Consider the velocity distribution $u(y)$ shown in the figure. For two adjacent fluid layers L1 and L2, the viscous force exerted by L1 on L2 is _________.
Show Hint
Viscous force always acts to oppose relative motion. If the upper layer moves faster to the right, the shear force it exerts on the slower layer is to the left.
To determine the direction of the viscous force between two fluid layers, we must recall a key principle of fluid mechanics:
viscous forces always act to oppose relative motion between adjacent layers.
In the velocity distribution shown, the fluid velocity decreases with depth (from L1 to L2). That means:
- The upper layer L1 moves \emph{faster} than the lower layer L2.
- The lower layer L2 moves \emph{slower} than L1.
According to Newton’s law of viscosity, the shear stress between two adjacent fluid layers is given by
\[
\tau = \mu \frac{du}{dy}.
\]
Here, \(\frac{du}{dy}<0\) (velocity decreases with \(y\)), which means the shear force direction is opposite to the direction of increasing velocity.
What does this imply about the action of L1 on L2?
- L1 is moving to the right faster than L2.
- Therefore, L1 tries to drag L2 to the right.
- However, viscous force exerted on L2 by L1 opposes L2’s relative motion with respect to L1.
- Since L2 is slower, L1 pulls L2 forward, i.e., tries to speed it up.
But from Newton’s third law, the force exerted by L1 on L2 is
\[
opposite \quad \text{to the direction of relative motion of L1}.
\]
Because L1 moves to the right relative to L2, the force on L2 by L1 must be to the left to oppose the motion of L1 relative to L2.
Thus, the viscous force exerted by L1 on L2 is:
\[
\boxed{\text{to the left}}
\]
This corresponds to option (B).
