flow \
rates.\
\>", "Text"],

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Cell["Pressure drop in terms of flow rates", "Subsubsection"],

Cell[TextData[{
  "In an experimental or process flow, the flow rates for the fluids are \
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Cell["Start with the average equation for the lower fluid, ", "Text"],

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  "Solve this for the pressure drop.  Note that if  ",
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Then we get the stratified pressure drop in a useful form for later \
calculations.  \
\>", "Text"],

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Cell["\<\
Use the pressure drop in the equation for the average velocity of \
the second fluid to get a useful relation for its average velocity. \
\>", 
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Cell["Friction velocity", "Subsubsection"],

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  "It is useful to also calculate the interfacial friction velocity.  It is \
often termed, v*, and is defined as  ",
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  "We choose a channel is ",
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wall.  We presume that we can describe a dispersed flow with an average \
viscosity and density.  Note that this works sometimes, but in real flows \
there are several problems.  First, it should be recognized, as mentioned \
above, that it may not be possible to construct a uniformly dispersed \
(homogeneous) flow at all flow rates.  Second even if the flow exists, there \
can be an effective \"slip\" between the phases so that a \"drift flux\" \
model is needed (see ",
  StyleBox[
  "Two-phase flows and heat transfer with applications to nuclear reactor \
design problems",
    FontSlant->"Italic"],
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little slip between the phases, the average viscosity of the mixture can be a \
complicated and often unknown function (ibid p22; M. Ishii & N. Zuber ",
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  " ",
  StyleBox["J",
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  " pp843-854, 1979).  Finally, even when the flow is nominally dispersed, it \
may not be homogeneous so that people solve averaged versions of the \
Navier-Stokes equations for each phase with appropriate closures.   "
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Cell["Equations for a steady, laminar single phase flow", "Subsubsection"],

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The single differential equation and no slip boundary conditions \
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\>", "Text"],

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We might be interested in the friction factor, which is defined as \
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\>", "Text"],

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It could also be of interest to derive the wall friction velocity.  \
\
\>", "Text"],

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          ButtonStyle->"Hyperlink"], TraditionalForm]]]
}], "Text"],

Cell[CellGroupData[{

Cell["Problem description", "Subsection"],

Cell["\<\
We cannot easily compute the exact relation for this alternating \
geometry even for laminar flow, although a number of solutions exist for \
bubbles in capillaries at low Reynolds number.  We will instead restrict our \
solution to the case where the slugs and bubbles are long compared to the \
diameter (not like the picture) so that each region is in fully-developed \
laminar  flow.  We can then get an average pressure drop using the volumetric \
flowrate to tell us the fraction of the time that each region is \
contributing.   \
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Equations for a steady, laminar single phase flow", "Subsubsection"],

Cell[BoxData[
    \(slugeq1 = \[Mu]\ \[PartialD]\_{y, 2}u[y] - dpdx\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    FormBox[
      RowBox[{
        RowBox[{"\[Mu]", " ", 
          RowBox[{
            SuperscriptBox["u", "\[DoublePrime]",
              MultilineFunction->None], "(", "y", ")"}]}], "-", "dpdx"}], 
      TraditionalForm]], "Output"],

Cell[TextData[{
  "We choose a channel that is  ",
  StyleBox["H",
    FontSlant->"Italic"],
  " high with fixed wall boundary conditions and the origin on the bottom \
wall. "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(slugans1 = DSolve[{slugeq1 == 0, u[0] == 0, u[H] == 0}, u[y], y]\)], 
  "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`{{u(y) \[Rule] 
          \(dpdx\ y\^2\)\/\(2\ \[Mu]\) - \(dpdx\ H\ y\)\/\(2\ \[Mu]\)}}\)], 
  "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Velocity profile", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(slugans2 = 
      u[y] /. slugans1\[LeftDoubleBracket]1\[RightDoubleBracket]\)], "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm
    \`\(dpdx\ y\^2\)\/\(2\ \[Mu]\) - \(dpdx\ H\ y\)\/\(2\ \[Mu]\)\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Average velocity", "Subsubsection"],

Cell["We need the average velocity to get the flow rate.  ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(uslugave = \(\[Integral]\_0\%H slugans2 \[DifferentialD]y\)\/H\)], 
  "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`\(-\(\(dpdx\ H\^2\)\/\(12\ \[Mu]\)\)\)\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Pressure drop", "Subsubsection"],

