## Differential Shell Momentum Balance in Rectangular Cartesian Coordinates

## Transport Phenomena – Fluid Mechanics Theory

## Differential Shell Momentum Balance in Rectangular Cartesian Coordinates

L and width W, which is at an angle β to the vertical. End effects are neglected assuming the dimension of the system in the x-direction is relatively very small compared to those in the y-direction (W) and the z-direction (L).
Figure. Differential rectangular slab (shell) of fluid of thickness Δx used in z-momentum balance for flow in rectangular Cartesian coordinates. The y-axis is pointing outward from the plane of the computer screen.
z-direction, v = 0, _{x}v = 0, and only _{y}v exists. For small flow rates, the viscous forces prevent continual acceleration of the fluid. So, _{z}v is independent of _{z}z and it is meaningful to postulate that velocity v = _{z}v(_{z}x) and pressure p = p(z). The only nonvanishing components of the stress tensor are τ = _{xz}τ, which depend only on _{zx}x.
Consider now a thin rectangular slab (shell) perpendicular to the z-momentum In − Out + Generation = Accumulation v(_{z}x) is the same at both ends of the system, the convective terms cancel out because (ρ v Δ_{z} v_{z} Wx)|_{z = 0} = (ρ v Δ_{z} v_{z} Wx)|_{z = L}. Only the molecular term (L W τ ) remains to be considered, whose ‘in’ and ‘out’ directions are taken in the positive direction of the _{xz}x-axis. Generation of z-momentum occurs by the pressure force acting on the surface [p W Δx] and gravity force acting on the volume [(ρ g cos β) L W Δx].
The different contributions may be listed as follows: - rate of
*z*-momentum in by viscous transfer across surface at*x*is (*L W τ*)|_{xz}_{x} - rate of
*z*-momentum out by viscous transfer across surface at*x*+ Δ*x*is (*L W τ*)|_{xz}_{x + Δx} - rate of
*z*-momentum in by overall bulk fluid motion across surface at*z*= 0 is (*ρ v*_{z}v_{z}*W*Δ*x*)|_{z = 0} - rate of
*z*-momentum out by overall bulk fluid motion across surface at*z = L*is (*ρ v*_{z}v_{z}*W*Δ*x*)|_{z = L} - pressure force acting on surface at
*z*= 0 is*p*_{0}*W*Δ*x* - pressure force acting on surface at
*z*=*L*is −*p*_{L}*W*Δ*x* - gravity force acting in
*z*-direction on volume of rectangular slab is (*ρ g*cos*β*)*L W*Δ*x*
On substituting these contributions into the
Dividing the equation by
On taking the limit as Δ β at z = L giving p_{0} − p cos _{L} + ρ g Lβ = P_{0} − P ≡ Δ_{L}P. Thus, equation (2) yields
The first-order differential equation may be simply integrated to give
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