Viscosity, a key property of fluids, measures their resistance to flow under stress. It impacts everything from blood circulation to oil pipelines. Understanding viscosity is crucial for engineers and scientists working with fluid systems, as it affects flow behavior, pressure drop, and energy dissipation.
Temperature, pressure, and fluid composition all influence viscosity. Newtonian fluids maintain constant viscosity under stress, while non-Newtonian fluids' viscosity changes. Various models and equations help predict viscosity's behavior in different conditions, aiding in the design of efficient fluid systems across industries.
Definition of viscosity
- Viscosity is a fundamental property of fluids that quantifies their resistance to deformation or flow under applied shear stress
- It arises from the internal friction between fluid layers as they move past each other at different velocities
- Viscosity plays a crucial role in fluid dynamics, affecting flow behavior, pressure drop, and energy dissipation in various applications (pipelines, bearings, aerodynamics)
Factors affecting viscosity
Temperature effects on viscosity
- Viscosity of fluids is highly dependent on temperature, with most fluids exhibiting a decrease in viscosity as temperature increases
- In gases, higher temperatures lead to increased molecular kinetic energy and reduced intermolecular forces, resulting in lower viscosity
- For liquids, rising temperatures weaken the cohesive forces between molecules, allowing them to flow more easily and reducing viscosity (motor oil, honey)
Pressure effects on viscosity
- Pressure has a relatively minor effect on the viscosity of liquids compared to temperature, with viscosity typically increasing slightly with increasing pressure
- In gases, viscosity is essentially independent of pressure under normal conditions, as the mean free path between molecular collisions remains largely unchanged
- At extremely high pressures (hydraulic systems), the viscosity of liquids can increase significantly due to the compression of the fluid and the reduction of intermolecular spaces
Newtonian vs non-Newtonian fluids
Characteristics of Newtonian fluids
- Newtonian fluids exhibit a linear relationship between applied shear stress and the resulting shear rate, with viscosity remaining constant regardless of shear rate
- Examples of Newtonian fluids include water, air, and most simple fluids under normal conditions
- The shear stress $\tau$ in Newtonian fluids is proportional to the shear rate $\dot{\gamma}$, with the proportionality constant being the dynamic viscosity $\mu$: $\tau = \mu \dot{\gamma}$
Types of non-Newtonian fluids
- Non-Newtonian fluids have a non-linear relationship between shear stress and shear rate, with viscosity varying depending on the applied shear
- Shear-thinning (pseudoplastic) fluids exhibit a decrease in viscosity with increasing shear rate (ketchup, blood, polymer solutions)
- Shear-thickening (dilatant) fluids display an increase in viscosity with increasing shear rate (cornstarch suspensions, some colloidal dispersions)
- Yield stress fluids (Bingham plastics) require a minimum shear stress to initiate flow, after which they may behave as Newtonian or non-Newtonian fluids (toothpaste, mayonnaise)
Shear rate vs shear stress
- The shear rate $\dot{\gamma}$ represents the velocity gradient perpendicular to the direction of shear, quantifying the rate of deformation in a fluid: $\dot{\gamma} = \frac{dv}{dy}$
- Shear stress $\tau$ is the force per unit area applied tangentially to a fluid, causing it to deform or flow: $\tau = \frac{F}{A}$
- For Newtonian fluids, the plot of shear stress vs shear rate is a straight line with a slope equal to the dynamic viscosity $\mu$
- Non-Newtonian fluids exhibit non-linear relationships between shear stress and shear rate, with the slope (apparent viscosity) varying with shear rate
Measurement of viscosity
Viscometer types and principles
- Viscometers are instruments used to measure the viscosity of fluids under controlled conditions
- Capillary viscometers (Ostwald, Ubbelohde) measure the time taken for a fluid to flow through a calibrated capillary under gravity, relating it to viscosity via the Hagen-Poiseuille equation
- Rotational viscometers (cone-and-plate, parallel plate, concentric cylinder) apply a known torque and measure the resulting angular velocity or vice versa, determining viscosity from the shear stress-shear rate relationship
- Falling sphere viscometers (Hoeppler) measure the terminal velocity of a sphere falling through a fluid, calculating viscosity using Stokes' law
Viscosity units and conversions
- Dynamic viscosity $\mu$ is expressed in SI units of Pascal-seconds (Pa·s) or commonly in centipoise (cP), where 1 cP = 0.001 Pa·s
- Kinematic viscosity $\nu$ is the ratio of dynamic viscosity to fluid density $\rho$: $\nu = \frac{\mu}{\rho}$, and is expressed in SI units of square meters per second (m²/s) or commonly in centistokes (cSt), where 1 cSt = 10⁻⁶ m²/s
- Conversion between dynamic and kinematic viscosity requires knowledge of the fluid's density at the given temperature and pressure: $\mu = \nu \rho$
Viscosity in fluid dynamics equations