Cell["\<\
Rearrange the average velocity equation to get the pressure drop.\
\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dpdx\_slug = dpdx /. \(Solve[uslugave == U\&_, dpdx]\)[\([1]\)]\)], 
  "Input",
  AspectRatioFixed->True],

Cell[BoxData[
    \(TraditionalForm\`\(-\(\(12\ \[Mu]\ U\&_\)\/H\^2\)\)\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Friction factor", "Subsubsection"],

Cell[TextData[{
  "It is convenient to construct a friction factor  ",
  StyleBox["\[Dash] ",
    FontSize->11],
  "Reynolds number relation from this.  "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\nftemp = \(-H\)\ \(\(dpdx\_slug/2\)/\[Rho]\)/U\&_\^2\ \ \)\)], 
  "Input"],

Cell[BoxData[
    \(TraditionalForm\`\(6\ \[Mu]\)\/\(H\ \[Rho]\ U\&_\)\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Which is the expected ",
  StyleBox["f",
    FontSlant->"Italic"],
  " = 6/",
  StyleBox["Re",
    FontSlant->"Italic"],
  " for a channel flow.  "
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Friction velocity", "Subsubsection"],

Cell["It is also useful to derive the friction velocity.  ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(vstar\_slug\  = \ 
      Sqrt[\[Mu] \((D[slugans2, y] /. y -> 0)\)/\[Rho]] /. 
        dpdx -> dpdx\_slug\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`\@6\ \@\(\(\[Mu]\ U\&_\)\/\(H\ \[Rho]\)\)\)], "Output"]
}, Open  ]]
}, Closed]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Pressure drop comparisons", "Subtitle"],

Cell["\<\
We now need to compare the pressure drop for the different \
configurations to see what happens.  To show the importance of the regime, \
the ratio of pressure drop between two different regimes will be plotted as a \
function of flow rate.  \
\>", "Text"],

Cell[CellGroupData[{

Cell["Dispersed / Slug", "Subsection"],

Cell[CellGroupData[{

Cell["Pressure drop ratio", "Subsubsection"],

Cell["\<\
The pressure drop ratio for dispersed flow to slug flow is given as\
\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dpdx\_disp/dpdx\_slug\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`\(\[Mu]\_mix\ U\&_\_mix\)\/\(\[Mu]\ U\&_\)\)], "Output"]
}, Open  ]],

Cell["\<\
Sensible values for the velocities and viscosities are needed in \
this expression.  \
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Dispersed relations", "Subsubsection"],

Cell["For the mixture velocity, we just use the total flow rates", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Umix\  = \ 
      \(\(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L + \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G\)\/H
        \)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(\(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G + \(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L\)\/H
      \)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Averages will be used for the viscosity and density.  A straight volume \
average should be good for the density.  This will not be good for the \
viscosity and we will use a mass fraction weighting for viscosity.  \n\nThe \
volume averaged expression  is ",
  Cell[BoxData[
      \(\(\[Xi]\_average\  = \)\)]],
  " volfrac(L) ",
  Cell[BoxData[
      \(\[Xi]\_L\)]],
  " + volfrac(G) ",
  Cell[BoxData[
      \(\[Xi]\_G\)]],
  ".  The volume fraction for, say, the L phase is ",
  Cell[BoxData[
      \(q\_L\/\(q\_L\  + \ q\_G\)\)]],
  ". \n\nThe expression for the density is,  "
}], "Text"],

Cell[BoxData[
    \(\(\[Rho]mix = 
      \(\(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L\ \[Rho]\_L\)\/\((
            \(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L + \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G)
            \) + \(\(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G\ \[Rho]\_G\ 
          \)\/\((\(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L + 
              \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G)\); \)\)], "Input",
  AspectRatioFixed->True],

Cell[TextData[{
  "For the viscosity we will use an expression (from  ",
  StyleBox[
  "Two-phase flows and heat transfer with applications to nuclear reactor \
design problems",
    FontSlant->"Italic"],
  ", ed. J. J. Ginoux, Hemisphere, 1978, p22), attributed to Cicchitti et \
al.. It is essentially a mass fraction weighting and assumes that the more \
viscous fluid is also more dense.  (Of course this is not always true!)  If \
you don't like this one, check out the above-mentioned Ishii and Zuber paper. \
 "
}], "Text"],

