Finite Control Volume – Conservation of Linear Momentum

Key Concept: The Reynolds Transport Theorem provides an expression for conservation

of linear momentum using a finite control volume. The jist of it is that the the linear momentum stored in the control volume plus the net linear momentum flux across the control surface sums to the forces acting on the control surface and within the control volume.

In a Nutshell: For conservation of linear momentum, the change in linear momentum of the

system must equal the net force on the system. D/Dt ∫ V ρ dV = Σ F_sys

sys

Use of Reynolds Transport Theorem gives the integral form for conservation of linear momentum.

∂/∂t ∫ ρ V dV + ∫ V ρ V . n dS = Σ F

cv cs cv = control volume, cs = control surface

rate of change of linear momentum in cv + momentum flux across cs = net force on contents

where ρ is the mass density of the fluid, F_x and F_y are forces acting on cv

∂/∂t = the time rate of change

dV = the element of volume within the control volume

V = the absolute 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

Σ F = forces acting on contents within the control volume and on the control surface

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