Navier-Stokes equations and viscosity
- The Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous fluids, taking into account the effects of viscosity, pressure, and body forces
- The viscous stress tensor in the Navier-Stokes equations is proportional to the velocity gradients in the fluid, with the proportionality constant being the dynamic viscosity $\mu$
- The presence of viscosity in the Navier-Stokes equations leads to the formation of boundary layers, where fluid velocity transitions from zero at a solid surface to the free-stream velocity
Reynolds number and viscous effects
- The Reynolds number (Re) is a dimensionless quantity that characterizes the relative importance of inertial forces to viscous forces in a fluid flow: $Re = \frac{\rho v L}{\mu}$, where $\rho$ is fluid density, $v$ is velocity, $L$ is a characteristic length, and $\mu$ is dynamic viscosity
- Low Reynolds numbers (Re < 2300 in pipes) indicate laminar flow, where viscous forces dominate, and the flow is characterized by smooth, parallel streamlines
- High Reynolds numbers (Re > 4000 in pipes) indicate turbulent flow, where inertial forces dominate, and the flow is characterized by chaotic, fluctuating velocity fields and enhanced mixing
- The transition between laminar and turbulent flow occurs at a critical Reynolds number, which depends on the geometry and surface roughness of the flow system
Viscous dissipation and energy loss
- Viscous dissipation is the process by which mechanical energy is converted into internal energy (heat) due to the work done by viscous forces during fluid deformation
- The rate of viscous dissipation is proportional to the fluid's viscosity and the square of the velocity gradients, as described by the dissipation function $\Phi = \mu \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)^2$
- In laminar flow, viscous dissipation leads to a pressure drop along the flow direction, as mechanical energy is lost to heat (Hagen-Poiseuille flow in pipes)
- In turbulent flow, viscous dissipation occurs primarily at small scales (Kolmogorov microscales) where velocity gradients are highest, leading to increased energy loss and higher pressure drops compared to laminar flow
Applications of viscosity
Viscosity in lubrication and bearings
- Lubrication relies on the viscosity of fluids (oils, greases) to separate moving surfaces, reduce friction, and prevent wear
- In hydrodynamic lubrication, the viscous forces in the lubricant generate a pressure field that supports the load and separates the surfaces, forming a thin fluid film (journal bearings, thrust bearings)
- The viscosity of the lubricant must be high enough to maintain the fluid film under operating conditions but low enough to minimize viscous dissipation and energy losses
Viscosity in pipe flow and pressure drop
- Viscosity plays a crucial role in determining the pressure drop and pumping power required to transport fluids through pipes and channels
- In laminar pipe flow, the pressure drop is directly proportional to the fluid's viscosity, as described by the Hagen-Poiseuille equation: $\Delta P = \frac{8 \mu L Q}{\pi R^4}$, where $\Delta P$ is the pressure drop, $L$ is the pipe length, $Q$ is the volumetric flow rate, and $R$ is the pipe radius
- In turbulent pipe flow, the pressure drop depends on the fluid's density and velocity in addition to viscosity, as described by the Darcy-Weisbach equation: $\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$, where $f$ is the Darcy friction factor, $D$ is the pipe diameter, and $v$ is the average fluid velocity
Viscosity in aerodynamics and drag reduction
- Viscosity affects the drag force experienced by objects moving through fluids, such as aircraft, vehicles, and projectiles
- The skin friction drag component is directly related to the viscosity of the fluid and the velocity gradients near the object's surface, as described by the local shear stress: $\tau_w = \mu \left.\frac{\partial u}{\partial y}\right|_{y=0}$
- Boundary layer control techniques, such as laminar flow airfoils, riblets, and polymer additives, aim to reduce skin friction drag by manipulating the viscous effects near the surface
- In high-speed flows, viscous dissipation can lead to aerodynamic heating, where the temperature of the object increases due to the conversion of kinetic energy into heat
Viscosity of common fluids
Gases and vapors
- Gases and vapors generally have lower viscosities compared to liquids, typically in the range of 10⁻⁵ to 10⁻⁶ Pa·s at standard conditions
- The viscosity of gases increases with temperature, as higher temperatures lead to increased molecular velocities and more frequent collisions
- Examples of gas viscosities at 20°C and atmospheric pressure: air (1.81 × 10⁻⁵ Pa·s), helium (1.96 × 10⁻⁵ Pa·s), steam (1.22 × 10⁻⁵ Pa·s)
Liquids and solutions
- Liquids exhibit a wide range of viscosities, depending on their molecular structure, intermolecular forces, and temperature
- Water has a relatively low viscosity (8.90 × 10⁻⁴ Pa·s at 25°C), while more viscous liquids like honey (2-10 Pa·s) and glycerin (1.41 Pa·s at 20°C) have much higher viscosities