Cell[BoxData[
    \(\(\(\[Mu]mix = \ 
      \(\(\(\[Rho]\_L\) \(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L\)\/\((
                \(\[Rho]\_L\) \(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L + 
                  \(\[Rho]\_G\) \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G)\)\) 
          \[Mu]\_L\  + 
        \(\(\[Rho]\_G\) \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G\)\/\((
              \(\[Rho]\_L\) \(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L + 
                \(\[Rho]\_G\) \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G)\)\ \[Mu]\_G\ ; 
    \)\ \n\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[Mu]mixsimp = Simplify[\[Mu]mix]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(Re\_G\ \[Mu]\_G\%2 + Re\_L\ \[Mu]\_L\%2\)\/\(Re\_G\ \[Mu]\_G + 
        Re\_L\ \[Mu]\_L\)\)], "Output"]
}, Open  ]],

Cell["\<\
Now substitute the averages into the expression for the dispersed \
pressure drop. \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(disptemp = 
      FullSimplify[
        dpdx\_disp /. {\[Mu]\_mix -> \[Mu]mixsimp, \n\t\t\tU\&_\_mix -> Umix}]
        \)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(12\ \((Re\_G\ \[Mu]\_G\%2 + Re\_L\ \[Mu]\_L\%2)\)\ 
            \((\(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G + 
                \(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L)\)\)\/\(H\^3\ 
            \((Re\_G\ \[Mu]\_G + Re\_L\ \[Mu]\_L)\)\)\)\)\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Slug relations", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dpdx\_slug\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`\(-\(\(12\ \[Mu]\ U\&_\)\/H\^2\)\)\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "From the relation for ",
  Cell[BoxData[
      \(dpdx\_slug\)]],
  ", we see that we need an average of the product of ",
  Cell[BoxData[
      \(TraditionalForm\`\(\[Mu]\ \)\)]],
  "and ",
  Cell[BoxData[
      \(TraditionalForm\`\(\ U\&_\)\)]],
  ".  The way this works is that the pressure drop for each region is just ",
  Cell[BoxData[
      \(\(\(-12\)\ \[Mu]\_i\ U\_i\)\/H\^2\)]],
  " but the ",
  Cell[BoxData[
      \(U\_i\)]],
  " is increased over its single phase velocity by the loss of flow area \
owing to the presence of the other phase.  So in a region of L, the pressure \
drop is higher than if no G were present.  Of course, the entire pipe is not \
filled with L, it is only ",
  Cell[BoxData[
      \(U\_L\/\((U\_L + U\_G)\)\)]],
  "  full of L.  So we can combine this to get the final result. \n\nThe \
fraction still available to fluid L should be"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(XL = 1 - U\_G\/\((U\_L + U\_G)\)\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1 - U\_G\/\(U\_G + U\_L\)\)], "Output"]
}, Open  ]],

Cell["The fraction available for fluid G is then", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(XG = 1 - U\_L\/\((U\_L + U\_G)\)\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1 - U\_L\/\(U\_G + U\_L\)\)], "Output"]
}, Open  ]],

Cell["This gives a pressure drop that becomes", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dpslug = \ 
      \(\(-12\)\/H\^2\) 
        \((\(\(\(\[Mu]\_L\) U\_L\)\/XL\) U\_L\/\((U\_L + U\_G)\) + \ 
            \(\(\(\[Mu]\_G\) U\_G\)\/XG\) U\_G\/\((U\_L + U\_G)\))\)\)], 
  "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(12\ 
            \((\(\[Mu]\_G\ U\_G\%2\)\/\(\((U\_G + U\_L)\)\ 
                    \((1 - U\_L\/\(U\_G + U\_L\))\)\) + 
                \(U\_L\%2\ \[Mu]\_L\)\/\(\((U\_G + U\_L)\)\ 
                    \((1 - U\_G\/\(U\_G + U\_L\))\)\))\)\)\/H\^2\)\)\)], 
  "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(slugtemp = 
      Simplify[dpslug /. {U\_L -> \(Re\_L\ \[Mu]\_L\)\/\(H\ \[Rho]\_L\), 
            U\_G -> \(Re\_G\ \[Mu]\_G\)\/\(H\ \[Rho]\_G\)}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(12\ 
            \((Re\_G\ \[Rho]\_L\ \[Mu]\_G\%2 + Re\_L\ \[Mu]\_L\%2\ \[Rho]\_G)
              \)\)\/\(H\^3\ \[Rho]\_G\ \[Rho]\_L\)\)\)\)], "Output"]
}, Open  ]],

