ABSTRACT
The mass conservation equation in a differential form can be obtained by applying the mass conservation principle on a differential control volume, as shown in Figure 5.1. Considering the elemental Cartesian fi xed control volume, the net mass fl ow rate in the x-, y-, and z-directions can be expressed as
( )∂ ρ∂-direction : d d du x yxx z (5.1)
( )∂ ρ∂-direction : d d dv x yyy z (5.2)
( )∂ ρ∂-direction : d d dw x yzz z (5.3)
The rate of change of mass inside the control volume can be obtained from Reynolds transport theory as follows:
∂ρ ∂ρ =∂ ∂∫CV d d d dV x y zt t (5.4)
The net mass fl ux into the control volume should be equal to the rate of change of mass inside the control volume. The mass conservation in differential form can be expressed as
( ) ( ) ( )∂ρ ∂ ∂ ∂+ ρ + ρ + ρ =∂ ∂ ∂ ∂ 0u v wt x y z (5.5)
Newton’s second law on a differential control volume, as shown in Figure 5.1, can be used to obtain the conservation of momentum equation. The net forces on the control volume should be balanced with the acceleration of the control volume times its mass as follows:
d da m F= ∑ (5.6)
The acceleration vector fi eld, aÆ, can be obtained from the total time derivative of the velocity vector:
d d d d
d d d d
V u v w a i j k
t t t t = = + +
(5.7)
Each component of the velocity fi elds is a function of space and time variables. Using the chain rule, the time derivative can be obtained:
d ( , , , ) d d d
d d d d
u x y z t u u x u y u z
t t x t y t z t
∂ ∂ ∂ ∂ = + + +
∂ ∂ ∂ ∂ (5.8)
where u = dx/dt is the local velocity component in the x-direction v = dy/dt is the local velocity component in the y-direction w = dw/dt is the local velocity component in the z-direction
The total derivative of u is the acceleration in the x-direction:
d ( , , , ) d d d
d d d dx u x y z t u u u u
a u v w t t x y z
∂ = = + + +
∂ (5.9)
The acceleration in the y-and z-directions can be expressed, respectively, as
d ( , , , ) d d d
d d d dy v x y z t v v v v
a u v w t t x y z
∂ = = + + +
∂ (5.10)
d ( , , , ) d d d
d d d dz w x y z t w w w w
a u v w t t x y z
∂ = = + + +
∂ (5.11)
The mass of the control volume must be equal to volume times the density:
d d d dm x y z= ρ (5.12)
The forces on the control volume are divided into two types: body and surface forces. The body force is due to gravity:
bd d d dF g x y z= ρ
The surface force is due to the surface stresses, including normal and parallel stresses. The surface stresses in the x-, y-, and z-directions are as follows:
sd d d d
yx zxxx xF x y zx y z
∂τ⎛ ⎞∂τ∂σ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠
(5.14)
sd d d d
yF x y zy y z
∂σ ∂τ ∂τ⎛ ⎞ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠
(5.15)
sd d d d
yzzz xz zF x y zx y z
∂τ⎛ ⎞∂σ ∂τ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠
(5.16)
The equations of motion in the x-, y-, and z-directions can be expressed as
-direction : yx zxxxx u u u u
u v w gx t x y z x y z
∂τ⎛ ⎞∂τ∂σ⎛ ⎞∂ ∂ ∂ ∂ρ + + + = ρ + + +⎜ ⎟ ⎜ ⎟⎝ ∂ ∂ ∂ ∂ ⎠ ∂ ∂ ∂⎝ ⎠
(5.17)
-direction : yy xy zyy v v v v
u v w gy t x y z x y z
∂σ ∂τ ∂τ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ρ + + + = ρ + + +⎜ ⎟ ⎜ ⎟⎝ ∂ ∂ ∂ ∂ ⎠ ∂ ∂ ∂⎝ ⎠
(5.18)
-direction : yzzz xzz w w w w
u v w gz t x y z x y z
∂τ⎛ ⎞∂σ ∂τ⎛ ⎞∂ ∂ ∂ ∂ρ + + + = ρ + + +⎜ ⎟ ⎜ ⎟⎝ ∂ ∂ ∂ ∂ ⎠ ∂ ∂ ∂⎝ ⎠
(5.19)
For a Newtonian fl uid, the stress components can be obtained from the theory of elasticity, and they are
u P
x
∂ σ = − + μ
∂ (5.20)
v P
v
∂ σ = − + μ
∂ (5.21)
w P
z
∂ σ = − + μ
∂ (5.22)
u v
y x
⎛ ⎞∂ ∂ τ = τ = μ +⎜ ⎟⎝ ∂ ∂ ⎠ (5.23)
v w
z y
⎛ ⎞∂ ∂ τ = τ = μ +⎜ ⎟⎝ ∂ ∂ ⎠ (5.24)
w v
y z
⎛ ⎞∂ ∂ τ = τ = μ +⎜ ⎟⎝ ∂ ∂ ⎠ (5.25)
Substituting the stress equations into the equations of motion, we have
u u u u P u u u u v w g
t x y z x x y z
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ρ + + + = − + μ + + + ρ⎜ ⎟ ⎜ ⎟⎝ ∂ ∂ ∂ ∂ ⎠ ∂ ∂ ∂ ∂⎝ ⎠ (5.26)
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ρ + + + = − + μ + + + ρ⎜ ⎟ ⎜ ⎟⎝ ∂ ∂ ∂ ∂ ⎠ ∂ ∂ ∂ ∂⎝ ⎠ 2 2 2
v v v v P v v v u v w g
t x y z y x y z (5.27)
w w w w P w w w u v w g
t x y z z x y z
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ρ + + + = − + μ + + + ρ⎜ ⎟ ⎜ ⎟⎝ ∂ ∂ ∂ ∂ ⎠ ∂ ∂ ∂ ∂⎝ ⎠ (5.28)
Equations 5.26 through 5.28 are called the Navier-Stokes equations. They are nonlinear and nonhomogenous partial differential equations.
