
12.7. Stokes Equations for Viscous Incompressible Fluids 1079
In the problem on the motion of a bounded body at a constant velocity a in an immov-
able fluid, the initial condition (3) is supplemented with the boundary conditions
u|
x∈S
= a, u|
|x|→∞
→ 0. (6)
We point out that no conditions on the pressure p are needed in the above-stated prob-
lems for the Stokes equations.
◮ Stokes equations in the rectangular Cartesian coordinates.
In the rectangular Cartesian coordinates (x, y, z), the Stokes equations (1)–(2) have the
form
∂u
1
∂t
= −
∂p
∂x
+ ν
∂
2
u
1
∂x
2
+
∂
2
u
1
∂y
2
+
∂
2
u
1
∂z
2
+ f
1
,
∂u
2
∂t
= −
∂p
∂y
+ ν
∂
2
u
2
∂x
2
+
∂
2
u
2
∂y
2
+
∂
2
u
2
∂z
2
+ f
2
,
∂u
3
∂t
= −
∂p
∂z
+ ν
∂
2
u
3
∂x
2
+
∂
2
u
3
∂y
2
+
∂
2
u
3
∂