Cell["This is, of course, the same as the obvious answer, ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dpslug2 = 
      FullSimplify[
        \((\ \(\(-12\)\/H\^2\) \((\(\[Mu]\_L\) U\_L + \ \(\[Mu]\_G\) U\_G)\))
            \) /. {U\_L -> \(Re\_L\ \[Mu]\_L\)\/\(H\ \[Rho]\_L\), 
            U\_G -> \(Re\_G\ \[Mu]\_G\)\/\(H\ \[Rho]\_G\)}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(12\ 
            \((Re\_G\ \[Rho]\_L\ \[Mu]\_G\%2 + Re\_L\ \[Mu]\_L\%2\ \[Rho]\_G)
              \)\)\/\(H\^3\ \[Rho]\_G\ \[Rho]\_L\)\)\)\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["The dispersed- slug ratio", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dispslugratio = Simplify[disptemp/slugtemp]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(\((Re\_G\ \[Mu]\_G\%2 + Re\_L\ \[Mu]\_L\%2)\)\ 
        \((Re\_L\ \[Mu]\_L\ \[Rho]\_G + Re\_G\ \[Mu]\_G\ \[Rho]\_L)\)\)\/\(\((
          Re\_G\ \[Mu]\_G + Re\_L\ \[Mu]\_L)\)\ 
        \((Re\_G\ \[Rho]\_L\ \[Mu]\_G\%2 + Re\_L\ \[Mu]\_L\%2\ \[Rho]\_G)
          \)\)\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Limits of the expression", "Subsubsection"],

Cell[CellGroupData[{

Cell["This result gives all of the expected limits.  ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(ratiotest\  = \ 
      FullSimplify[
        dispslugratio /. {Re\_L -> \ U\_L\ \(\[Rho]\_L\) H/\[Mu]\_L, \n\t\t\t
            Re\_G -> \ U\_G\ \(\[Rho]\_G\) H/\[Mu]\_G}]\n\t\t\t\)\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(\((U\_G + U\_L)\)\ 
        \((U\_G\ \[Mu]\_G\ \[Rho]\_G + U\_L\ \[Mu]\_L\ \[Rho]\_L)\)\)\/\(\((
          U\_G\ \[Mu]\_G + U\_L\ \[Mu]\_L)\)\ 
        \((U\_G\ \[Rho]\_G + U\_L\ \[Rho]\_L)\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(ratiotest2 = 
      FullSimplify[ratiotest /. \n\t\t\t{\ \[Mu]\_G -> \[Mu]\_L}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Limit[dispslugratio, Re\_G -> 0]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Limit[dispslugratio, Re\_G -> Infinity]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1\)], "Output"]
}, Open  ]]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Plots of the ratio", "Subsubsection"],

Cell[CellGroupData[{

Cell[TextData[{
  "Air-water in a 2.54 cm channel, ",
  Cell[BoxData[
      \(Re\_L\)]],
  " =100.  \nThe pressure drop ratio for dispersed flow divided by slug flow \
gives a difference of a factor greater than 30.   The liquid flow rate is \
held constant as the gas flow is increased.  "
}], "Text",
  CellTags->"dp30"],

Cell[" Pressure drop comparison for dispersed vs. slug flow ", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(Plot[
      Evaluate[dispslugratio /. {Re\_L -> 100, \[Mu]\_L ->  .01, \ 
            \[Mu]\_G ->  .00018, \n\t\t\t\[Rho]\_L -> 1, \ 
            \[Rho]\_G -> 1/899, H -> 1}], {Re\_G, 1, \ 10000}, 
      AxesLabel -> {\*"\"\<\!\(Re\_G\)\>\"", "\<dispersed/slug ratio\>"}]; 
    \)\)], "Input",
  CellTags->"dp30"],

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}, Open  ]]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Dispersed - slug observations", "Subsubsection"],

Cell["\<\
The first observation, that was probably obvious, is that the \
character of the flow can change with the relative values of the two flow \
rates.  

