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kyleniemeyer committed Sep 30, 2021
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13 changes: 7 additions & 6 deletions book/content/bvps/eigenvalue.ipynb
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"## Example: mass-spring system\n",
"\n",
"Let's analyze the motion of masses connected by springs in a system:\n",
"<figure>\n",
" <center>\n",
" <img src=\"../images/mass-spring-system.jpg\" alt=\"mass-spring system\" style=\"width: 400px;\"/>\n",
" <figcaption>Figure: System with two masses connected by springs</figcaption>\n",
" </center>\n",
"</figure>\n",
"\n",
":::{figure-md} fig-mass-spring\n",
"<img src=\"../../images/mass-spring-system.jpg\" alt=\"mass-spring system\" class=\"bg-primary mb-1\" width=\"400px\">\n",
"\n",
"System with two masses connected by springs\n",
":::\n",
"\n",
"First, we need to write the equations of motion, based on doing a free-body diagram on each mass:\n",
"\\begin{align}\n",
"m_1 \\frac{d^2 x_1}{dt^2} &= -k x_1 + k(x_2 - x_1) \\\\\n",
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24 changes: 11 additions & 13 deletions book/content/bvps/finite-difference.ipynb
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Expand Up @@ -539,12 +539,11 @@
"\n",
"Let's now consider a more complicated example: heat transfer through an extended surface (a fin).\n",
"\n",
"<figure>\n",
" <center>\n",
" <img src=\"../images/fin.png\" alt=\"Heat transfer fin\" style=\"width: 400px;\"/>\n",
" <figcaption>Figure: Geometry of a heat transfer fin</figcaption>\n",
" </center>\n",
"</figure>\n",
":::{figure-md} fig-fin\n",
"<img src=\"../../images/fin.png\" alt=\"Heat transfer fin\" class=\"bg-primary mb-1\" width=\"400px\">\n",
"\n",
"Geometry of a heat transfer fin\n",
":::\n",
"\n",
"In this situation, we have the temperature of the body $T_b$, the temperature of the ambient fluid $T_{\\infty}$; the length $L$, width $w$, and thickness $t$ of the fin; the thermal conductivity of the fin material $k$; and convection heat transfer coefficient $h$.\n",
"\n",
Expand All @@ -556,12 +555,11 @@
"\n",
"Our goal is to solve for the temperature distribution $T(x)$. To do this, we need to set up a governing differential equation. Let's do a control volume analysis of heat transfer through the fin:\n",
"\n",
"<figure>\n",
" <center>\n",
" <img src=\"../images/fin-control-volume.png\" alt=\"Control volume for heat transfer fin\" style=\"width: 300px;\"/>\n",
" <figcaption>Figure: Control volume for heat transfer through the fin</figcaption>\n",
" </center>\n",
"</figure>\n",
":::{figure-md} fig-control-volume\n",
"<img src=\"../../images/fin-control-volume.png\" alt=\"Control volume for heat transfer fin\" class=\"bg-primary mb-1\" width=\"300px\">\n",
"\n",
"Control volume for heat transfer through the fin\n",
":::\n",
"\n",
"Given a particular volumetric slice of the fin, we can define the heat transfer rates of conduction through the fin and convection from the fin to the air:\n",
"\\begin{align}\n",
Expand Down Expand Up @@ -652,7 +650,7 @@
],
"mimetype": "text/x-octave",
"name": "matlab",
"version": "0.16.7"
"version": "0.16.11"
}
},
"nbformat": 4,
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13 changes: 6 additions & 7 deletions book/content/bvps/shooting-method.ipynb
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Expand Up @@ -146,12 +146,11 @@
"\n",
"We can use the shooting method to solve a famous fluids problem: the [Blasius boundary layer](https://en.wikipedia.org/wiki/Blasius_boundary_layer).\n",
"\n",
"<figure>\n",
" <center>\n",
" <img src=\"../images/boundary-layer.png\" alt=\"Laminar boundary layer, from https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg\" style=\"width: 400px;\"/>\n",
" <figcaption>Figure: Laminar boundary layer, taken from <a href=\"https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg\">https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg</a></figcaption>\n",
" </center>\n",
"</figure>\n",
":::{figure-md} fig-boundary-layer\n",
"<img src=\"../../images/boundary-layer.png\" alt=\"Laminar boundary layer, from https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg\" class=\"bg-primary mb-1\" width=\"400px\">\n",
"\n",
"Laminar boundary layer, taken from <https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg>\n",
":::\n",
"\n",
"To get to a solveable ODE, we start with the conservation of momentum equation (i.e., Navier–Stokes equation) in the $x$-direction:\n",
"\\begin{equation}\n",
Expand Down Expand Up @@ -452,7 +451,7 @@
],
"mimetype": "text/x-octave",
"name": "matlab",
"version": "0.16.7"
"version": "0.16.11"
}
},
"nbformat": 4,
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4 changes: 2 additions & 2 deletions book/content/numerical-methods/stability.ipynb
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Expand Up @@ -396,9 +396,9 @@
],
"mimetype": "text/x-octave",
"name": "matlab",
"version": "0.16.7"
"version": "0.16.11"
}
},
"nbformat": 4,
"nbformat_minor": 2
"nbformat_minor": 4
}
15 changes: 5 additions & 10 deletions book/content/quizzes/quiz3-BVPs.md
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Expand Up @@ -9,12 +9,11 @@ r \frac{d^2 T}{dr^2} + \frac{dT}{dr} - rm^2 (T - T_{\infty}) = 0 \;,
where $r$ is the radial distance from the centerline (the independent
variable) and $m^2$ is a constant that depends on the heat transfer coefficient, thermal conductivity, and thickness of the annulus. Assuming we choose a spatial step size $\Delta r$,

