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\begin{document}

\Large
\noindent \textbf{Problem Set 6} \\
\normalsize
ME 444: Continuum Mechanics \\
D.~H.\ Kelley \\
\numpoints~points \\

\begin{questions}

    \question[8] Thermal convection is well-modeled by the Boussinesq equations: 
        \begin{align}
            \frac{\partial u_j}{\partial x_j} &= 0, \\
            \frac{\partial u_j}{\partial t} + u_i \frac{\partial u_j}{\partial x_i} &= -\frac{1}{\rho_0} \frac{\partial p^\prime}{\partial x_j}+\frac{\rho^\prime}{\rho_0}g_j + \nu \frac{\partial^2 u_j}{\partial x_i \partial x_i}, \\
            \frac{\partial T}{\partial t} + u_i \frac{\partial T}{\partial x_i} &= \frac{k}{C_p \left( \rho_0+\rho^\prime \right)} \frac{\partial^2 T}{\partial x_i \partial x_i}.
        \end{align}
        Here $p^\prime = p - p_0$ is the fluctuation from the reference pressure $p_0$, and $\rho^\prime = \rho - \rho_0$ is the fluctuation from reference density $\rho_0$. If speed $U$, size $L$, temperature $T_0$, reference pressure $p_0$, and material properties (reference density $\rho_0$, viscosity $\nu$, and the ratio $k/C_p$ of thermal conductivity to specific heat) are taken as controls, how many dimensionless parameters characterize the system, according to the Buckingham Pi theorem? Write the Boussinesq equations in dimensionless form and identify the relevant dimensionless parameters. Stating their mathematical form is more important than knowing their historical names. 

    \question Surface waves restored by both gravitational acceleration $g$ and surface tension $\sigma$ in a fluid layer of depth $H$ and density $\rho$ have frequency
        \[
        \omega = \sqrt{k \left( g+\frac{\sigma k^2}{\rho} \right) \tanh kH},
        \]
        where $k$ is the wave number. \label{prob:SurfWaveGravTens} % Kundu 7.10 
    \begin{parts} 
        \part[3] Calculate their group velocity and plot its variation with wavelength $\lambda$ (where $\lambda = 2\pi/k$). 
        \part[3] Assuming deep water ($H \gg \lambda$), show that the group velocity is 
            \[ 
            c_g = \frac{1}{2} \sqrt{\frac{g}{k_z} } \frac{1+3 \sigma k_z^2 / \rho g}{\sqrt{1+\sigma k_z^2/\rho g}}.
            \]
        \part[3] Show that this becomes minimum at a wave number given by 
            \[
            \frac{\sigma k_z^2}{\rho g} = \frac{2}{\sqrt{3}}-1.
            \]
        \part[2] Calculate the minimum group velocity for cool water ($\rho = 1000$~kg/m$^3$ and $\sigma = 0.074$~N/m). 
    \end{parts}

    \question[5] An acoustic wave in an inviscid fluid without body forces and with bulk modulus $\kappa$ has pressure field $p^\prime = p_0 \sin \left( \bm{k} \cdot \bm{x} - \omega t \right)$, where $p_0$ is a constant pressure, $\bm{k} = k \ehat_1$ is the wave vector, $\bm{x}$ is position, $\omega$ is frequency, and $t$ is time. Is the wave longitudinal or transverse? Determine the corresponding displacement field $\bm{u}$. 

\question Consider a situation like the one in Prob.~\ref{prob:SurfWaveGravTens}, but with one fluid layer of depth $H$ and density $\rho_2$ lying beneath a lighter, infinitely thick fluid of density $\rho_1$. Then, small perturbations lead to waves with phase speed 
    \[
    c = \pm \sqrt{\frac{\frac{g}{k} \left( \rho_2 - \rho_1 \right) + \sigma k}{\rho_1 + \rho_2 \coth kH}},
    \]
    where $g = 9.8~\mathrm{m/s}^2 > 0$ is the acceleration due to gravity and $k$ is the wave number. Now turn the problem upside-down, to consider a more-dense fluid \emph{above} a less-dense fluid. The phase speed $c$ remains unchanged except that $g$ is now negative. With this configuration, we can understand why drops form when a liquid is splashed on the underside of a flat surface. This is the Rayleigh-Taylor instability, which also has primary importance for inertial-confinement fusion. % Rephrased from Kundu 11.2
    \begin{parts}
        \part[2] Are long or short waves more unstable?
        \part[2] Does a professional painter want interior ceiling paint with high or low surface tension? 
        \part[2] For a smooth finish, should the painter apply thin or thick coats of paint? 
        \part[3] Assuming the liquid has the properties of water ($\sigma = 0.072$~N/m, $\rho_2 = 10^3$~kg/m$^3$) and that the lighter fluid is air, what is the longest neutrally stable wavelength on the underside of a horizontal surface?
    \end{parts}

    \question Consider a Kelvin-Voigt material with damping coefficient $\eta$ and spring constant $G$, connected in series with a dashpot with damping coefficient $\eta_0$ (an arrangement sometimes called ``the standard linear fluid''). % Lai 8.4
        \begin{parts}
            \part[3] Determine the force-displacement relationship. That is, write an ordinary differential equation that relates the force $S$ applied to the system to the strain $\varepsilon$ experienced by the system, in terms of the given coefficients. 
        \part[6] Determine the relaxation function. That is, solve the equation from part (a) to determine $\varepsilon(t)$ if the stress is $S = S_0 H(t)$, where $H(t)$ is the Heaviside function. \emph{Hint: What are the initial strain and initial strain rate?}
        \end{parts}

    \question[2] What topic on this problem set was easy? What topic was hard? What topic was impossible? 

\end{questions}
\end{document}