For gas-liquid cases, we see that dispersed flow always has a larger pressure \
drop that slug.  This is evidently because the viscosity of the mixture is \
always higher than the less viscous fluid and thus the pressure drop is \
always increased over a single phase flow of the less viscous fluid.  In \
contrast, the alternating flow has regions of the low viscosity fluid in \
laminar  flow which gives the lowest possible pressure drop for a requisite \
fraction of the flow distance.

For oil-water flows, dispersed flow has a lower pressure drop.  This is \
evidently because of the density weighting of the mixture viscosity function \
which keeps the viscosity below its volume fraction value.

For all of the test cases, there is an intermediate  maximum and an \
intermediate minimum that appear to depend on the viscosity ratio.  Let's see \
if we can make some sense of this.    \
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Analysis", "Subsubsection"],

Cell["\<\
The Reynolds number is not an intrinsically important parameter in \
this problem so let us do some rearranging.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dpmax1 = 
      dispslugratio /. {Re\_L -> \ U\_L\ \(\[Rho]\_L\) H/\[Mu]\_L, \n\t\t\t
          Re\_G -> \ U\_G\ \(\[Rho]\_G\) H/\[Mu]\_G}\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(\((H\ U\_G\ \[Mu]\_G\ \[Rho]\_G + H\ U\_L\ \[Mu]\_L\ \[Rho]\_L)\)\ 
        \((H\ U\_G\ \[Rho]\_G\ \[Rho]\_L + H\ U\_L\ \[Rho]\_G\ \[Rho]\_L)
          \)\)\/\(\((H\ U\_G\ \[Rho]\_G + H\ U\_L\ \[Rho]\_L)\)\ 
        \((H\ U\_G\ \[Mu]\_G\ \[Rho]\_G\ \[Rho]\_L + 
            H\ U\_L\ \[Mu]\_L\ \[Rho]\_G\ \[Rho]\_L)\)\)\)], "Output"]
}, Open  ]],

Cell["\<\
Now get the pressure drop ratio completely in terms of parameter \
ratios.  Note we could have done the whole problem this way by using the \
appropriate nondimensionalization at the beginning.   \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dpmax2 = 
      FullSimplify[
        dpmax1 /. {\[Mu]\_G -> \ m\ \[Mu]\_L, \ U\_G\  -> \ \[Psi]\ U\_L, \n
            \t\t\t\t\[Rho]\_G -> r\ \[Rho]\_L}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(\((\[Psi] + 1)\)\ \((m\ r\ \[Psi] + 1)\)\)\/\(\((m\ \[Psi] + 1)\)\ 
        \((r\ \[Psi] + 1)\)\)\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "If the viscosities are equal,  ",
  Cell[BoxData[
      \(TraditionalForm\`m\)]],
  "=1, then the models give the same result as they should."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[dpmax2 /. m -> 1]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Likewise, if the densities of the fluids are equal, ",
  Cell[BoxData[
      \(TraditionalForm\`r\)]],
  "=1, the pressure drop ratio is unity.  "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[dpmax2 /. r -> 1]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "For ",
  StyleBox["r",
    FontSlant->"Italic"],
  " not equal to unity, we see that we have the same behavior as above using \
this simpler expression and we now find that there is a maximum.  Note that \
in the limit of gas/liquid flowrate ratio, \[Psi]-> \[Infinity], the pressure \
ratio returns to unity.  "
}], "Text"],

Cell[CellGroupData[{

Cell["We need to load a package to do a log plot", "Text"],

Cell[BoxData[
    \(<< \ Graphics`Graphics`\)], "Input"]
}, Open  ]],

Cell["Here is air-water.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(LogLogPlot[dpmax2 /. {m ->  .02, r ->  .001}, {\[Psi],  .01, 100000}, 
      AxesLabel -> {"\<flow ratio\>", "\<dispersed/slug ratio\>"}]; \)\)], 
  "Input"],

Cell[GraphicsData["PostScript", "\<\
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  "<1 and ",
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  "<1, there is a maximum at high flow rates.  For ",
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  "<1, ",
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Cell["Now take the derivative to find the extremum, ", "Text"],