<figure>
<center>
<img src="../images/annular-fin.png" alt="Annular fin" style="width: 400px;"/>
<figcaption>Figure: Annular fin</figcaption>
</center>
</figure>
:::{figure-md} markdown-fig
<img src="../../images/annular-fin.png" alt="Annular fin" class="bg-primary mb-1" width="400px">

Annular fin
:::

a.) Write the finite-difference representation of the ODE (that applies at a location $r_i$), using central differences.

Expand Down Expand Up @@ -138,7 +137,3 @@ z_2 (1.0) &= z_2 (0.5) + \left( 4z_1(0.5) \right) 0.5 = 6.0
so for solution 3: $y(1) = 3$ which is the target.

So our answer is $y'(0) = 3$.

```python

```
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13 changes: 7 additions & 6 deletions docs/_sources/content/bvps/eigenvalue.ipynb
View file
Original file line number Diff line number Diff line change
Expand Up @@ -302,12 +302,13 @@
"## Example: mass-spring system\n",
"\n",
"Let's analyze the motion of masses connected by springs in a system:\n",
"<figure>\n",
" <center>\n",
" <img src=\"../images/mass-spring-system.jpg\" alt=\"mass-spring system\" style=\"width: 400px;\"/>\n",
" <figcaption>Figure: System with two masses connected by springs</figcaption>\n",
" </center>\n",
"</figure>\n",
"\n",
":::{figure-md} fig-mass-spring\n",
"<img src=\"../../images/mass-spring-system.jpg\" alt=\"mass-spring system\" class=\"bg-primary mb-1\" width=\"400px\">\n",
"\n",
"System with two masses connected by springs\n",
":::\n",
"\n",
"First, we need to write the equations of motion, based on doing a free-body diagram on each mass:\n",
"\\begin{align}\n",
"m_1 \\frac{d^2 x_1}{dt^2} &= -k x_1 + k(x_2 - x_1) \\\\\n",
Expand Down
24 changes: 11 additions & 13 deletions docs/_sources/content/bvps/finite-difference.ipynb
View file
Original file line number Diff line number Diff line change
Expand Up @@ -539,12 +539,11 @@
"\n",
"Let's now consider a more complicated example: heat transfer through an extended surface (a fin).\n",
"\n",
"<figure>\n",
" <center>\n",
" <img src=\"../images/fin.png\" alt=\"Heat transfer fin\" style=\"width: 400px;\"/>\n",
" <figcaption>Figure: Geometry of a heat transfer fin</figcaption>\n",
" </center>\n",
"</figure>\n",
":::{figure-md} fig-fin\n",
"<img src=\"../../images/fin.png\" alt=\"Heat transfer fin\" class=\"bg-primary mb-1\" width=\"400px\">\n",
"\n",
"Geometry of a heat transfer fin\n",
":::\n",
"\n",
"In this situation, we have the temperature of the body $T_b$, the temperature of the ambient fluid $T_{\\infty}$; the length $L$, width $w$, and thickness $t$ of the fin; the thermal conductivity of the fin material $k$; and convection heat transfer coefficient $h$.\n",
"\n",
Expand All @@ -556,12 +555,11 @@
"\n",
"Our goal is to solve for the temperature distribution $T(x)$. To do this, we need to set up a governing differential equation. Let's do a control volume analysis of heat transfer through the fin:\n",
"\n",
"<figure>\n",
" <center>\n",
" <img src=\"../images/fin-control-volume.png\" alt=\"Control volume for heat transfer fin\" style=\"width: 300px;\"/>\n",
" <figcaption>Figure: Control volume for heat transfer through the fin</figcaption>\n",
" </center>\n",
"</figure>\n",
":::{figure-md} fig-control-volume\n",
"<img src=\"../../images/fin-control-volume.png\" alt=\"Control volume for heat transfer fin\" class=\"bg-primary mb-1\" width=\"300px\">\n",
"\n",
"Control volume for heat transfer through the fin\n",
