hey pf!
i was hoping one of you could help me with deriving the navier stokes equations. i know the equations state something like this [tex]\sum \vec{F}=m\vec{a}[/tex] and for fluids we have something like this for the forces: [tex]\underbrace{-\iint Pd\vec{s}}_{\text{pressure}}+\underbrace{\vec{F}}_{\text{viscous forces}}-\underbrace{\iint \rho\vec{V}\vec{V}\cdot d\vec{s}}_{\text{momentum leaving flux}}[/tex] and for the acceleration terms we have [tex]\underbrace{\frac{\partial}{\partial t}\iiint \rho\vec{V} dv}_{\text{time rate of change of momentum}}[/tex]
now ultimately we can use the divergence theorem, some vector/tensor identities, and arrive at the navier stokes equation: [tex]\rho \frac {D \vec{V}}{Dt} = - \nabla P + \mu \nabla^2 \vec{V}[/tex]
my question is, can someone please explain to me how to arrive from my [itex]\vec{F}[/itex] viscous force to the [itex]\mu \nabla^2 \vec{V}[/itex]
i'm good with the rest, but i'm just not sure how to start from an integral calculus perspective.
thanks!
i was hoping one of you could help me with deriving the navier stokes equations. i know the equations state something like this [tex]\sum \vec{F}=m\vec{a}[/tex] and for fluids we have something like this for the forces: [tex]\underbrace{-\iint Pd\vec{s}}_{\text{pressure}}+\underbrace{\vec{F}}_{\text{viscous forces}}-\underbrace{\iint \rho\vec{V}\vec{V}\cdot d\vec{s}}_{\text{momentum leaving flux}}[/tex] and for the acceleration terms we have [tex]\underbrace{\frac{\partial}{\partial t}\iiint \rho\vec{V} dv}_{\text{time rate of change of momentum}}[/tex]
now ultimately we can use the divergence theorem, some vector/tensor identities, and arrive at the navier stokes equation: [tex]\rho \frac {D \vec{V}}{Dt} = - \nabla P + \mu \nabla^2 \vec{V}[/tex]
my question is, can someone please explain to me how to arrive from my [itex]\vec{F}[/itex] viscous force to the [itex]\mu \nabla^2 \vec{V}[/itex]
i'm good with the rest, but i'm just not sure how to start from an integral calculus perspective.
thanks!