( ) Lagrangian Control Volume < > < l < l = ( , , ) l l l One set of symbol options to indicate a vector linear velocity through control volume > The reference space moves and the stuff in that space moves because the space is moving. < l acceleration of gravity < l = ( , , ) l x l y l z < g l x acceleration of control volume < a ( ) Other perspective has l > = l y m Mass flow rate the stuff moving through the non-moving control volume. = (t) l l z l x < < Gravitational force on the object m g = F Velocity in x direction I = 1 n < F = resultant force Example 7 Develop the mass transport model for material referenced to the following Lagrangian control volume +x +z +y < a The actual force that is pulling the control volume up n Balance statement: < resultant force F < F < F < a = + = [ m(t) ] n 1 2 I = 1 l z m [ m(t) ] < g l < [ m(t) ] < a ( -0, -0, -[m(t)] ) g z + ( +0, +0, ) + m = = ( 0, 0, +[m(t)] ) a z < g This vector equation contains three scalar equations. (one for each of the direction components of the vector) (i) For the x direction: ( 0) + (0) = 0 -[m(t)] g z l z m + (ii) For the y direction: ( 0) + (0) = 0 a z = l z m [m(t)] (iii) For the x direction: (-[m(t)] ) g z + = ( 0, 0, +[m(t)] ) a z l z m dt d v (t) z = - g z + l z m Note: [m(t)] m(t) = m t=0 + m t v (t) z ( ) d( ) = - g z + dt m(t) = m + m t [m(t)]

Lagrangian Control Volume Example 7 Develop the mass transport model for material referenced to the following Lagrangian control volume +x +z +y < a The actual force that is pulling the control volume up n Balance statement: < resultant force F < F < F < a = + = [ m(t) ] n 1 2 I = 1 l z m [ m(t) ] < g l < [ m(t) ] < a ( -0, -0, -[m(t)] ) g z + ( +0, +0, ) + m = = ( 0, 0, +[m(t)] ) a z < g This vector equation contains three scalar equations. (one for each of the direction components of the vector) (i) For the x direction: ( 0) + (0) = 0 g z l z m -[m(t)] + (ii) For the y direction: ( 0) + (0) = 0 a z = l z m [m(t)] (iii) For the x direction: (-[m(t)] ) g z + = ( 0, 0, +[m(t)] ) a z l z m dt d v (t) z = - g z + [m(t)] l z m - g + = dt d( ) v (t) ) ( Note: [m(t)] m(t) = m t=0 + m t m(t) = m + m t scalar the magnitude of l z m m mass flow rate d( ) v (t) z ( ) = - g z + dt l z linear velocity [m(t)] [m(t)] l z m acceleration l z m force l z m momentum