r*f
发帖数: 731 | 1 还有一个，再帮帮我吧
write the expression for the kinetic energy of a fluid
flow occupying a volume V by a surface S. If the flow is
irrotational and incompressible prove that the kinetic
energy is given by
{\rho}/2*{\intergral_s{\phi*{d{\phi}/dn}dA}
where \rho is density, \phi is the velocity potential.
d in d{\phi}/dn means partial differential.
//bow again | l**n
发帖数: 67 | 2
irrotational->\vec{v} = \div{phi} which is defined by you
incompressible->\laplace{phi}=0???(no source,sink)
by definition, kinetic energy is
\int_V \rho/2*(\grad{phi})^2 dV
= \rho/2*\int_V{(\grad{phi})^2}dV
= \rho/2*\int_V{\phi*\laplace{phi}+(\grad{phi})^2}dV
using Green's first identity
= \rho/2*\int_s{\phi*\vec{n}.\grad{\phi}dA}
= \rho/2*\int_s{\phi*\partial{\phi}/\partial{n}}dA
The green's first identity is:
for any well-behaved scalar function a,b,
\int_V{a\laplace{b}+\grad{a}.\grad{b})dV
【在 r*f 的大作中提到】
: 还有一个，再帮帮我吧
: write the expression for the kinetic energy of a fluid
: flow occupying a volume V by a surface S. If the flow is
: irrotational and incompressible prove that the kinetic
: energy is given by
: {\rho}/2*{\intergral_s{\phi*{d{\phi}/dn}dA}
: where \rho is density, \phi is the velocity potential.
: d in d{\phi}/dn means partial differential.
: //bow again
| l**n
发帖数: 67 | 3
sorry,This should follow from the mass conservation.\rho is
constant????
\int_s{a*\vec{n}.\grad{b}}dA
【在 l**n 的大作中提到】
:
: irrotational->\vec{v} = \div{phi} which is defined by you
: incompressible->\laplace{phi}=0???(no source,sink)
: by definition, kinetic energy is
: \int_V \rho/2*(\grad{phi})^2 dV
: = \rho/2*\int_V{(\grad{phi})^2}dV
: = \rho/2*\int_V{\phi*\laplace{phi}+(\grad{phi})^2}dV
: using Green's first identity
: = \rho/2*\int_s{\phi*\vec{n}.\grad{\phi}dA}
: = \rho/2*\int_s{\phi*\partial{\phi}/\partial{n}}dA
| r*f
发帖数: 731 | 4
(\grad{phi}^2)=\phi*\laplace{\phi}+(\grad{phi}^2) ???
Is there any type error?
【在 l**n 的大作中提到】
:
: sorry,This should follow from the mass conservation.\rho is
: constant????
: \int_s{a*\vec{n}.\grad{b}}dA
| r*f
发帖数: 731 | 5
Oh, I see.
div(\phi*grad(\phi))=\phi*laplace(\phi)+\grad(\phi)^2
since incompressible, div(v)=0,i.e., laplace(\phi)=0
so, we can get the result.
【在 r*f 的大作中提到】
:
: (\grad{phi}^2)=\phi*\laplace{\phi}+(\grad{phi}^2) ???
: Is there any type error?
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