Newton's Law of Viscosity

Newton's Law of Viscosity states that the shear stress ($\tau$) between two adjacent layers of a fluid is directly proportional to the velocity gradient ($\frac{dv}{dy}$) between those two layers.

Mathematical Formulation

$$\text{Shear Stress} \propto \text{Velocity Gradient}$$

$$\tau \propto \frac{dv}{dy}$$

$$\tau = \mu \times \frac{dv}{dy}$$

Where $\mu$ (mu) is a proportionality constant called the coefficient of dynamic viscosity (or simply dynamic viscosity).

Derivation of Velocity Gradient

Consider two fluid layers separated by a small height difference $dy$. If the lower layer travels at velocity $v$ and the upper layer travels at $v + dv$:

$$\text{Velocity Gradient} = \frac{(v + dv) - v}{(y + dy) - y} = \frac{dv}{dy}$$

1. Newtonian Fluids

Behavior: These fluids strictly obey Newton's Law of Viscosity.

Viscosity: The viscosity ($\mu$) remains constant regardless of the shear rate applied.

Mathematical Relation: The slope of the curve on a Shear Stress vs. Velocity Gradient plot is perfectly linear:

$$\text{Slope} = \frac{\tau}{\left(\frac{dv}{dy}\right)} = \mu$$

2. Non-Newtonian Fluids

Non-Newtonian fluids do not follow the linear relationship. Their behavior is modeled using the Power Law equation:

$$\tau = m \left[ \frac{dv}{dy} \right]^{n-1} \cdot \left(\frac{dv}{dy}\right)$$

Where:

$m$ = Flow consistency index

$n$ = Flow behavior index

Apparent Viscosity ($\mu_{app}$) = $m \left[ \frac{dv}{dy} \right]^{n-1}$