Bernoulli Theorem

Discussion in 'Hydrodynamics and Aerodynamics' started by gerar, Mar 9, 2013.

  1. gerar
    Joined: Feb 2013
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    gerar Junior Member

    I am trying to understand the Bernoulli's theorem:

    [​IMG] is constant along a streamline

    The explanation of parameters is found here, in (2.13):

    http://www3.kis.uni-freiburg.de/~peter/teach/hydro/hydro02.pdf


    I got that:

    [​IMG]

    For a steady flow: ([​IMG]) and then:

    [​IMG] with the scalar: [​IMG]

    now, I didn't understand the next steps:

    Taking the “dot product” of [​IMG] the left hand side vanishes, as [​IMG] is perpendicular to u and we get:

    [​IMG]

    This implies that H is constant along a streamline

    Can someone explain me this thing in other words please?
     
  2. fisher563
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    fisher563 New Member

    In the reduction of the equation thay make a scalar product, which is signed with a dot. The equation is multiplyed with u vektor. When the multiplicand is perpendicular, the product is zero. When not, you have the scalar quantities and the cos angle of them to multiply. See: vector algebra.
     
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  3. markdrela
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    markdrela Senior Member

    The last step is missing a few words. It should say:
    "Taking the dot product of this equation with u the left hand side vanishes, ..."
    i.e.
    u . { (del x u) x u = -del H }
    or
    0 = -u . del H

    Another way to interpret the equation
    (del x u) x u = -del H
    is as follows:

    1) If the flow is irrotational so that del x u = 0 , then del H = 0 , or equivalently H is constant everywhere.

    2) If the flow is rotational so that del x u =/ 0 , then del H is nonzero but perpendicular to the velocity u , and hence H is constant along a streamline.
     
  4. fredrosse
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    fredrosse USACE Steam

    Also note that this equation is for flow without friction.
     

  5. TANSL
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    TANSL Senior Member

    Extracted from Wikipedia :
    "Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both its dynamic pressure and kinetic energy, and a decrease in its static pressure and potential energy"
     
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