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Streamlines - Continuum mechanics

  1. Nov 26, 2014 #1
    1. The problem statement, all variables and given/known data
    In Cartesian coordinates ##x##, ##y##, where ##x## is the horizontal and ##y## the vertical coordinate,
    the velocity in a small-amplitude standing surface wave on water of depth ##h## is given
    by;
    $$v_x = v_0 sin(\omega t) cos(kx) cosh[k(y + h)]$$
    $$v_y = v_0 sin(\omega t) sin(kx) sinh[k(y + h)]$$
    where ##v_0##, ##\omega## and ##k## are constants. Find the equation of streamlines written in the
    form ##F(x, y) = const##.

    2. Relevant equations
    $$\frac{dx_i}{d\lambda}=v_i (\lambda,t)$$

    3. The attempt at a solution
    Look being honest I have no idea what to do, I noticed that;
    $$\frac{dx}{d\lambda}=v_0 sin(\omega t) cos(kx) cosh[k(y + h)]$$
    $$\frac{dy}{d\lambda}=v_0 sin(\omega t) sin(kx) sinh[k(y + h)]$$
    I tried doing;
    $$\frac{dy}{d\lambda} \frac{d\lambda}{dx}=\frac{1}{tan(kx)tanh[k(y+h)]}$$
    I don't believe that is the correct way to do it, I think im supposed to try and write them as parametric equations but im not sure how.
     
  2. jcsd
  3. Nov 26, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    If [itex]F[/itex] is to be constant on streamlines, then its gradient must be orthogonal to [itex]v[/itex]. Thus you need to solve [tex]
    \frac{\partial F}{\partial x} = v_y, \\
    \frac{\partial F}{\partial y} = -v_x.[/tex]
     
  4. Nov 26, 2014 #3
    Oh thankyou! it appears i wasnt thinking about streamlines at all, not sure what i was trying to do
     
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