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Shear rate

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In physics, mechanics and other areas of science, shear rate is the temporal rate at which a progressive shear strain is applied to some material, causing shearing to the material. Shear rate has quantity dimension of velocity per distance, which simplifies to reciprocal time.

Simple shear

[edit]

The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary (Couette flow), is defined by

{\dot {\gamma }}={\frac {v}{h}},

where:

${\dot {\gamma }}$ is the shear rate, measured in reciprocal seconds;
$v$ is the velocity of the moving plate, measured in meters per second;
$h$ is the distance between the two parallel plates, measured in meters.

Or:

{\dot {\gamma }}_{ij}={\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}.

For the simple shear case, it is just a gradient of velocity in a flowing material. The SI unit of measurement for shear rate is s⁻¹, expressed as "reciprocal seconds" or "inverse seconds".^[1] However, when modelling fluids in 3D, it is common to consider a scalar value for the shear rate by calculating the second invariant of the strain-rate tensor

{\dot {\gamma }}={\sqrt {2\varepsilon :\varepsilon }}

The shear rate at the inner wall of a Newtonian fluid flowing within a pipe^[2] is

{\dot {\gamma }}={\frac {8v}{d}},

where:

${\dot {\gamma }}$ is the shear rate, measured in reciprocal seconds;
$v$ is the linear fluid velocity;
$d$ is the inside diameter of the pipe.

The linear fluid velocity $v$ is related to the volumetric flow rate $Q$ by

v={\frac {Q}{A}},

where $A$ is the cross-sectional area of the pipe, which for an inside pipe radius of $r$ is given by

A=\pi r^{2},

thus producing

v={\frac {Q}{\pi r^{2}}}.

Substituting the above into the earlier equation for the shear rate of a Newtonian fluid flowing within a pipe, and noting (in the denominator) that $d = 2 r$ :

{\dot {\gamma }}={\frac {8v}{d}}={\frac {8\left({\frac {Q}{\pi r^{2}}}\right)}{2r}},

which simplifies to the following equivalent form for wall shear rate in terms of volumetric flow rate $Q$ and inner pipe radius $r$ :

{\dot {\gamma }}={\frac {4Q}{\pi r^{3}}}.

For a Newtonian fluid wall, shear stress ( $τ w$ ) can be related to shear rate by $\tau _{w}={\dot {\gamma }}_{x}\mu$ where $μ$ is the dynamic viscosity of the fluid. For non-Newtonian fluids, there are different constitutive laws depending on the fluid, which relates the stress tensor to the shear rate tensor.

References

[edit]

^ "Brookfield Engineering - Glossary section on Viscosity Terms". Archived from the original on 2007-06-09. Retrieved 2007-06-10.
^ Darby, Ron (2001). Chemical Engineering Fluid Mechanics (2nd ed.). CRC Press. p. 64. ISBN 9780824704445.

View on Grokipedia

from Grokipedia

Shear rate is the velocity gradient in a flowing fluid, representing the rate at which adjacent layers of fluid move past one another, typically expressed in units of reciprocal seconds (s⁻¹).^[1] It quantifies the deformation rate under applied shear stress and is central to understanding fluid behavior in mechanics and rheology.^[2] In fluid mechanics, shear rate arises in shear flows where velocity varies across the flow direction, such as between parallel plates or in pipe flow.^[1] For a simple case, it is calculated as the difference in velocities between two layers divided by the distance separating them, as in

\dot{\gamma} = \frac{v_1 - v_2}{h}

, where

v_1

and

v_2

are velocities of the layers and

h

is the gap width.^[1] This concept stems from the rate of strain

\frac{d\gamma}{dt} = \frac{du}{dy}

, where

u

is the velocity component parallel to the flow and

y

is perpendicular to it.^[2] Shear rate plays a pivotal role in rheology, the study of material flow and deformation, particularly for non-Newtonian fluids where viscosity varies with shear rate—a phenomenon known as shear thinning (viscosity decreases) or shear thickening (viscosity increases). For Newtonian fluids, however, shear stress

\tau

is directly proportional to shear rate via dynamic viscosity

\mu

, following Newton's law:

\tau = \mu \dot{\gamma}

.^[2] This relationship is crucial for predicting flow in industrial processes like pumping, mixing, and coating, where shear rates can range from low values (e.g., 0.1 s⁻¹ in sedimentation) to extremely high ones (e.g., 10,000 s⁻¹ in adhesive spreading).^[3] Beyond engineering applications, shear rate influences biological systems, such as blood flow in vessels where it affects platelet adhesion and thrombosis risk,^[4] and in geophysics for modeling lava flows or drilling muds.^[5]^[1] Accurate measurement of shear rate using rheometers, often via rotational geometries like cone-and-plate, ensures reliable characterization of complex fluids, aiding in formulation optimization and process design.^[3]

