Reynolds Transport Theorem

 

 

Key Concept:  The Reynolds Transport Theorem provides a way to transfer equations

for conservation of mass, momentum, and energy from the Lagrangian point of view to the Eulerian point of view using a control surface and a control volume.

 

 

In a Nutshell:  You can study fluid motion either by following fluid particles (system, the Lagrangian

point of view) or by observing fluid flow past a control volume (finite volume, Eulerian point of view). 

Let  B  denote extensive property related to fluid flow such as mass, linear momentum, angular

momentum, or energy.  Let  b  denote  comparable intensive property such as mass per unit mass,

linear momentum per unit mass, angular momentum per unit mass, or energy per unit mass.  Then Reynolds transport theorem is:           

 

 

              DB/dt |sys =  ∂/∂t  ∫ ρb dV  +  ∫ ρb V . n dS      

                                         cv              cs

 

        

                  cv = control volume,    cs  =  control surface

 

where  B       =  the extensive property (contained in the system, fixed quantity)

            b       =   the intensive property = property per unit mass

            ρ       =   the mass density of the fluid

        ∂/∂t      =    the time rate of change 

         dV      =    the element of volume within the control volume

         V        =   the fluid velocity crossing the control surface

         n         =    the unit outward normal to the control surface

     V . n      =    the normal component of velocity crossing the control surface (dot product)

       dS         =     the element of area on the control surface

 

Physical Interpretation of Reynolds Transport Theorem

 

DB/dt  represents the time rate of change of the arbitrary extensive property,  B

 

∂/∂t  ∫ ρb dV   represents the time rate of change of  B within the control volume
      

cv

  ∫ ρb V . n dS     represents the flux of  B  across the control surface

cs