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| − | $\newcommand\p{\partial}$
| + | *[[Homework/1|Homework 1]] |
| − | | + | *[[Homework/2|Homework 2]] |
| − | <ol>
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| − | <li>'''Change in moments captured by a diffusion-advection equation.''' This problem illustrates how the diffusion coefficient $D$ controls the change in second moment of the solution and how $v$ controls the change in the first moment of the solution.
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| − | <ol>
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| − | <li>Show that the non-dimensional equation \[ \frac{\p w}{\p \tau} = \frac{\p^2 w}{\p z^2}\] (on the whole real line) is satisfied by the function \[w(z,\tau)=\frac{1}{\sqrt{4\pi \tau}} \exp\left(-\frac{z^2}{4\tau}\right)\] which has a zero mean for all time and linearly increasing variance.</li>
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| − | <li>Use changes of variables to turn the solution in part (a) into a solution to the dimensional equations
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| − | <ol>
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| − | <li>$\displaystyle \frac{\p u}{\p t} = D \frac{\p^2 u}{\p x^2}$</li>
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| − | <li>$\displaystyle \frac{\p u}{\p t} = D \frac{\p^2 u}{\p x^2}-v\frac{\p u}{\p x}$</li>
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| − | </ol></li>
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| − | </ol></li>
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| − | <li>'''Conserved quantity.''' Describe a physical system (a real one if possible but a made-up one is ok) that is well-modeled by the following system of equations for the functions $u(x,t), v(x,t), p(t)$ and $q(t)$:
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| − | \[ \frac{\p u}{\p t} = D_1 \frac{\p^2 u}{\p x^2} - c_1 \frac{\p u}{\p x} - a u + b v, \]
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| − | \[ \frac{\p v}{\p t} = D_2 \frac{\p^2 v}{\p x^2} - c_2 \frac{\p v}{\p x} + a u - b v, \]
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| − | \[ -D_1 \frac{\p u}{\p x}(0,t) + c_1 u(0,t) = -\alpha \frac{V_1}{A} (u(0,t)-p), \quad
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| − | -D_1 \frac{\p u}{\p x}(L,t) + c_1 u(L,t) = 0, \]
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| − | \[ -D_2 \frac{\p v}{\p x}(0,t) + c_2 v(0,t) = 0, \quad
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| − | -D_2 \frac{\p v}{\p x}(L,t) + c_2 v(L,t) = \beta \frac{V_2}{A} (v(L,t)-q), \]
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| − | \[ \frac{d p}{d t} = \alpha (u(0,t)-p) , \quad p(0)=p_0, \]
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| − | \[ \frac{d q}{d t} = \beta (v(L,t)-q) , \quad q(0)=0, \]
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| − | where $D_i, a, b, c_i, \alpha$, and $\beta$ are all constants. Give a physically reasonable constraint for any parameter that needs one to prevent unreasonable (e.g. negative) values of $u,v, p$ or $q$. Show that some quantity is conserved throughout the system. That is, find a function $Q(t)$ with the property that $dQ/dt = 0$.</li>
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| − | <li>'''Telegrapher's Equation.''' Consider a collection of particles that can move to the left or to the right at a constant velocity \(c\). Switching from left-moving to right-moving occurs with a rate constant \(\alpha_1\) and switching from right-moving to left-moving occurs with rate constants \(\alpha_2\).
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| − | <ol>
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| − | <li>Write down equations for the density of left- and right- movers ($u(x,t)$ and $v(x,t)$ respectively). Derive a single equation for $Y(x,t)=u(x,t)+v(x,t)$. $Z(x,t)=u(x,t)-v(x,t)$ will be useful along the way as will some mixed partial derivatives.</li>
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| − | <li>Nondimensionalize using arbitrary time and space scales \(\tau\) and \(L\). What are the characteristic time and space scales for the equation? </li>
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| − | <li>Under what temporal scaling does the single-time-derivative term dominate the double-time-derivative term? Under what spatial scaling do the diffusion and transport terms balance? Under what temporal and spatial scaling is the equation (approximately) an advection-diffusion equation? What condition on the \(\alpha_i\) must hold for such a scaling to exist?</li>
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| − | </ol>
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| − | </li>
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| − | <li>'''First passage time calculation.''' Consider a particle diffusing within the interval $[0,L]$, and under the influence of a linear potential (with diffusion coefficient $D$ and constant velocity $v$). The left edge is reflecting and the right edge is absorbing. Derive a boundary value problem for the '''mean''' first passage time (MFPT) for a particle starting at position $x$ from the equation for the full distribution discussed in class. Assuming that $\epsilon=vL/D$ is small, approximate the MFPT using a Taylor expansion to high enough order so that the $v$ dependence is not lost. List all space and time scales in the problem and describe the small-$\epsilon$ regime in terms of these scales.</li>
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| − | </ol>
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