Fundamentals

Definition

Shear rate is the velocity gradient within a deforming fluid or solid, quantifying the rapidity with which adjacent layers of the material slide relative to one another.^[6] In continuum mechanics, it functions as a fundamental kinematic quantity that describes the rate of shear deformation in materials subjected to tangential forces.^[7] This concept originated in the context of viscous flow studies during the 19th century, with key formalizations by Claude-Louis Navier in 1822 and George Gabriel Stokes in 1845, who incorporated velocity gradients into the equations governing fluid motion, thereby extending Isaac Newton's 1687 postulate on the proportionality of viscous resistance to relative motion in fluids.^[8]^[9] Shear rate differs from shear stress, which is the tangential force per unit area acting on a surface, and from shear strain, which measures the angular distortion or change in angle between originally perpendicular material lines.^[10]^[11] As a rate quantity, it captures the time-dependent aspect of deformation, with units typically expressed as inverse seconds (s⁻¹).^[12] A straightforward example is the configuration of fluid between two parallel plates, one fixed and the other translating at constant speed parallel to the first, where the resulting linear variation in velocity across the gap exemplifies pure shearing motion without rotational components.^[13]

Units and Dimensions

The SI unit of shear rate is the reciprocal second (s⁻¹), derived from the ratio of velocity, measured in meters per second (m/s), to a characteristic length scale, such as the distance between fluid layers in meters (m).^[14] This unit reflects the fundamental nature of shear rate as a velocity gradient, quantifying how rapidly adjacent fluid elements slide past one another. The dimensional formula for shear rate is [T⁻¹], where T denotes time, indicating that it is a kinematic quantity solely dependent on temporal scales and independent of length in homogeneous flows.^[14] This time-inverse dimension arises because shear rate normalizes velocity differences over spatial separations, canceling length dimensions and emphasizing its role as a rate of deformation per unit time. In non-SI systems, such as the centimeter-gram-second (cgs) framework, shear rate retains the unit of reciprocal seconds (s⁻¹), consistent with the SI derivation since velocity in cm/s divided by distance in cm yields the same temporal inverse. Shear rates span a wide range across flow regimes, providing scale for engineering applications. Low shear rates, typically below 1 s⁻¹, characterize creeping flows where viscous forces dominate and inertial effects are negligible, such as in slow sedimentation or microfluidic channels.^[15] In contrast, high shear rates exceeding 10⁶ s⁻¹ occur near the wall in turbulent boundary layers of high-Reynolds-number flows, like those in aerospace or high-speed aerodynamics, where sharp velocity gradients drive intense mixing and heat transfer. These extremes highlight shear rate's utility in dimensional analysis for predicting flow behavior without specific geometric details.

Shear Rate in Fluid Mechanics

Simple Shear Flow

Simple shear flow constitutes a fundamental type of unidirectional fluid motion in which the velocity field varies solely in the direction perpendicular to the primary flow direction, exhibiting no acceleration and lacking rigid-body rotational components.^[16] This configuration represents an idealized case where fluid layers slide parallel to one another, enabling the isolation of shear effects without complicating influences from curvature or pressure-driven variations.^[17] The shear rate in simple shear flow is quantified by the velocity gradient perpendicular to the flow, expressed as

\dot{\gamma} = \frac{du}{dy},

where

u

denotes the velocity component aligned with the flow direction and

y

is the coordinate transverse to it.^[2] This measure captures the rate at which adjacent fluid layers deform relative to each other.^[18] For an incompressible Newtonian fluid under low Reynolds number conditions, where inertial effects are negligible, the velocity profile arises from the continuity equation and the simplified momentum balance. The continuity equation for incompressible flow,

\nabla \cdot \mathbf{v} = 0

, is inherently satisfied by a unidirectional velocity field

u(y)

with no variations in the flow direction. The steady-state x-momentum equation, neglecting pressure gradients and body forces, reduces to

\frac{d}{dy} \left( \mu \frac{du}{dy} \right) = 0

, implying a constant shear rate since viscosity

\mu

is uniform. Integrating twice yields a linear velocity profile

u(y) = \dot{\gamma} y + c

, with constants determined by boundary conditions.^[17] A canonical example is planar Couette flow, occurring between two infinite parallel plates separated by a gap

h

, where the lower plate remains stationary and the upper plate translates at constant speed

V

. The resulting uniform shear rate is

\dot{\gamma} = \frac{V}{h}.

