Final Exam



  1. Consider the following initial value problem

    \begin{eqnarray*} \epsilon y'' + 2y' +2y &=&0 \hspace*{.2in} 0 < x<1 \hspace*{.2in} \epsilon \ll 1 \\ y(0)&=&0 \\ y(1)&=&1. \end{eqnarray*}



    As show in class using matched asymptotic expansions the leading order approximation is

    \begin{displaymath}y \sim e^{1-x}-e^{1-2x/\epsilon } .\end{displaymath}

    Find the second term in the approximation, i.e. the O$(\epsilon )$ term. Simplify your answer.

  2. Consider the simple case of steady two-dimensional boundary-layer flow with the flow parallel to a solid wall of length $l$. Let the velocity outside the boundary layer be uniform and equal to $U$. The pressure is likewise uniform just outside the boundary layer and is approximately uniform throughout the boundary layer. For steady flow this reduces the boundary layer equations in the $x$-direction to
    $\displaystyle u\frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}$ $\textstyle =$ $\displaystyle \nu \frac{\partial^2 u}{\partial y^2}$ (1)
    $\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}$ $\textstyle =$ $\displaystyle 0.$  

    To non-dimensionalize these equations we can use a simplifying assumption. Consider the fact that the velocity in the boundary layer does not depend on the solid boundary downstream, thus the boundary layer thickness does not depend on $l$. This fact will hold if the non-dimensional $u^*$ depends on $x^*$ and $y^*$ in the form

    $\displaystyle \eta =\frac{y^*}{\sqrt{x^*}} = \sqrt{\frac{U}{\nu x}}y.$     (2)

    This implies that
    $\displaystyle u$ $\textstyle =$ $\displaystyle Uf'(\eta )$ (3)
    $\displaystyle v$ $\textstyle =$ $\displaystyle \frac 12 \sqrt{\frac{\nu U}{x}} \left( \nu f' -f \right).$ (4)

    where $f$ is a dimensionless parameter.

    Transform ([*]) into a differential equation in $f$ using ([*]), ([*]) and ([*]). What would you expect the solution $u$ to look like as a function of $\eta $?