":::\n",
"\n",
"Given a particular volumetric slice of the fin, we can define the heat transfer rates of conduction through the fin and convection from the fin to the air:\n",
"\\begin{align}\n",
Expand Down Expand Up @@ -652,7 +650,7 @@
],
"mimetype": "text/x-octave",
"name": "matlab",
"version": "0.16.7"
"version": "0.16.11"
}
},
"nbformat": 4,
Expand Down
13 changes: 6 additions & 7 deletions docs/_sources/content/bvps/shooting-method.ipynb
View file
Original file line number Diff line number Diff line change
Expand Up @@ -146,12 +146,11 @@
"\n",
"We can use the shooting method to solve a famous fluids problem: the [Blasius boundary layer](https://en.wikipedia.org/wiki/Blasius_boundary_layer).\n",
"\n",
"<figure>\n",
" <center>\n",
" <img src=\"../images/boundary-layer.png\" alt=\"Laminar boundary layer, from https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg\" style=\"width: 400px;\"/>\n",
" <figcaption>Figure: Laminar boundary layer, taken from <a href=\"https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg\">https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg</a></figcaption>\n",
" </center>\n",
"</figure>\n",
":::{figure-md} fig-boundary-layer\n",
"<img src=\"../../images/boundary-layer.png\" alt=\"Laminar boundary layer, from https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg\" class=\"bg-primary mb-1\" width=\"400px\">\n",
"\n",
"Laminar boundary layer, taken from <https://commons.wikimedia.org/wiki/File:Laminar_boundary_layer_scheme.svg>\n",
":::\n",
"\n",
"To get to a solveable ODE, we start with the conservation of momentum equation (i.e., Navier–Stokes equation) in the $x$-direction:\n",
"\\begin{equation}\n",
Expand Down Expand Up @@ -452,7 +451,7 @@
],
"mimetype": "text/x-octave",
"name": "matlab",
"version": "0.16.7"
"version": "0.16.11"
}
},
"nbformat": 4,
Expand Down
4 changes: 2 additions & 2 deletions docs/_sources/content/numerical-methods/stability.ipynb
View file
Original file line number Diff line number Diff line change
Expand Up @@ -396,9 +396,9 @@
],
"mimetype": "text/x-octave",
"name": "matlab",
"version": "0.16.7"
"version": "0.16.11"
}
},
"nbformat": 4,
"nbformat_minor": 2
"nbformat_minor": 4
}
15 changes: 5 additions & 10 deletions docs/_sources/content/quizzes/quiz3-BVPs.md
View file
Original file line number Diff line number Diff line change
Expand Up @@ -9,12 +9,11 @@ r \frac{d^2 T}{dr^2} + \frac{dT}{dr} - rm^2 (T - T_{\infty}) = 0 \;,
where $r$ is the radial distance from the centerline (the independent
variable) and $m^2$ is a constant that depends on the heat transfer coefficient, thermal conductivity, and thickness of the annulus. Assuming we choose a spatial step size $\Delta r$,

<figure>
<center>
<img src="../images/annular-fin.png" alt="Annular fin" style="width: 400px;"/>
<figcaption>Figure: Annular fin</figcaption>
</center>
</figure>
:::{figure-md} markdown-fig
<img src="../../images/annular-fin.png" alt="Annular fin" class="bg-primary mb-1" width="400px">

Annular fin
:::

a.) Write the finite-difference representation of the ODE (that applies at a location $r_i$), using central differences.

Expand Down Expand Up @@ -138,7 +137,3 @@ z_2 (1.0) &= z_2 (0.5) + \left( 4z_1(0.5) \right) 0.5 = 6.0
so for solution 3: $y(1) = 3$ which is the target.

So our answer is $y'(0) = 3$.

```python

```
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