^[16] This setup produces a linear velocity profile

u(y) = \frac{V}{h} y

, directly linking the imposed motion to the deformation rate.^[18] In Newtonian fluids, the shear stress

\tau

relates linearly to the shear rate via

\tau = \mu \dot{\gamma}

, establishing the proportionality that defines viscous behavior in such flows.^[2]

General Shear Flows

In general shear flows, the shear rate is represented tensorially as the rate-of-strain tensor, which is the symmetric portion of the velocity gradient tensor. This tensor, denoted as

\dot{\epsilon}

, is defined as

\dot{\epsilon} = \frac{1}{2} \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^T \right)

, where

\mathbf{u}

is the velocity field. The off-diagonal components of

\dot{\epsilon}

capture the shear deformation, while the diagonal elements represent extensional rates. This formulation generalizes the concept beyond unidirectional flows, accounting for the full deformation in multi-dimensional scenarios.^[19] In three-dimensional flows, the shear rate tensor can be decomposed into its magnitude and principal directions to analyze local deformation characteristics. The magnitude of the shear rate is typically computed as

\dot{\gamma} = \sqrt{2 \dot{\epsilon} : \dot{\epsilon}}

γ˙=2ϵ˙:ϵ˙

, providing a scalar measure of the overall shearing intensity. Principal directions are obtained by diagonalizing $\dot{\epsilon}$ , revealing the orientations of maximum and minimum strain rates. For isotropic analysis, the invariants of $\dot{\epsilon}$ —such as the first invariant (trace, related to dilatation) and the second invariant (related to shear intensity)—are used to characterize flow behavior without dependence on coordinate systems, aiding in the identification of regions dominated by shear versus extension.^[20]^[21] Representative examples illustrate the spatial variation of shear rate in general flows. In Poiseuille flow through a circular pipe, the shear rate varies radially, with the local value given by $\dot{\gamma}(r) = \frac{4 Q r}{\pi R^4}$ , where $Q$ is the volumetric flow rate and $R$ is the pipe radius; at the wall ( $r = R$ ), it reaches $\dot{\gamma}_w = \frac{4 Q}{\pi R^3}$ , decreasing linearly toward the centerline. In boundary layer flows over a flat plate, the shear rate varies streamwise, with the wall shear rate $\dot{\gamma}_w$ decreasing as $\dot{\gamma}_w \propto x^{-1/2}$ for laminar conditions, reflecting the thickening of the boundary layer downstream. These variations highlight how geometry and flow constraints lead to non-uniform shearing.^[22]^[23] Inhomogeneities in shear rate, such as gradients across the flow field, can induce secondary flows or instabilities. For instance, radial variations in shear rate within curved channels or non-uniform velocity profiles may generate centrifugal forces that drive cross-stream circulations, as seen in Dean vortices. These effects arise from the coupling between the velocity gradient and flow geometry, promoting transverse motions without invoking turbulent mechanisms.^[24] In computational fluid dynamics (CFD) simulations, local shear rates are calculated by numerically evaluating the velocity gradient tensor from discretized velocity fields, enabling detailed mapping in complex geometries. Methods like finite volume or finite element approaches solve the Navier-Stokes equations to obtain $\nabla \mathbf{u}$ , from which $\dot{\epsilon}$ is derived pointwise; this is essential for predicting shear-induced phenomena in engineering applications such as mixers or biofluid transport.^[25]^[26]

Rheological Implications

Newtonian Fluids

In Newtonian fluids, the relationship between shear stress

\tau

and shear rate

\dot{\gamma}

is linear, as described by the constitutive equation

\tau = \mu \dot{\gamma}

, where

\mu

is the dynamic viscosity, a material constant independent of the shear rate. This equation implies that the fluid's resistance to flow remains proportional to the rate of deformation, with no dependence on the magnitude or history of the shear. The linearity holds under constant temperature and pressure, making Newtonian fluids predictable for many engineering calculations.^[27] The physical basis for this behavior stems from molecular momentum transfer, particularly in dilute gases and simple liquids. According to kinetic theory, viscosity arises from the diffusion of momentum across velocity gradients, where faster-moving molecules from a higher-velocity layer collide with slower ones in an adjacent layer, transferring momentum downward. An approximate expression from this theory is

\mu \approx \rho \lambda v

, where

\rho

is the fluid density,

\lambda

is the mean free path of molecules, and

v

is the average molecular thermal speed; this underscores how viscosity in Newtonian fluids results from random molecular motions rather than structured interactions.^[28]^[29] This linear stress-rate relation was first hypothesized by Isaac Newton in his 1687 work Philosophiæ Naturalis Principia Mathematica, where he proposed that the frictional resistance in fluids is proportional to the velocity gradient between fluid layers. Common examples include water and air at room temperature, which exhibit constant viscosity over wide shear rate ranges, such as from

