Saramito-Herschel-Bulkley Model

• The Saramito-Herschel-Bulkley model is a constitutive framework for elastoviscoplastic materials that integrates elastic deformation below yield with viscoplastic flow above yield.
• It captures unique phenomena such as residual stresses, negative wakes, and transition zones, providing a smooth continuous response absent in classic Herschel-Bulkley formulations.
• Key parameters like shear modulus, consistency index, and yield stress govern the model's ability to simulate complex multiphase and porous media flows.

Searching arXiv for recent and foundational papers on the Saramito-Herschel-Bulkley model and closely related EVP flow studies. The Saramito-Herschel-Bulkley (SHB) model is a constitutive framework for elastoviscoplastic (EVP) materials that extends the Herschel-Bulkley law by incorporating elasticity. In the formulation used across recent benchmark, porous-media, and multiphase studies, the model is designed to represent solid-like response below yield, viscoplastic flow above yield, and shear-thinning through the power-law index. It is therefore used for materials such as Carbopol gels and for flows in which purely viscoplastic models fail to capture elastic storage, negative wakes, or history-dependent stress fields (Esposito et al., 2024, Mousavi et al., 2024).

1. Constitutive structure

In SHB formulations used in finite-volume and DNS studies, the total extra stress is commonly decomposed into a solvent part and an EVP part,

$$\mathbf{T}'=\mathbf{T}_s+\mathbf{T}, \qquad \mathbf{T}_s=\eta_s\left(\nabla\mathbf{u}+\nabla\mathbf{u}^T\right),$$

while the EVP stress obeys an upper-convected evolution law with a yield-activated dissipative term. One form used in planar contraction studies is

$$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$

where $G$ is the shear elastic modulus, $k$ the consistency index, $n$ the Herschel-Bulkley index, $\tau_0$ the yield stress, $$\mathbf{D}=0.5(\nabla\mathbf{u}+\nabla\mathbf{u}^T)$$, and $$|\mathbf{T}_d|=\sqrt{0.5\,\mathbf{T}_d:\mathbf{T}_d}$$ is the von Mises-type equivalent stress of the deviatoric part (Mousavi et al., 2024).

Equivalent notational variants appear in lid-driven cavity, drop, and wake simulations. A representative form is

$$\frac{1}{G}\stackrel{\nabla}{\boldsymbol{\tau}}+ \left(\frac{\max(0,\tau_d-\tau_y)}{k}\right)^{1/n} \frac{1}{\tau_d}\boldsymbol{\tau} = \dot{\boldsymbol{\gamma}},$$

with the upper-convected derivative

$$\stackrel{\nabla}{\boldsymbol{\tau}}= \frac{\partial \boldsymbol{\tau}}{\partial t} +\mathbf{u}\cdot\nabla\boldsymbol{\tau} -(\nabla\mathbf{u})^T\cdot\boldsymbol{\tau} -\boldsymbol{\tau}\cdot\nabla\mathbf{u}.$$

In this representation, the model can be interpreted mechanically as a spring in series with a viscoplastic Herschel-Bulkley branch, so that total deformation rate is split into elastic and viscoplastic contributions (Syrakos et al., 2018).

The key rheological parameters have the roles stated explicitly in the numerical studies: $$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$0 sets the stiffness of the solid phase below yield, $$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$1 sets viscosity magnitude in the yielded regime, $$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$2 controls shear-thinning or shear-thickening, and $$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$3 sets the threshold for yielding. The yield criterion is written in terms of the second invariant of the deviatoric stress, and flow proceeds when the local stress magnitude exceeds the yield stress (Esposito et al., 2024).

2. Regimes, limiting cases, and physical interpretation

The model is built to distinguish unyielded and yielded material states. Below yield, studies describe the material as a viscoelastic solid or, in the buoyant-drop formulation, as a Kelvin-Voigt solid. At and above yield, it behaves as a viscoelastic liquid with Herschel-Bulkley-type dissipation; in porous-media simulations this yielded state is described as a viscoelastic Oldroyd-B fluid for stresses higher than the yield stress (Vita et al., 2018, Esposito et al., 2024).