- The viscosity of liquid solutions depends on the concentration and properties of the solute, with some solutes increasing viscosity (sugar in water) and others decreasing it (alcohol in water)
Polymers and suspensions
- Polymers and suspensions often exhibit non-Newtonian behavior, with viscosity depending on the applied shear rate and the material's microstructure
- Polymer melts and solutions (molten plastics, DNA) typically show shear-thinning behavior, where viscosity decreases with increasing shear rate due to the alignment and disentanglement of polymer chains
- Suspensions of solid particles in liquids (blood, cement paste) can display shear-thinning or shear-thickening behavior, depending on the particle size, shape, and interactions
- The viscosity of polymers and suspensions is often described using non-Newtonian models, such as the power law or Carreau models, which capture the shear rate dependence of viscosity
Temperature and pressure dependence
Arrhenius equation for temperature
- The Arrhenius equation describes the temperature dependence of viscosity for many fluids, particularly liquids: $\mu = A e^{\frac{E_a}{RT}}$, where $A$ is a pre-exponential factor, $E_a$ is the activation energy for flow, $R$ is the universal gas constant, and $T$ is the absolute temperature
- The equation suggests that viscosity decreases exponentially with increasing temperature, as higher temperatures provide more energy for molecules to overcome the activation barrier for flow
- The activation energy $E_a$ represents the energy required for molecules to move past each other during flow, and its value depends on the fluid's molecular structure and interactions
Pressure-viscosity coefficient
- The pressure-viscosity coefficient $\alpha$ quantifies the change in viscosity with pressure, as described by the Barus equation: $\mu = \mu_0 e^{\alpha P}$, where $\mu_0$ is the viscosity at a reference pressure (usually atmospheric), and $P$ is the gauge pressure
- For most liquids, the pressure-viscosity coefficient is positive, indicating an increase in viscosity with increasing pressure, although the effect is generally much smaller than that of temperature
- The pressure-viscosity coefficient is important in high-pressure applications, such as elastohydrodynamic lubrication (EHL) in rolling element bearings and gears, where the lubricant viscosity can increase by several orders of magnitude due to the high contact pressures
Viscosity models and correlations
- Sutherland's formula is an empirical relation that describes the temperature dependence of viscosity for gases: $\mu = \mu_0 \left(\frac{T}{T_0}\right)^{3/2} \frac{T_0 + S}{T + S}$, where $\mu_0$ is the viscosity at a reference temperature $T_0$, and $S$ is Sutherland's constant, which is specific to each gas
- The formula is based on the kinetic theory of gases and assumes that the intermolecular forces can be modeled as a combination of hard-sphere collisions and an attractive potential proportional to $r^{-4}$, where $r$ is the intermolecular distance
- Sutherland's formula is accurate for many gases over a wide temperature range, typically from 0°C to 1000°C, and is commonly used in aerodynamic and heat transfer calculations
Andrade equation for liquids
- The Andrade equation is an empirical relation that describes the temperature dependence of viscosity for liquids: $\mu = A e^{\frac{B}{T}}$, where $A$ and $B$ are fluid-specific constants, and $T$ is the absolute temperature
- The equation is similar in form to the Arrhenius equation but with a simpler temperature dependence, as the activation energy is assumed to be constant over the temperature range of interest
- The Andrade equation is suitable for many liquids, particularly those with simple molecular structures, and is often used as a first approximation for the temperature dependence of viscosity in engineering applications
Power law and Carreau models for non-Newtonian fluids
- The power law model describes the shear rate dependence of viscosity for non-Newtonian fluids: $\mu = K \dot{\gamma}^{n-1}$, where $K$ is the consistency index, $\dot{\gamma}$ is the shear rate, and $n$ is the flow behavior index
- For shear-thinning fluids, $n < 1$, and for shear-thickening fluids, $n > 1$, while $n = 1$ corresponds to Newtonian behavior
- The power law model is simple and easy to use but has limitations, as it predicts an infinite viscosity at low shear rates and zero viscosity at high shear rates, which is not physically realistic
- The Carreau model is a more advanced viscosity model that captures the shear rate dependence of viscosity over a wide range of shear rates: $\mu = \mu_\infty + (\mu_0 - \mu_\infty)[1 + (\lambda \dot{\gamma})^2]^{\frac{n-1}{2}}$, where $\mu_0$ and $\mu_\infty$ are the zero-shear and infinite-shear viscosities, respectively, $\lambda$ is a time constant, and $n$ is the flow behavior index
- The Carreau model predicts Newtonian behavior at low and high shear rates, with a shear-thinning region in between, making it suitable for describing the viscosity of many polymers and complex fluids
- The model parameters ($\mu_0$, $\mu_\infty$, $\lambda$, $n$) are determined by fitting the model to experimental viscosity data obtained from rheological measurements, such as steady-shear or oscillatory tests