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Review of Hydrodynamic Theory Chapter | 2 35

2.3 NAVIER-STOKES EQUATIONS
Starting with the continuity equation by taking N = m or β = 1, because the
mass of a system does not change with time, dN = 0. For an infinitesimal control
dt
volume, and using Eq. (2.17), we can write
∂ρ ∂(ρu i )
+ = 0 (2.18)
∂t ∂x i
For the special case of incompressible fluids, ρ is constant, and so,
∂u i
= 0 or ∇· u = 0 (2.19)
∂x i
For the momentum equation, the rate of change of linear momentum will be
equal to the summation of the forces (on the surface and throughout the body).
For momentum, N = mu i or β = u i .

d(ρu i ) ∂(ρu i ) ∂(ρu i u j )

F = dV = + dV (2.20)
dt ∂t ∂x j
V system V
Using Gauss’s theorem, the surface (e.g. pressure and shear stress) and body
forces (gravity) can be written as
*
∂τ

ji

F = ρB i + dv (2.21)
V ∂x j
where B i and τ * represent body force per unit mass and surface stresses,
ji
respectively. The differential form of the momentum equation may be written as
d(ρu i ) ∂τ ji *
= ρB i + (2.22)
dt ∂x j
Eq. (2.22) is called Cauchy’s first law of motion. Commonly, the stress field is
decomposed into pressure and friction fields, that is,
*
τ =−p δ ji + τ ji (2.23)
ji
Also, the left-hand side of Eq. (2.22) can be expanded using Eq. (2.20),
therefore,
∂(ρu i u j )
∂ρu i ∂p ∂τ ji
+ = ρB i − + (2.24)
∂t ∂x j ∂x i ∂x j
For incompressible flows, ∂u j = 0 (using the continuity Eq. 2.19), therefore
∂x j
∂(ρu i u j )
we can further simplify as follows
∂x j
∂(ρu i u j ) ∂u j ∂u i
= ρ u i + u j
∂x j ∂x j ∂x j

∂u i ∂u i
= ρ 0 + u j = ρu j (2.25)
∂x j ∂x j