Cell[CellGroupData[{

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Cell[BoxData[
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            \((r\ \[Psi] + 1)\)\^2\)\)\)\)], "Output"]
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Cell[TextData[
"We are fortunate that we get a simple result for the flow rate ratio, \[Psi] \
of the extreme point,  "], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[
"We see that \[Psi], the flow rate ratio, for the maximum pressure drop will \
be large for air-water flows and possibly closer to unity for oil-water flows \
for the minimum.  The pressure drop value of the maximum or minimum is given \
by substituting this into the pressure drop ratio expression, "], "Text"],

Cell[CellGroupData[{

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Cell[BoxData[
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Cell["\<\
It is seen that this value can be arbitrarily large, or somewhat \
smaller than 1.  Most engineering designs cannot stand a factor of 20 \
uncertainty in a key parameter.\
\>", "Text"],

Cell["\<\
Consider gas-liquid density ratio.  The air-water viscosity ratio \
would be 0.018, giving a factor of 30 of difference in the pressure \
prediction of the two models. \
\>", "Text"],

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Cell[CellGroupData[{

Cell["Conclusions for dispersed verus slug flows", "Subsubsection"],

Cell["\<\
The dispersed flow has a higher pressure drop than the model slug \
flow if the less viscous fluid has a lower density than the more viscous \
fluid.  This difference can easily be a factor of 30 for reasonable values of \
the viscosity and velocity ratios.  Dispersed flow gives a modest reduction \
in pressure drop (~10-15%) occurs (compared to slug) if the more viscous \
fluid has a lower density.  It should be emphasized that the model for the \
mixture viscosity of the dispersed flow can significantly change these  \
results (Check this yourself).  \
\>", "Text",
  CellTags->"dispersedslugconclusions"],

Cell[TextData[{
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    ButtonData:>"preview",
    ButtonStyle->"Hyperlink"],
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Cell["\<\
We can examine the viscosity relation for dispersed flow to gain \
some physical insight into why the dispersed flow can have a much higher \
pressure drop for air-water.  

Change the mixture viscosity into ratios,\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(visc1 = 
      \[Mu]mix /. {Re\_L -> \ U\_L\ \(\[Rho]\_L\) H/\[Mu]\_L, \n\t\t\t
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Cell[BoxData[
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    \`\(H\ U\_G\ \[Mu]\_G\ \[Rho]\_G\)\/\(H\ U\_G\ \[Rho]\_G + 
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      \(H\ U\_L\ \[Mu]\_L\ \[Rho]\_L\)\/\(H\ U\_G\ \[Rho]\_G + 
          H\ U\_L\ \[Rho]\_L\)\)], "Output"]
}, Open  ]],

Cell["Now compare the mixture to the liquid viscosity, ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \(TraditionalForm\`\(m\ r\ \[Psi] + 1\)\/\(r\ \[Psi] + 1\)\)], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
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We see that the mixture viscosity is staying close to the liquid \
value even when the flow has a high volume fraction of the gas.  Thus while \
the alternating flow has regions of low pressure drop, (the gas), the \
dispersed flow never gets this benefit. \
\>", "Text"],

Cell["\<\
In contrast for oil-water, since water is less viscous and more \
dense, the mixture viscosity does not increase too quickly as the oil flow is \
increased so there is a range where the dispersed pressure drop is not as \
large as the alternating flow.\
\>", "Text"],

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Cell[CellGroupData[{

Cell["Dispersed / Stratified", "Subsection"],

Cell[CellGroupData[{

Cell["Pressure drop ratio", "Subsubsection"],

Cell["Here is the ratio of dispersed flow to stratified flow", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dpdx\_disp/dpdx\_strat\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(h\^2\ \((\((h\^2 + 2\ H\ h - 3\ H\^2)\)\ \[Mu]\_L - h\^2\ \[Mu]\_G)
          \)\ \[Mu]\_mix\ \[Rho]\_L\ U\&_\_mix\)\/\(H\^2\ Re\_L\ \[Mu]\_L\%2\ 
        \((\((h - H)\)\ \[Mu]\_L - h\ \[Mu]\_G)\)\)\)], "Output"]
}, Open  ]],

Cell["\<\
We now need sensible values for the velocities and viscosities.  \
\
\>", "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Dispersed relations", "Subsubsection"],

