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\begin{document}

\Large
\noindent \textbf{Problem Set 5} \\
\normalsize
ME 444: Continuum Mechanics \\
D.~H.\ Kelley \\
\numpoints~points \\

\begin{questions}

    \question[3] Show that the rate of deformation tensor $D_{ij} = (\partial v_i / \partial x_j + \partial v_j / \partial x_i)/2$ is objective. % Lai 5.105

    \question[5] A material deforms according to
    \begin{eqnarray*}
        r &=& \lambda_1 R \\
        \theta &=& \Theta + K Z \\
        z &=& \lambda_3 Z,
    \end{eqnarray*}
    where the current configuration is expressed in cylindrical coordinates $\{r,\theta,z\}$, the reference configuration is expressed in cylindrical coordinates $\{R,\Theta,Z\}$, and the quantities $\lambda_1$, $\lambda_3$, and $K$ are constants, with $\lambda_1^2 \lambda_3 = 1$. Calculate the left Cauchy-Green deformation tensor $\bm{B}$. \emph{Hint: See Lai section 3.29.} % Lai 5.113

    \question For strain given in Cartesian coordinates by the deformation gradient $\bm{F}$, where $F_{11}=\lambda_1$, $F_{22}=\lambda_2$, $F_{33}=\lambda_3$, and all other $F_{ij}=0$, % adapted from Mase & Mase 8.1
        \begin{parts}
            \part[3] Calculate the principal scalar invariants of the right Cauchy-Green strain tensor $\bm{C}$. 
            \part[3] Calculate the first two principal scalar invariants if the material is also incompressible. 
        \end{parts}

        \question[6] For the simple shear with respect to Cartesian coordinates specified by 
        \begin{eqnarray*}
            x_1 &=& X_1 + k X_2 \\
            x_2 &=& X_2 \\
            x_3 &=& X_3,
        \end{eqnarray*}
        where $k$ is a constant, determine the stress $\bm{T}$ for a Mooney-Rivlin material with shear modulus $G$ and fit parameter $\beta$, in the absence of applied pressure $p$. % Dill 2.9

    \question A Newtonian fluid with viscosity $\mu = 0.96~\mathrm{mPa} \cdot \mathrm{s}$ flows with velocity field $\bm{v} = k (x_1^2-x_2^2) \hat{\bm{e}}_1 - 2 k x_1 x_2 \hat{\bm{e}}_2$, where $k=1~\mathrm{s}^{-1}\mathrm{m}^{-1}$, expressed in Cartesian coordinates $(x_1,x_2,x_3)$. At the point $(x_1,x_2,x_3)=(1,2,1)$~m and along the plane with normal direction $3\hat{\bm{e}}_1 + 4\hat{\bm{e}}_2$, % Lai 6.16
        \begin{parts}
            \part[3] Determine the amount by which the normal compressive stress exceeds the pressure $p$. 
            \part[3] Determine the magnitude of the shear stress. 
        \end{parts}

        \question A liquid with kinematic viscosity $\nu$ flows down a plane inclined at an angle $\theta$ with respect to the horizontal with volume flow rate $Q$ (per unit distance across the plane), as sketched in Fig.~\ref{fig:slopeflow}. The fluid layer has thickness $h$, and its top surface is free. The $\hat{\bm{e}}_1$ and $\hat{\bm{e}}_2$ directions are parallel and perpendicular to the plane, respectively. \label{prob:slopeflow}

        \begin{parts}
            \part[3] What are the boundary conditions for fluid velocity and pressure? 
            \part[3] Assume that the flow is steady, two-dimensional, and fully-developed. State the mathematical implications of those assumptions. 
            \part[5] Using the simplifying assumptions discussed above, solve the mass and momentum equations in order to give an expression for the height $h$ in terms of $Q$, $\nu$, $g$, and $\theta$, where $g$ is the acceleration due to gravity. 
        \end{parts}

        \begin{figure}[h]
            \begin{minipage}{.49\textwidth}
                \begin{center}
                    \includegraphics[width=2.5in]{slopeflow.pdf}
                    \caption{Problem~\ref{prob:slopeflow}.}
                    \label{fig:slopeflow}
                \end{center}
            \end{minipage}
            \begin{minipage}{.49\textwidth}
            \end{minipage}
        \end{figure}

    \question Two horizontal, infinite plates are separated by two incompressible Newtonian fluids of density $\rho_B > \rho_A$ and viscosity $\mu_B$ and $\mu_A$. The two fluids do not mix. The plates move steadily with velocities $V \ehat_1$ and $-V \ehat_1$, as shown. The lower plate lies at $x_2=0$, the fluid interface lies at $x_2=h$, and the upper plate lies at $x_2=H$. 

    \includegraphics{TwoLayerCouette.pdf}
    \begin{parts}
        \part[2] State the boundary conditions at the infinite plates. 
        \part[2] State the two boundary conditions at the fluid interface ($x_2=h$). \emph{Hint: The no-slip condition must hold, and Newton's third law must be true for shear stresses.}
        \part[8] Assuming the flow to be steady, two-dimensional, and fully-developed, solve for the velocity $\bm{v}$ and pressure $p$ in each fluid layer. As a reference, take the pressure to be $p_0$ at $x_2=0$. Do not neglect gravity. 
    \end{parts}

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

\end{questions}
\end{document}