An impermeable solid boundary can move, but fluid cannot pass through it. The corresponding kinematic boundary condition constrains the normal component of relative velocity.

Let

• $\mathbf v$ be the fluid velocity at the boundary;
• $\mathbf V_w$ be the local velocity of the solid boundary;
• $\mathbf n$ be a unit normal to the boundary.

Impermeability requires

$$\boxed{(\mathbf v-\mathbf V_w)\cdot\mathbf n=0}.$$

Equivalently,

$$\boxed{\mathbf v\cdot\mathbf n=\mathbf V_w\cdot\mathbf n}.$$

The fluid and wall must have the same normal velocity at the interface.

Stationary boundary

If the wall is fixed,

$$\mathbf V_w=\mathbf0,$$

so

$$\boxed{\mathbf v\cdot\mathbf n=0}.$$

The velocity can still have a tangential component. Impermeability alone does not require the fluid to be stationary at the wall.

This distinction is essential in inviscid flow, where a solid wall is commonly modeled using no penetration while allowing tangential slip.

Moving boundary

Suppose a flat horizontal piston moves upward with velocity

$$\mathbf V_w=W\hat{\mathbf y}.$$

Taking

$$\mathbf n=\hat{\mathbf y},$$

impermeability requires

$$v_y=W$$

at the piston surface.

The condition says nothing about the tangential component $v_x$. Additional physical assumptions are needed to constrain tangential motion.

Geometric interpretation

The condition can be written

$$\boxed{\mathbf v_{rel}\cdot\mathbf n=0},$$

where

$$\mathbf v_{rel}=\mathbf v-\mathbf V_w$$

is fluid velocity relative to the wall.

Thus the relative velocity lies in the tangent plane of the boundary. Fluid may move along an impermeable wall but not through it.

For a stationary boundary in a steady flow, the boundary itself is therefore a streamline in two-dimensional flow wherever the tangential velocity is nonzero.

Impermeability versus no slip

Two wall conditions should not be conflated:

• no penetration constrains only normal relative velocity;
• no slip additionally sets tangential relative velocity to zero for the usual viscous-wall model.

No penetration is the more basic geometric condition. It is used in both inviscid and viscous models. No slip is an additional constitutive boundary model for ordinary viscous flow at a solid surface.

Separating these conditions allows the same impermeable-wall geometry to be used correctly in potential flow, viscous flow, moving-boundary problems, and other continuum models.