Cell["For the mixture velocity, we just use the total flow rates", "Text"],

Cell[BoxData[
    \(\(Umix\  = \ 
      \(\(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L + \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G\)\/H; 
    \)\)], "Input"],

Cell[TextData[{
  "Averages will be used for the viscosity and density.  A straight volume \
average should be good for the density.  This will not be good for the \
viscosity and we will use a mass fraction weighting for viscosity.  \n\nThe \
volume averaged expression  is ",
  Cell[BoxData[
      \(\(\[Xi]\_average\  = \)\)]],
  " volfrac(L) ",
  Cell[BoxData[
      \(\[Xi]\_L\)]],
  " + volfrac(G) ",
  Cell[BoxData[
      \(\[Xi]\_G\)]],
  ".  The volume fraction for, say, the L phase is ",
  Cell[BoxData[
      \(q\_L\/\(q\_L\  + \ q\_G\)\)]],
  ". \n\nThe expression for the density is,  "
}], "Text"],

Cell[BoxData[
    \(\(\[Rho]mix = 
      \(\(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L\ \[Rho]\_L\)\/\((
            \(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L + \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G)
            \) + \(\(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G\ \[Rho]\_G\ 
          \)\/\((\(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L + 
              \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G)\); \)\)], "Input",
  AspectRatioFixed->True],

Cell[TextData[{
  "For the viscosity we will use an expression (from  ",
  StyleBox[
  "Two-phase flows and heat transfer with applications to nuclear reactor \
design problems",
    FontSlant->"Italic"],
  ", ed. J. J. Ginoux, Hemisphere, 1978, p22), attributed to Cicchitti et \
al.. It is essentially a mass fraction weighting and assumes that the more \
viscous fluid is also more dense.  (Of course this is not always true!)  If \
you don't like this one, check out the above-mentioned Ishii and Zuber paper. \
 "
}], "Text"],

Cell[BoxData[
    \(\(\(\[Mu]mix = \ 
      \(\(\(\[Rho]\_L\) \(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L\)\/\((
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                  \(\[Rho]\_G\) \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G)\)\) 
          \[Mu]\_L\  + \n\t\t\t\t\t\t\t\ 
        \(\(\[Rho]\_G\) \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G\)\/\((
              \(\[Rho]\_L\) \(Re\_L\ \[Mu]\_L\)\/\[Rho]\_L + 
                \(\[Rho]\_G\) \(Re\_G\ \[Mu]\_G\)\/\[Rho]\_G)\)\ 
          \[Mu]\_G\ \ ; \)\n\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Simplify[\[Mu]mix]\)], "Input"],

Cell[BoxData[
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    \`\(Re\_G\ \[Mu]\_G\%2 + Re\_L\ \[Mu]\_L\%2\)\/\(Re\_G\ \[Mu]\_G + 
        Re\_L\ \[Mu]\_L\)\)], "Output"]
}, Open  ]],

Cell["\<\
Now substitute the averages into the expression for the dispersed \
pressure drop. \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(disptemp = 
      Simplify[dpdx\_disp /. {\[Mu]\_mix -> \[Mu]mix, \n\t\t\t
            U\&_\_mix -> Umix}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(12\ \((Re\_G\ \[Mu]\_G\%2 + Re\_L\ \[Mu]\_L\%2)\)\ 
            \((Re\_L\ \[Mu]\_L\ \[Rho]\_G + Re\_G\ \[Mu]\_G\ \[Rho]\_L)
              \)\)\/\(H\^3\ \((Re\_G\ \[Mu]\_G + Re\_L\ \[Mu]\_L)\)\ 
            \[Rho]\_G\ \[Rho]\_L\)\)\)\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Stratified relations", "Subsubsection"],