10^{-3}

10^{3}

^{-1}

, encompassing typical flows in pipes, atmospheric conditions, and low-speed aerodynamics.^[30]^[27]

Non-Newtonian Fluids

Non-Newtonian fluids are characterized by an apparent viscosity

\eta

that varies as a function of the shear rate

\dot{\gamma}

, in contrast to the constant viscosity of Newtonian fluids. This variation leads to distinct flow behaviors, including shear-thinning (pseudoplastic), where

\eta

decreases with increasing

\dot{\gamma}

, and shear-thickening (dilatant), where

\eta

increases with

\dot{\gamma}

. Shear-thinning is more common and occurs in fluids where structural elements, such as particles or polymers, align or deform under shear, reducing resistance to flow. Shear-thickening arises from mechanisms like particle jamming or hydrodynamic clustering at high rates.^[31]^[32] A foundational model for describing these behaviors is the power-law (Ostwald-de Waele) model, which relates shear stress

\tau

to shear rate via

\tau = K \dot{\gamma}^n

, where

K

is the consistency index and

n

is the flow behavior index. For shear-thinning fluids,

n < 1

, resulting in an apparent viscosity

\eta = K \dot{\gamma}^{n-1}

that diminishes with

\dot{\gamma}

; for shear-thickening,

n > 1

, causing

\eta

to rise. This empirical model, originally proposed for colloidal suspensions, effectively captures the non-linear stress-strain rate relationship over a range of shear rates in many polymeric and particulate systems, though it may fail at very low or high

\dot{\gamma}

where Newtonian plateaus emerge.^[33]^[34] Other models extend this framework to account for yield stresses, below which the fluid behaves as a solid. The Bingham plastic model incorporates a yield stress

\tau_0

followed by linear flow: for

\tau > \tau_0

\tau = \tau_0 + \mu_p \dot{\gamma}

, where

\mu_p

is the plastic viscosity; below

\tau_0

\dot{\gamma} = 0

. This describes materials like drilling muds that resist flow until a critical stress is exceeded. The Herschel-Bulkley model generalizes both power-law and Bingham behaviors:

\tau = \tau_0 + K \dot{\gamma}^n

for

\tau > \tau_0

, combining yield stress with non-linear flow, and is widely applied to complex fluids exhibiting both yielding and shear-dependent viscosity.^[35]^[36] Representative examples illustrate these behaviors in practical contexts. Blood exhibits shear-thinning primarily due to the deformation and disaggregation of red blood cells at increasing

\dot{\gamma}

, facilitating efficient circulation from low-flow veins to high-shear arteries. Polymer melts, such as molten polystyrene, and paints also display shear-thinning, with viscosity dropping under processing conditions to enable molding or application. Ketchup, a yield-stress fluid, remains static at low

\dot{\gamma}

but flows readily above a critical shear rate, approximately 100 s

^{-1}

, exemplifying pseudoplasticity in food products. These models highlight implications for flow design, such as optimizing pumping efficiency. Additionally, some non-Newtonian fluids show time-dependent effects like thixotropy, a reversible decrease in

\eta

under sustained shear due to structural breakdown, observed in paints and inks.^[37]^[38]^[39]

