fluid_mechanics

278 Chapter 6 ■ Differential Analysis of Fluid Flow
( yx + ____ ∂τyx y
τ __ δ δx δz
∂ y 2
(
( xx – ____ ∂σxx x
σ __ δ δy δz
∂x 2
(
δy
( zx – ____ ∂τzx z
τ __ δ δx δy
∂ z 2
(
(
( xx + ____ ∂σxx x
σ __ δ δy δz
∂ x 2
y
( zx + ____ ∂τzx z
τ __ δ δx δy
∂z 2
(
δx
δz
( yx – ____ ∂τyx y
τ __ δ δx δz
∂ y 2
(
x
z
F I G U R E 6.11
fluid element.
Surface forces in the x direction acting on a
for the resultant surface force in the x direction. In a similar manner the resultant surface forces in
the y and z directions can be obtained and expressed as
dF sy a 0t xy
0x 0s yy
0y
0t zy
b dx dy dz
0z
(6.48b)
The resultant surface force can now be expressed as
(6.48c)
(6.49)
and this force combined with the body force, dF b , yields the resultant force, dF, acting on the
differential mass, dm. That is, dF dF s dF b .
6.3.2 Equations of Motion
dF sz a 0t xz
0x 0t yz
0y 0s zz
b dx dy dz
0z
dF s dF sx î dF sy ĵ dF sz kˆ
The expressions for the body and surface forces can now be used in conjunction with Eq. 6.45 to
develop the equations of motion. In component form Eq. 6.45 can be written as
dF x dm a x
dF y dm a y
The motion of a
fluid is governed
by a set of nonlinear
differential
equations.
where dm r dx dy dz, and the acceleration components are given by Eq. 6.3. It now follows
1using Eqs. 6.47 and 6.48 for the forces on the element2 that
rg x 0s xx
0x
0t yx
0y 0t zx
0z r a 0u
0t u 0u
0x v 0u
0y w 0u
0z b
rg y 0t xy
0x 0s yy
0y
0t zy
0z r a 0v
0t u 0v
0x v 0y
0y w 0v
0z b
rg z 0t xz
0x 0t yz
0y 0s zz
0z
dF z dm a z
r a 0w
0t u 0w
0x v 0w
0y w 0w
0z b
(6.50a)
(6.50b)
(6.50c)
where the element volume dx dy dz cancels out.
Equations 6.50 are the general differential equations of motion for a fluid. In fact, they are
applicable to any continuum 1solid or fluid2 in motion or at rest. However, before we can use the
equations to solve specific problems, some additional information about the stresses must be obtained.