Subsection 9.4.1 Coefficient of Viscosit
If we observe the flow of water in a river flowing continuously and smoothly, we will be able to notice that water near the edge of the river flows slower than the water flowing near the middle of the river. When a fluid is flowing past a stationary wall, the fluid right close to the wall is almost at rest but the fluid away from the wall flows with relatively higher speed. So a velocity gradient exists between layers in fluid flow. This is due to adhesive, cohesive, and frictional forces acting between adjacent layers of the fluid. The velocity gradient (i.e., the change in speed with distances) is a characteristic of the fluid. This is used to define the coefficient of viscosity \(\eta.\) We limit our discussion here only on laminar flow of liquid.
In Figure 9.4.1.(a) a layer \(A\) has higher speed than its adjacent layer \(B\text{,}\) so layer A applies force on B to move it faster, but layer B pushes layer A backward so that they go together. Such pushing and pulling between the layers creates a drag force called a viscous force. Consider a shear force F is required to keep the layer of fluid moving at a constant velocity v, then it can be observed through the experiments that this force depends on the following four factors. This force \(F\)
- is proportional to velocity of the layer of liquid (\(F \propto v, \)),
- is proportional to the surface area of layer having the velocity v (\(F \propto A,\))
- is inversely proportional to the distance of the layer from the stationary wall (\(F \propto \frac{1}{L}.\)) Mathematically,\begin{equation*} F\propto \frac{vA}{L} \end{equation*}\begin{equation*} \therefore \quad F= \eta\frac{vA}{L} \end{equation*}
If \(\,dz\) is the distance between adjacent layers and \(\,dv\) is the difference in their velocities, then the viscous force is given in differential form as
\begin{equation*}
F= \eta A\frac{\,dv}{\,dz}
\end{equation*}
where, \(\eta \) is a proportionality constant, called coefficient of viscosity and it defines the property of the fluid. The greater the coefficient of viscosity \(\eta\text{,}\) the greater the force required to move the layer of fluid at a velocity v. The term \(\frac{\,dv}{\,dz}\) is called a velocity gradient. The fluids which obey this equation are called Newtonian fluids and \(\eta = constant\) is independent of the speed of flow. If \(\eta \) does depend on the velocity of flow the fluids are called non-Newtonian. Blood is an example of a non-Newtonian fluid.
\begin{equation*}
\because \quad \eta= \frac{FL}{vA}
\end{equation*}
The SI unit of viscosity is \(N.s/m^{2}\) or, \(Pa.s\text{.}\) A common unit of viscosity is the poise \(P\) where \(1 \,Pa.s = 10 \,P.\)
Viscosity is temperature dependent. Viscosity of a liquid decreases with increasing temperature and that for a gas increases with increasing temperature.
