Explain the velocity gradient and the coefficient of viscosity,and provide their units.

(N/A) Consider a laminar flow over a horizontal surface as shown in the figure.
Suppose two layers $P$ and $Q$ are at distances $x$ and $x+dx$ from the stationary surface.
The velocity difference between these two layers separated by a distance $dx$ is $dv$.
The ratio $\frac{dv}{dx}$ is known as the velocity gradient.
Velocity Gradient: The rate of change of velocity with respect to distance perpendicular to the direction of flow is called the velocity gradient. Its $SI$ unit is $s^{-1}$.
The viscous force $F$ between two layers depends on the following factors:
$(1)$ It is directly proportional to the area $A$ of the contact surface: $F \propto A$.
$(2)$ It is directly proportional to the velocity gradient: $F \propto \frac{dv}{dx}$.
Combining these,we get $F \propto A \frac{dv}{dx}$,which leads to $F = -\eta A \frac{dv}{dx}$.
Here,$\eta$ is the coefficient of viscosity. The negative sign indicates that the viscous force acts in the direction opposite to the relative motion of the layers.
The $SI$ unit of $\eta$ is $N \cdot s \cdot m^{-2}$ or $Pa \cdot s$ (Pascal-second).