A recurring motivation for SHB is that classical Herschel-Bulkley or Bingham models do not represent elastic deformation below yield and do not provide a smooth solid-fluid transition. SHB is used precisely to supply an elastic regime below yield, a continuous transition between solid and fluid response, viscoelasticity after yield, and an objective stress evolution through the upper-convected derivative. One numerical study further states that, unlike Bingham or Herschel-Bulkley, stress and strain histories are uniquely determined everywhere, even if material points are unyielded (Esposito et al., 2024).

The principal limiting cases reported across the literature are summarized below.

Limit	Result	Source
$$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$4	Oldroyd-B model	(Vita et al., 2018)
$$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$5	Bingham plastic	(Vita et al., 2018)
$$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$6	Herschel-Bulkley recovered	(Mousavi et al., 11 Jan 2025)
Small $$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$7	Herschel-Bulkley viscoplasticity recovered	(Corrochano et al., 5 Aug 2025)

These reductions explain why SHB is often presented as a unifying EVP model: depending on parameter choice, it interpolates between viscoelastic, viscoplastic, and Newtonian limits. A plausible implication is that SHB is most valuable when both yield stress and elastic memory are dynamically relevant; where one of those mechanisms is negligible, simpler constitutive laws may suffice.

3. Distinctive mathematical behavior

A central distinction from classical Herschel-Bulkley flow is the status of the unyielded region. In the lid-driven cavity study, the Herschel-Bulkley model yields perfectly rigid unyielded zones with indeterminate stress, whereas SHB yields an elastic solid with unique stress that depends on the initial condition. The same study reports that, for identical driving, SHB admits multiple steady states distinguished by different trapped residual stress fields in unyielded material (Syrakos et al., 2018).

Another model-specific feature is the appearance of transition zones. In cavity flow, between stationary unyielded and yielded regions there appears a region in which the strain rate is vanishingly small while the stress approaches the yield stress extremely slowly; the study notes that the region may transition only infinitely slowly to an unyielded state. This is absent from the corresponding classical Herschel-Bulkley solution and is emphasized again in confined-cylinder EVP flow, where an elongated yielded area can be sandwiched between two unyielded areas downstream of the cylinder (Syrakos et al., 2018, Mousavi et al., 11 Jan 2025).

The cessation problem also separates SHB from classical viscoplasticity. After the lid is halted in cavity flow, the Herschel-Bulkley model predicts cessation in finite time, whereas SHB does not reach complete cessation in finite time. Instead, elastic energy and kinetic energy exchange through damped oscillations, and yielded pockets can persist. Wall slip changes that picture by maintaining non-zero strain rate and accelerating dissipation of residual oscillations (Syrakos et al., 2018).

These features matter computationally. Several studies state that transient simulation is required even when the goal is nominally steady flow, because elastic memory, slow stress relaxation, and evolving yield surfaces are intrinsic parts of the model dynamics (Mousavi et al., 2024, Syrakos et al., 2018).

4. Canonical benchmark flows and instability structure

SHB has become a benchmark constitutive law in canonical geometries. In the lid-driven cavity, a finite-volume method on cell-centred structured and unstructured grids was developed specifically for SHB flows, with momentum-interpolation pressure stabilization, “both sides diffusion” stabilization for velocity, the CUBISTA convection scheme for stress, and second-order temporal discretization with adaptive time step. The resulting solutions showed plug regions and stationary unyielded zones similar in broad topology to Herschel-Bulkley flow, but with elastic asymmetry, residual stress, and transition regions unique to SHB (Syrakos et al., 2018).

In the 4 to 1 planar contraction, the model predicts unyielded material in the concave corners and along the mid-plane of both channels, while the contraction region itself remains yielded because all stress components are larger there. When the Bingham or Weissenberg numbers are below critical values, a steady state is reached; increasing either quantity expands the unyielded regions and shifts them in the flow direction. Above critical values, the transient no longer approaches steady state monotonically: damped or sustained oscillations, complex stress fields without a plane of symmetry, and purely elastic instability near the contraction entrance are reported under creeping conditions (Mousavi et al., 2024).

In flow around a confined cylinder, SHB predicts several classes of yielded and unyielded structures: unyielded regions around the plane of symmetry ahead of and behind the cylinder, small islands above and below the cylinder, and polar caps at stagnation points. A distinctive EVP prediction is an elongated yielded region downstream of the cylinder, sandwiched between two unyielded areas. For Carbopol 0.1% with blockage ratio $$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$8, this appears above a critical elastic modulus $$\overset{\triangledown}{\mathbf{T}}+ \max\left(0,\frac{|\mathbf{T}_d|-\tau_0}{k}\right)^{1/n}\mathbf{T} = 2G\mathbf{D},$$9 Pa. The drag coefficient increases with yield stress and blockage ratio but decreases with material elasticity; unyielded regions expand with increasing yield stress and also with increasing elasticity because the material can deform more before yielding (Mousavi et al., 11 Jan 2025).

