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Notation (v*gradient operator)v in Navier-Stokes

  1. Jul 3, 2011 #1
    Notation "(v*gradient operator)v" in Navier-Stokes

    What does [itex]\left( \textbf{v} \cdot \nabla \right) \textbf{v}[/itex] mean, assuming knowledge of the gradient operator? And, specifically, how would that be expanded? In general, I'm ignorant of the notation [itex]\left( f \left( y, \frac{d}{dx}, \frac{d^2}{dx^2}, \ldots \right) y \right)[/itex] where, for example, [itex]f \left( y, \frac{d}{dx}, \frac{d^2}{dx^2}, \ldots \right)[/itex] could be [itex]y + \frac{d}{dx}[/itex].

    ec8991a09ac20811fda72211cb486792.png
     
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  3. Jul 4, 2011 #2

    HallsofIvy

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    Re: Notation "(v*gradient operator)v" in Navier-Stokes

    That is, by the way, the "del" or "nabla" operator. It is specifically only [itex]\nabla f(x,y,z[/itex], with f a scalar valued function, that is the "gradient". [itex]\nabla\cdot \vec{v}[/itex] is the "gradient" or grad [itex]\vec{v}[/itex] and [itex]\nabla\times\vec{v}[/itex] is the curl [itex]\vec{v}[/itex].

    In any case
    [tex]\vec{v}\cdot\nabla= v_x\frac{\partial }{\partial x}+ v_y\frac{\partial }{\partial y}+ v_z\frac{\partial }{\partial z}[/tex]
    so that
    [tex]\vec{v}\cdot\nabla \vec{v}= \left(v_x\frac{\partial }{\partial x}+ v_y\frac{\partial }{\partial y}+ v_z\frac{\partial }{\partial z}\right)v_x\vec{i}[/tex][tex]+ \left(v_x\frac{\partial }{\partial x}+ v_y\frac{\partial }{\partial y}+ v_z\frac{\partial }{\partial z}\right)v_y\vec{j}[/tex][tex]+ \left(v_x\frac{\partial }{\partial x}+ v_y\frac{\partial }{\partial y}+ v_z\frac{\partial }{\partial z}\right)v_z\vec{k}[/tex]
    [tex]= \left(v_x\frac{\partial v_x}{\partial x}+ v_y\frac{\partial v_x}{\partial y}+ v_z\frac{\partial v_x}{\partial z}\right)\vec{i}[/tex][tex]+ \left(v_x\frac{\partial v_y}{\partial x}+ v_y\frac{\partial v_y}{\partial y}+ v_z\frac{\partial v_y}{\partial z}\right)\vec{j}[/tex][tex]+ \left(v_x\frac{\partial v_z}{\partial x}+ v_y\frac{\partial v_z}{\partial y}+ v_z\frac{\partial v_z}{\partial z}\right)\vec{k}[/tex]
     
  4. Jul 4, 2011 #3
    Re: Notation "(v*gradient operator)v" in Navier-Stokes

    Oh, I didn't understand it as well as I thought I did. Thanks.

    Just wondering, is [itex]\left( y + \frac{d}{dx} \right)y[/itex] meaningless, or can it be expanded? If it can, what is it?

    Thanks for your time.
     
  5. Jul 4, 2011 #4

    hunt_mat

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    Re: Notation "(v*gradient operator)v" in Navier-Stokes

    it can be expanded as:
    [tex]
    y+\frac{dy}{dx}
    [/tex]
     
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