1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Help with Bernoulli Equation

  1. Aug 21, 2005 #1
    Was wondering if it was possible to derive the best possible shape of a boat hull to achieve maximum speed? As it is the equation on how to calculate the speed of a sphere moving in water... or else im just totally wrong and you can bluntly ignore this post :tongue:
     
  2. jcsd
  3. Aug 21, 2005 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    That's an extremely difficult problem. Since you are trying to find the shape- i.e. a function- that maximizes a property it is a "Calculus of Variations" problem.
     
  4. Aug 21, 2005 #3

    saltydog

    User Avatar
    Science Advisor
    Homework Helper

    Guys, I'm just interested in this Ok; know very little about the Calculus of variation. Could someone at least formulate the problem? Let me begin with a conceptual approach:

    Assume we have the shape of the boat in the form of a surface above the x-y plane. Now it seems to me that this surface can be a very nicely behaved function. The speed of the boat, I'll just guess will be a function of frictional forces which in turn are dependent on the shape of the surface as well as the boundary which I'll call [itex]\Omega[/itex]. Now, let me take a leap of faith:

    We wish to minimize the following integral which sums up all the frictional forces on the hull:

    [tex]I=\int\int_{\Omega} G[x,y,f(x,y),f_x,f_y]ds[/tex]

    Where G is some functional relationship of these forces to the shape of the hull (and I suppose it's slopes as well) and our objective is to find f(x,y) which minimizes the integral (constrained by realistic limits of course, such as manufacturing ones).

    I know for the simple case of a function of a single variable f(x), wishing to minimize the integral:

    [tex]I=\int_a^b F(x,y,y')dx[/tex]

    we can do some calculus and come up with Euler's equation which must be satisfied:

    [tex]\frac{\partial F}{\partial y}-\frac{d}{dx} \frac{\partial F}{\partial y'}=0[/tex]

    I suppose there is an equivalent one for a double integral? How about for just any old integral of that form say for:

    [tex]I=\int_0^1\int_0^1 (f+f_x+f_y)dxdy[/tex]

    How would I find the function f(x,y) which minimizes (or maximized) this integral)? Am I getting off-subject?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Help with Bernoulli Equation
Loading...