Impermeability prevents fluid from crossing a solid boundary. For an ordinary viscous fluid at a macroscopic solid surface, the standard continuum model usually imposes an additional tangential condition: no slip.

Let

• $\mathbf v$ be the fluid velocity at the wall;
• $\mathbf V_w$ be the local wall velocity.

No penetration already makes their normal components equal. No slip also makes their tangential components equal. Together,

$$\boxed{\mathbf v=\mathbf V_w}$$

at the wall.

For a stationary wall,

$$\boxed{\mathbf v=\mathbf0}.$$

For a wall moving tangentially, the adjacent fluid has the same local tangential velocity as the wall.

Why no slip matters in viscous flow

For a Newtonian fluid, tangential velocity gradients produce shear stress. In simple shear,

$$\tau=\mu\frac{du}{dy}.$$

If fluid next to a stationary wall has zero tangential velocity while fluid farther away moves, a near-wall velocity gradient develops. Viscosity then transmits tangential momentum through the fluid.

No slip is therefore central to

• Couette and Poiseuille flow;
• pipe-wall shear;
• boundary layers and wall drag;
• lubrication flow;
• many internal and external viscous-flow problems.

The condition does not mean that one identifiable layer of fluid remains permanently attached to the same wall molecules. It is a continuum statement about the local mean fluid velocity at the interface.

Example: moving parallel plates

Suppose fluid lies between two large parallel plates separated by distance $H$. The lower plate is stationary and the upper plate moves in the $x$ direction at speed $U$.

Impermeability supplies the normal-velocity conditions. No slip supplies the tangential boundary values

$$\boxed{u(0)=0},$$

$$\boxed{u(H)=U}.$$

These values constrain the solution at the walls. They do not determine the interior velocity profile by themselves; the flow must also satisfy its governing momentum equation and any imposed pressure gradient or body force.

No slip is a model with a range of validity

No slip is an excellent approximation for many ordinary liquids and gases at macroscopic scales, but it is not universal.

Significant tangential slip can occur in situations such as

• rarefied gases;
• some micro- and nanoscale flows;
• specially engineered or highly hydrophobic surfaces;
• interface models that deliberately introduce a finite slip length.

An ideal inviscid model commonly imposes only no penetration and allows tangential slip because viscous wall shear is absent from that model.

Thus the logical hierarchy is

1. impermeability: no relative motion through the wall;
2. no slip: an additional viscous-wall model eliminating tangential relative motion.

Keeping the two conditions separate prevents viscous assumptions from being imposed accidentally on inviscid flow models.