gibson:teaching:spring-2016:math445:lab9
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gibson:teaching:spring-2016:math445:lab9 [2016/03/30 19:15] gibson [Problem 7: inviscid fluid flow past a semicircular bump] |
gibson:teaching:spring-2016:math445:lab9 [2016/03/31 09:39] (current) vining |
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| **(b)** In a gentle rainstorm, water will flow down the mountains in the direction of steepest descent, i.e. along the negative of the gradient of $f$. Find the gradient of $f$ using elementary calculus, then make an $x,y$ plot with both contours of the mountain height $f$ and a quiver plot showing the direction of flow of rainwater. | **(b)** In a gentle rainstorm, water will flow down the mountains in the direction of steepest descent, i.e. along the negative of the gradient of $f$. Find the gradient of $f$ using elementary calculus, then make an $x,y$ plot with both contours of the mountain height $f$ and a quiver plot showing the direction of flow of rainwater. | ||
| - | ===== Problem 7: inviscid fluid flow along a square corner ===== | + | ===== Problem 7: inviscid fluid flow within a square corner ===== |
| - | The inviscid 2d fluid flow along a square corner with solid walls at $x=0$ and $y=0$ and the domain $x \geq 0$ and $y \geq 0$ has velocity components | + | The inviscid 2d fluid flow within a square corner with solid walls at $x=0$ and $y=0$ and the domain $x \geq 0$ and $y \geq 0$ has velocity components |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
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| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Make a quiver plot of this flow for $0 \leq x \leq 1$, $0 \leq y \leq 1$, and $v_0 = 1$. Label the axes and title the plot. | + | Make a quiver plot of this flow for $0 \leq x \leq 1$ and $0 \leq y \leq 1$, and with $v_0 = 1$. Label the axes and title the plot. |
| - | ===== Problem 7: inviscid fluid flow past a semicircular bump ===== | + | ===== Problem 8: inviscid fluid flow past a semicircular bump ===== |
| The inviscid 2d fluid flow past a semicircular bump of radius $a$ centered at the origin on a flat wall at $y=0$ has velocity components | The inviscid 2d fluid flow past a semicircular bump of radius $a$ centered at the origin on a flat wall at $y=0$ has velocity components | ||
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| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Make a quiver plot of this flow for $v_0=1$, $-3 \leq x \leq 3$ and $0 \leq y \leq 3$, excluding the interior of the bump, where $x^2 + y^2 < a^2$. Draw a blue curve that shows the surface of the semicircular bump. Make the $x$ and $y$ axis have the same scale, label the axes, and title the plot. | + | Make a quiver plot of this flow for $a=1$, $v_0=1$, $-3 \leq x \leq 3$ and $0 \leq y \leq 3$, excluding the interior of the bump, where $x^2 + y^2 < a^2$. Draw a blue curve that shows the surface of the semicircular bump. Make the $x$ and $y$ axis have the same scale, label the axes, and title the plot. |
| ===== Bonus ===== | ===== Bonus ===== | ||
gibson/teaching/spring-2016/math445/lab9.1459390544.txt.gz · Last modified: 2016/03/30 19:15 by gibson
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