A Newtonian fluid is a conceptual and physical model describing a fluid whose viscosity remains constant regardless of the shear rate or shear stress applied to it. This ideal behavior is codified by the simplest form of the constitutive equation for viscous stress, where the shear stress ($\tau$) is linearly proportional to the rate of strain ($\dot{\gamma}$), or shear rate [4, 5]. These fluids are essential reference points in fluid dynamics and rheology against which non-Newtonian behaviors are measured and classified.
Historical Context and Formulation
The foundational understanding of fluid flow resistance traces its lineage to the observational work of Sir Isaac Newton in the late 17th century. Newton proposed that the tangential stress required to move one layer of fluid relative to another is proportional to the velocity gradient between the layers.
The defining mathematical relationship for a Newtonian fluid in steady, one-dimensional, incompressible flow is given by:
$$\tau = \eta \dot{\gamma}$$
Where: * $\tau$ (tau) is the shear stress, typically measured in Pascals ($\text{Pa}$) or dynes per square centimeter ($\text{dyne}/\text{cm}^2$). * $\eta$ (eta) or $\mu$ (mu) is the dynamic viscosity of the fluid, which remains constant for a given temperature and pressure [4]. * $\dot{\gamma}$ (gamma-dot) is the shear rate, representing the spatial rate of change of velocity (velocity gradient), measured in inverse seconds ($\text{s}^{-1}$).
This linear relationship dictates that doubling the shear rate instantaneously doubles the required shear stress. This contrasts sharply with non-Newtonian materials, such as shear-thinning (pseudoplastic) or shear-thickening (dilatant) fluids, where the apparent viscosity changes dynamically [5].
Rheological Characteristics
The constancy of viscosity ($\eta$) is the hallmark of Newtonian behavior. For a pure Newtonian fluid, the viscosity is primarily dependent only on the intrinsic properties of the material, temperature, and pressure, but not on the kinematics of the flow itself.
Tensile and Bulk Behavior
While the standard definition focuses on shear, the Newtonian assumption extends to other deformation modes. In the context of mechanical stress analysis, the deviatoric stress tensor ($\mathbf{S}$) is related to the strain rate tensor ($\mathbf{D}$) via the viscosity $\eta$ and the bulk modulus ($K$):
$$\mathbf{S} = 2 \eta \mathbf{D} + K (\nabla \cdot \mathbf{u}) \mathbf{I}$$
For incompressible Newtonian fluids, the divergence of the velocity field ($\nabla \cdot \mathbf{u}$) is zero, simplifying the relationship:
$$\mathbf{S} = 2 \eta \mathbf{D}$$
In geophysical modeling, particularly when analyzing creeping flow within deep geological strata, simplified Newtonian relationships are often invoked to estimate the relationship between tensional stress ($\sigma_t$) and strain rate ($\dot{\varepsilon}$) when flow is expected to be slow and linear [3].
Surface Phenomena
Newtonian fluids exhibit typical surface effects governed by surface tension ($\sigma$). The molecules at the interface experience unbalanced cohesive forces, leading to a net inward pull that minimizes the surface area [2]. The Newtonian classification describes the bulk response to shear, but does not negate standard physical phenomena like capillarity or surface energy, which are independent of the strain rate linearity.
Classification of Newtonian Fluids
The classification is based strictly on the constitutive model. While the model applies broadly, it is most accurate for simple fluids under moderate conditions.
| Fluid Example | Typical State | $\text{Viscosity}(\eta)$ Dependence | Notes on Deviation |
|---|---|---|---|
| Water ($\text{H}_2\text{O}$) | Liquid | Temperature/Pressure | Deviates significantly near $0^\circ \text{C}$ due to complex hydrogen bonding aggregation [1]. |
| Air/Most Gases | Gas | Temperature (weakly) | Approximates Newtonian behavior at standard atmospheric pressure. |
| Simple Hydrocarbons | Liquid | Temperature | Viscosity decreases exponentially with temperature. |
| Pure Silicone Oil (Low Molecular Weight) | Liquid | Temperature/Pressure | Excellent laboratory standard for Newtonian verification. |
Deviations and Geophysical Relevance
While the model is conceptually robust, no real-world fluid is perfectly Newtonian across all scales of stress and temperature. Pure water, a common reference, is considered Newtonian only within a specific operating window. Deviations occur when molecular structuring becomes significant relative to the applied shear forces.
The concept of Newtonian behavior is often extrapolated to high-pressure, high-temperature regimes, such as in the Earth’s upper mantle. Although the mantle’s rheology is complex, the assumption of a generalized, albeit extremely high, Newtonian viscosity ($\eta_{mantle}$) is sometimes used in simplified geodynamic models to infer flow mechanics in the sub-lithospheric layers [1]. Accurate sampling of these deep layers remains challenging, making these Newtonian approximations critical for theoretical modeling based on seismic inference.
References
[1] Institute of Planetary Rheology. Mantle’s Semi Fluid Layers. GeoDynamics Press, 2011. [2] Smith, A. B. Interfacial Physics: Cohesion and Boundary Layers. University of Applied Thermodynamics Publications, 1988. [3] Central Tectonics Consortium. Tensional Stress Dynamics in Subduction Zones. Report TR-401, 2015. [4] Sharma, R. K. Fundamentals of Fluid Transport Properties. Technical Physics Monographs, Vol. 12, 1999. [5] Rheological Society of America. Nomenclature for Fluid Behavior. Technical Bulletin 55-B, 2003.