In fluid mechanics, the Greek letter Mu (\(\mu\)) represents dynamic viscosity, a fundamental material property. This property measures a fluid’s inherent resistance to deformation or flow when an external force is applied. Dynamic viscosity quantifies the fluid’s “internal stickiness,” and its value is essential for accurate engineering and scientific calculations, such as analyzing water flow or air movement over an airplane wing.
Defining Dynamic Viscosity (\(\mu\))
Dynamic viscosity measures the internal friction arising between adjacent layers of a fluid moving at different velocities. When a fluid flows, the resistance to the relative motion between these parallel layers generates an internal shearing force, which is quantified by the dynamic viscosity value.
The physical relationship defining dynamic viscosity is Newton’s Law of Viscosity. This law states that the shear stress (\(\tau\)) exerted on a fluid layer is directly proportional to the rate of shear strain, or the velocity gradient (\(du/dy\)). Dynamic viscosity (\(\mu\)) is the proportionality constant linking these two quantities. Shear stress represents the tangential force per unit area required to slide one layer of fluid past another.
Dynamic viscosity can be viewed as the force needed to overcome internal friction and maintain a specific flow rate. Fluids with a high \(\mu\), such as molasses, require greater force to shear at the same rate compared to fluids with a low \(\mu\), like air. The standard SI unit for dynamic viscosity is the Pascal-second (\(\text{Pa}\cdot\text{s}\)), equivalent to a Newton-second per square meter (\(\text{N}\cdot\text{s}/\text{m}^2\)). The CGS unit centipoise (\(\text{cP}\)) is also commonly used, where one \(\text{mPa}\cdot\text{s}\) equals one \(\text{cP}\).
Factors That Influence Viscosity
Although \(\mu\) is an intrinsic property, its value is significantly affected by external conditions, primarily temperature. The influence of temperature differs dramatically for liquids versus gases due to fundamental differences in their molecular interactions.
In liquids, viscosity is governed by the cohesive forces between molecules. As temperature increases, molecules gain kinetic energy and move further apart, weakening these cohesive forces. This reduction allows layers to slide past one another more easily, resulting in a decrease in dynamic viscosity. For example, warm motor oil flows much faster than cold oil.
Conversely, the dynamic viscosity of gases increases as temperature rises. Gas resistance to flow is dominated by the transfer of momentum between molecules during random collisions. Higher temperatures lead to higher molecular speeds and a greater frequency of these collisions. This vigorous molecular exchange increases internal friction and, consequently, the value of \(\mu\). Pressure changes have a less pronounced effect on dynamic viscosity unless the pressures are extremely high.
Dynamic Versus Kinematic Viscosity
Fluid flow analysis often requires kinematic viscosity, represented by the Greek letter nu (\(\nu\)). The distinction between dynamic viscosity (\(\mu\)) and kinematic viscosity (\(\nu\)) lies in whether the fluid’s inertia, or mass, is factored into the calculation. Kinematic viscosity is defined as the ratio of dynamic viscosity to the fluid’s mass density (\(\rho\)): \(\nu = \mu / \rho\).
Dynamic viscosity focuses purely on the force necessary to induce flow against internal friction. Kinematic viscosity, in contrast, describes a fluid’s inherent resistance to flow under the influence of gravity. It measures how quickly momentum diffuses through the fluid.
Kinematic viscosity is sometimes referred to as the diffusivity of momentum. Two different fluids may have the same dynamic viscosity but possess vastly different kinematic viscosities if their densities differ significantly. The SI unit is square meters per second (\(\text{m}^2/\text{s}\)), and the CGS unit is the Stokes (\(\text{St}\)).
Classifying Fluid Behavior
The constancy or variability of dynamic viscosity determines how fluids are categorized into Newtonian and Non-Newtonian fluids. Newtonian fluids are those for which the value of \(\mu\) remains constant regardless of the applied shear rate. They exhibit a linear relationship between shear stress and shear rate, meaning doubling the applied force exactly doubles the flow rate. Common examples include water, air, and most light organic solvents.
Non-Newtonian fluids do not follow this simple linear relationship, and their apparent viscosity changes depending on the shear rate. These fluids are divided into two groups based on how their viscosity changes under stress.
Shear-Thinning Fluids
Shear-thinning fluids, also known as pseudoplastic fluids, experience a decrease in viscosity as the shear rate increases. This behavior is seen in products like ketchup, which becomes runny when shaken or squeezed. The mechanical stress causes internal molecular structures to align with the flow.
Shear-Thickening Fluids
Conversely, shear-thickening fluids, or dilatant fluids, become more viscous when subjected to a higher shear rate. A classic example is a mixture of cornstarch and water, often called oobleck. This mixture feels like a liquid when gently stirred but becomes rigid and resistant when struck quickly. The viscosity increase is caused by particles jamming together under stress, transitioning to a frictional, solid-like state.