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Cell["\<\
Solution of the natural convection boundary-layer flow near a \
heated flat plate\
\>", "Title"],

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  "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\nmjm@nd.edu\nhttp://www.nd.edu/~mjm/\n\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, that if you use \
it, that this notice remain visible to other users. \nThere is no charge for \
copying and dissemination   \n\nVersion:  5/25/00\nMore recent versions of \
this notebook may be available at the web site:\n",
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Cell["\<\
Reference:  F. P. Incropera and D. P. DeWitt, Fundamentals of Heat \
and Mass Transfer, 4th ed. Wiley, 1996. \
\>", "Subsubsection"],

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Cell[TextData[{
  "This notebook is intended to give a first introduction to natural \
convection heat transfer from a solid body.  The application of interest is \
heat transfer from a heated sphere (a light bulb) that is enclosed in a clear \
plastic box.  The physical experiment is done at Notre Dame in the \
undergraduate chemical engineering lab (",
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  ").  The surface temperature of the light bulb is measured as a function of \
position around the light bulb.  These data show the development of a thermal \
boundary layer.\n\nThis notebook starts with the boundary layer equations for \
a heated, vertical flat plate under conditions of high Grashof number.  Using \
a similarity variable that involves the local Grashof number, the PDE's are \
reduced to ODE's.  The ODE's are solved for different Prandtl numbers to show \
the velocity and temperature profiles.   "
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I would enter the dynamic boundary condition from a key pad \
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Cell["\<\
Natural Convection Heat Transfer, Thermal Boundary Layer, \
Boundary-Layer Flow, Similarity Variable,  High Grashof Number Heat \
Transfer\
\>", "Text"]
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Cell[TextData[{
  "The problem of interest is a heated sphere in a fluid of infinite extent \
that is otherwise not flowing.  We expect that if the sphere is heated it \
will be at a higher temperature than the surrounding fluid, energy will be \
transferred to the fluid and the resulting temperature difference within the \
fluid will lead to a density gradient.  If the sphere is in a uniform gravity \
field, you did not want pay to put your sphere on ",
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shuttle, then the density gradient will lead to a body force acting on the \
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upward and replaced by higher density fluid.  The question is, will the fluid \
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road tar, we expect a low Grashof number and thus no flow.  For a lighted \
light bulb in air, the Grashof number can be calculated as ",
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  "  this will certainly cause a strong flow.\n\nFor the case of low ",
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pure condition limit.  Since the Nusselt number gives the ratio of total heat \
transfer to pure condition (we are not considering radiation), we expect that \
as the Grashof number gets large, there will be a significant fluid flow and \
thus significant convective enhancement.  "
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Cell["High Grashof number heat transfer", "Subsection"],

Cell["\<\
For a Grashof number that is large compared to unity, we expect \
that the resistance to heat transfer, alternatively the region where most of \
the temperature change occurs, will be confined to a thin region close to the \
surface.  If this region is thin compared to the macroscopic dimensions, we \
expect that boundary-layer approximations will give valid simplifications to \
the governing equations.  This will be the case.  Thus the boundary-layer \
equations will serve as the basic theory for the convection/conduction heat \
transfer from the heated sphere. \
\>", "Text",
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  "We recall from fluid dynamics that we expect boundary-layers to occur \
within the fluid close to solid surfaces at high Reynolds numbers.  The \
current interest where the Grashof number is large is analogous.  In a \
momentum boundary-layer the inertia and viscous forces are about the same \
order of magnitude.  The governing equation thus contains the dominant \
inertia and viscous terms.  As the Reynolds number is increased, the boundary \
layer gets thinner as \[Delta]/L ~ 1/Re^1/2.  This scaling for this can be \
determined from the form of the equations with out solving any differential \
equations.  The drag on the plate can be determined to order one accuracy \
simply by knowing the boundary layer thickness, \[Delta], and calculating \
\[Tau] = \[Mu] du/dy =~ \[Mu] U",
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boundary-layer equation for momentum.  The primary difference is that the \
energy equation does not have a term like dp/dx, unless there is a heat \
source within the boundary layer.  This is a source term and in the case of \
fluid flow is the reason the fluid is flowing (a momentum source).  \n\n\
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is that we can get the basic scaling of the solution from the form of the \
equations.  Thus once we have the equations, can figure out how to \
nondimensionalize them and finally turn the PDE's into ODE's, we will know \
how the Nusselt number scales with Grashof number without solving the \
equations! \n\nStrictly speaking boundary-layer theory is the first order \
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what we are solving) are the same for a sphere or a flat plate.  The main \
reason for this is because the layer is very thin compared to the radius of \
the sphere.  Thus \"the earth is flat\"! -- or at least curvature does not \
matter.  \n\nIf we derive the continuity, momentum and energy equations for \
flow past a heated flat plate we get "
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The basic mass, momentum and energy equations for a free convection \
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\>", "Text"]
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scaling can be collapsed onto the same curve.  The length scale for the \
normal direction within a boundary - layer is constructed from the physical \
properties of the flow and any other length or time variables that come to \
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Start by defining a similarity variable. We will just choose the \
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Define the velocities in terms of the stream function and \
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Now derive the 2 nonlinear ODE's that describe momentum and energy \
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It is easiest to do the substitutions starting with the original \
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\>", "Text"],

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Numerical solution of the coupled ODE's that result after a \
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 \nReference:  Numerical Recipes in Fortran, See also the notes from ChEg \
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The streamwise velocity and dimensionless temperature should be 0 \
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We can plot the temperature to see how linear the profile is.  Also \
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Cell[CellGroupData[{

Cell["Conclusions", "Subtitle"],

Cell[TextData[{
  "1.  For natural convection flows where the ",
  ButtonBox["Grashof",
    ButtonData:>"Grashof",
    ButtonStyle->"Hyperlink"],
  " number is larger, a ",
  ButtonBox["boundary layer",
    ButtonData:>"boundary_layer",
    ButtonStyle->"Hyperlink"],
  " can be expected close to a solid surface.  \n\n2.  For situations where a \
natural convection boundary-layer is occurring, heat transfer will be \
governed by ",
  ButtonBox["coupled energy and momentum equations",
    ButtonData:>"govern_eqs",
    ButtonStyle->"Hyperlink"],
  ". \n\n3.  For transport processes that occur on a semi-infinite domain, \
where there is ",
  StyleBox["no geometric length scale",
    FontVariations->{"Underline"->True}],
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(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)