Cell[TextData[{
  "We need to make the translation from ",
  Cell[BoxData[
      \(Re\_L\)]],
  " and ",
  Cell[BoxData[
      \(\(Re\_G\ \)\)]],
  "and we need to eliminate the ",
  Cell[BoxData[
      \(Re\_G\)]],
  "from the dispersed flow dpdx, shown just above.\n\nHere is an expression \
for ",
  Cell[BoxData[
      \(Re\_G\)]],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(regfunc\  = Ugexpress\ \((H - h)\)\ \[Rho]\_G/\[Mu]\_G\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(\((h - H)\)\ \((H - h)\)\ Re\_L\ \[Mu]\_L\%2\ 
            \((\((h - H)\)\^2\ \[Mu]\_L - h\ \((h - 4\ H)\)\ \[Mu]\_G)\)\ 
            \[Rho]\_G\)\/\(h\^2\ \[Mu]\_G\%2\ 
            \((h\^2\ \[Mu]\_G - \((h\^2 + 2\ H\ h - 3\ H\^2)\)\ \[Mu]\_L)\)\ 
            \[Rho]\_L\)\)\)\)], "Output"]
}, Open  ]],

Cell["Here again is the stratified pressure drop.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(strattemp = 
      dpdx\_strat /. {\ U\&_\_L -> \ Re\_L\ \ \(\[Mu]\_L/h\)/\[Rho]\_L}\)], 
  "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\(12\ Re\_L\ \[Mu]\_L\%2\ 
            \((\((h - H)\)\ \[Mu]\_L - h\ \[Mu]\_G)\)\)\/\(h\^2\ 
            \((\((h\^2 + 2\ H\ h - 3\ H\^2)\)\ \[Mu]\_L - h\^2\ \[Mu]\_G)\)\ 
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}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Useful form of the ratio", "Subsubsection"],

Cell["This is the ratio that we want.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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      Simplify[\((disptemp/strattemp)\) /. Re\_G -> regfunc]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(\((
          \((\[Mu]\_G\%2\ h\^4 - 
                2\ \((h\^3 - 2\ H\ h\^2 + 3\ H\^2\ h - 2\ H\^3)\)\ \[Mu]\_G\ 
                  \[Mu]\_L\ h + \((h - H)\)\^4\ \[Mu]\_L\%2)\)\ 
            \((h\ \[Mu]\_G\ 
                  \((\((h - 4\ H)\)\ \((h - H)\)\^2\ \[Rho]\_G - 
                      h\^3\ \[Rho]\_L)\) - 
                \((h - H)\)\ \[Mu]\_L\ 
                  \((\((h - H)\)\^3\ \[Rho]\_G - 
                      h\^2\ \((h + 3\ H)\)\ \[Rho]\_L)\))\))\)/
        \((H\^3\ \((h\ \[Mu]\_G + \((H - h)\)\ \[Mu]\_L)\)\ 
            \((\[Mu]\_G\%2\ \[Rho]\_L\ h\^4 - 
                \((h - H)\)\ \[Mu]\_G\ \[Mu]\_L\ 
                  \((\((h\^2 - 5\ H\ h + 4\ H\^2)\)\ \[Rho]\_G + 
                      h\ \((h + 3\ H)\)\ \[Rho]\_L)\)\ h + 
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}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Plots of the ratio", "Subsubsection"],

Cell[CellGroupData[{

Cell[TextData[{
  StyleBox["Oil-water flow in a 1 cm channel ",
    FontVariations->{"Underline"->True}],
  Cell[BoxData[
      \(Re\_L\)],
    FontVariations->{"Underline"->True}],
  StyleBox["=100",
    FontVariations->{"Underline"->True}],
  ".  Dispersed flow has a lower pressure drop at low oil flow rates but \
stratified becomes lower as the oil flow rate becomes large.  This is because \
the water ",
  StyleBox["lubricates",
    FontSlant->"Italic"],
  " the flow in the sense of reducing the stress near one wall.  (The effect \
would be quantitatively larger for a pipe geometry where the water could \
completely surround the oil.)  The reason that the pressure drop is higher \
for stratified at low water flow rates is that the ",
  StyleBox["water,",
    FontSlant->"Italic"],
  " in its stratified configuration, is taking up flow ",
  StyleBox["area",
    FontSlant->"Italic"],
  " that is not available to the more viscous phase."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

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Cell["\<\
Here we plot the viscosity of the dispersed phase.  It will be \
shown better below, but on the left side of the plot, the viscosity of the \
dispersed phase is decreasing while the pressure drop ratio is increasing.  \
This demonstrates that the lubrication effect occurs because of the flow \
geometry -- less viscous fluid in region of high shear reduces the pressure \
drop.\
\>", "Text"],

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