Measurement and Applications

Experimental Methods

Rotational viscometers are widely used to measure shear rate in controlled laboratory settings by applying a rotational motion to a fluid sample and recording the resulting torque. In the Couette geometry, consisting of two concentric cylinders with the inner one typically rotating at angular speed Ω, the shear rate γ̇ is approximated for narrow gaps as γ̇ = 2Ω R / (R - r), where R and r are the outer and inner radii, respectively; this setup provides a nearly uniform shear field suitable for low to moderate shear rates up to approximately 10³ s⁻¹.^[40] The cone-plate geometry, featuring a shallow cone rotating above a flat plate, ensures a constant shear rate across the gap, calculated as γ̇ = Ω / θ, where θ is the cone angle in radians, making it ideal for precise measurements of viscosity and normal stresses in small sample volumes.^[41] Capillary rheometers extend measurements to high shear rates, often exceeding 10⁵ s⁻¹, by forcing fluid through a narrow tube and using the pressure drop ΔP to compute wall shear stress τ_w = ΔP R / (2L), where R is the capillary radius and L its length; the apparent shear rate is then derived from the volumetric flow rate Q as γ̇_app = 4Q / (π R³).^[40] For non-Newtonian fluids, the Weissenberg-Rabinowitsch correction adjusts the apparent shear rate to the true wall shear rate: γ̇_w = (1/4) (4Q / π R³) [3 + d(ln Q)/d(ln τ_w)], accounting for the non-parabolic velocity profile due to shear-thinning or shear-thickening behaviors.^[42] This method is particularly valuable in industrial simulations of extrusion processes but requires additional Bagley correction for entrance effects at the capillary inlet.^[43] Other techniques include the falling sphere method, which applies Stokes' law to determine viscosity at low shear rates (typically <1 s⁻¹) by measuring the terminal velocity v_t of a sphere of radius a and density ρ_s falling through a fluid of density ρ_f: η = (2/9) (ρ_s - ρ_f) g a² / v_t, where the local shear rate around the sphere varies but remains minimal overall.^[44] Parallel plate squeeze flow, involving the compression of a sample between two approaching plates, generates shear rates from 10⁻³ to 10² s⁻¹ and is useful for soft solids or yield stress fluids, with shear rate estimated from the plate separation rate and radial flow assumptions.^[45] Calibration of these instruments presents challenges, including ensuring uniform shear distribution to avoid artifacts from non-ideal geometries. Edge effects in rotational setups, such as surface tension or fracture at the sample-air interface, can distort torque readings by up to 2% for Newtonian fluids, necessitating corrections like truncation or moated geometries.^[46] At high shear rates (>10⁴ s⁻¹), inertial forces introduce instabilities, such as Taylor vortices in Couette flow or secondary flows in capillaries, requiring damping or wider gaps to maintain laminar conditions.^[47] Modern rheometers incorporate optical techniques like particle image velocimetry (PIV) to map local shear rates non-invasively by tracking seeded particle displacements in the flow field, enabling validation of global measurements and detection of heterogeneities in complex fluids such as suspensions or shear-banding materials.^[48] PIV, often integrated with confocal microscopy, provides velocity gradients with resolutions down to 10⁻² s⁻¹, complementing traditional torque-based methods for spatially resolved rheology.^[49]

Practical Applications

In chemical engineering, shear rate plays a critical role in the pumping and mixing of non-Newtonian fluids such as slurries, where maintaining optimal shear rates of 10–100 s⁻¹ in pipelines helps prevent particle jamming and ensures efficient flow by reducing viscosity through shear-thinning effects.^[50]^[51] This range is particularly relevant for industrial slurries like those in oilfield operations, where insufficient shear can lead to settling and blockages, while excessive rates may increase energy costs.^[50] In biomedical applications, shear rates in arterial blood flow typically range from 100 to 1000 s⁻¹, influencing thrombosis risk by modulating platelet activation and von Willebrand factor unfolding, which promotes clot formation under pathological low-shear conditions near vessel walls.^[52]^[53] Similarly, in drug delivery systems, controlled shear rates are engineered in microfluidic devices to mimic vascular environments, enhancing targeted release while minimizing endothelial damage and improving therapeutic efficacy.^[54] During materials processing, such as polymer extrusion, high shear rates of 10³–10⁵ s⁻¹ induce molecular alignment and chain orientation, enhancing flow properties and mechanical strength in the final product by reducing viscosity and promoting anisotropic structures.^[55]^[56] This is essential in processes like injection molding and fiber spinning, where shear alignment contributes to improved tensile properties without excessive degradation.^[57] In the food industry, shear rate is leveraged for texture modification in sauces and dressings, where shear-thinning behavior—viscosity decreasing at higher rates—balances pourability during dispensing with a stable, non-dripping consistency at rest, achieved through ingredients like hydrocolloids.^[58]^[59] This rheological tuning ensures sensory appeal and prevents separation, as seen in products like ketchup, where low-shear viscosity maintains body while high-shear facilitates flow.^[60] Environmentally, in river sedimentation processes, low shear rates promote particle settling by allowing suspended sediments to aggregate and deposit, influencing sediment transport and bed formation in low-velocity flows typical of meandering channels.^[61] At these subdued rates, below 1–10 s⁻¹, flocculation enhances settling velocities, affecting water quality and habitat stability in aquatic systems.^[62] As of 2025, recent developments in 3D bioprinting highlight the role of tuned shear rates in bioinks, where moderate shear rates, typically 200–400 s⁻¹, during extrusion can preserve cell viability above 90% by minimizing mechanical stress on encapsulated cells, enabling viable tissue constructs for regenerative medicine.^[63]^[64] Shear-thinning formulations in these inks further support precise deposition while protecting cellular integrity throughout printing and post-processing.^[65]

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