In the two-dimensional wake of a circular cylinder at $G$0, SHB DNS combined with POD and HODMD identifies three regimes: periodic, transitional, and fully complex. Stronger plastic effects, especially with $G$1, increase flow complexity; increasing elasticity elongates the recirculation bubble and shifts it downstream; decreasing the solvent viscosity ratio enhances EVP effects. The study explicitly frames these dynamics as the interaction of inertia, elasticity, and yield stress in non-Newtonian wake flow (Corrochano et al., 5 Aug 2025).

5. Drops, bubbles, entrapment, and negative wakes

Multiphase applications have been among the clearest demonstrations of why SHB is used instead of purely viscoplastic models. For the buoyancy-driven motion of a Newtonian drop in an EVP material, numerical solutions obtained with a VOF method in Basilisk show quantitative agreement with earlier experimental and numerical results in terms of terminal velocity and drop shape. The study reports that incorporating elastic effects into the continuous phase is essential for reproducing experimentally observed features such as the negative wake and the teardrop shape. Small drops can remain entrapped when buoyancy is insufficient to fluidize the surrounding material, and the entrapment conditions depend on the interplay between capillarity and elasto-plasticity (Esposito et al., 2024).

The same work emphasizes parameter effects that are archetypal for SHB flows. Increasing yield stress suppresses motion and promotes entrapment. Lower elastic modulus strengthens elastic response, increases deformation, and promotes more prolate or teardrop-like shapes. Lower interfacial tension increases deformation and terminal velocity. These outcomes are presented as consequences of the coupled roles of elasticity, plasticity, and capillarity (Esposito et al., 2024).

For coaxially rising bubbles in a 0.1% Carbopol solution modeled with SHB, three interaction patterns are reported: bubble approach, bubble separation, and establishment of a constant distance. The constant-distance state is attributed to the coupling of a negative wake behind the leading bubble with a slight modification of the stresses at its rear pole. The study further states that the magnitude of that equilibrium distance is mainly determined by the elastic response of the surrounding medium, and that the same equilibrium can be achieved for other bubble-size pairs when the initial distance exceeds a critical value (Kordalis et al., 2024).

Taken together, these results establish a common theme: in buoyancy-driven EVP multiphase flow, SHB is used because purely viscoplastic constitutive laws do not capture elastic recoil, negative wakes, or the stress-history effects that determine trapping, deformation, and body-body interaction.

6. Porous media, adjacent formulations, and scope of applicability

In porous media, SHB has been used to study EVP transport beyond generalized-Newtonian Herschel-Bulkley laws. Numerical simulations through a symmetric array of cylinders report time-dependent flow even at low Reynolds numbers, with oscillations in the unyielded region at high Bingham numbers. The unyielded volume strongly increases with Bingham number and reaches up to 70% of the total volume for the highest values considered. The apparent permeability, normalized by the Newtonian case, is greater than $G$2 at low Bingham number and smaller than $G$3 at high Bingham number, reflecting the competition between elastic facilitation and dissipation around large unyielded regions (Vita et al., 2018).

This porous-media use should be distinguished from recent pure Herschel-Bulkley transport models. A Darcy-type law for Herschel-Bulkley fluids in porous media and a pore-network model for Herschel-Bulkley flow in disordered porous media both explicitly exclude elasticity and focus on viscoplastic transport, yielding thresholds, channelisation, and wall slip within HB rheology alone. One of these studies states that, in steady non-elastic regimes, the yield limit and Herschel-Bulkley macroscopic laws are recovered in the limit of vanishing elasticity in the Saramito framework (Chaparian, 4 Jul 2026, Fonte et al., 10 Mar 2026).

The SHB model should also be separated from other Herschel-Bulkley extensions that are not elastoviscoplastic. In a mathematical existence study for incompressible flows with stress given by the subgradient of a convex potential, the authors explicitly distinguish their purely viscous-plastic family from the Saramito model, noting that the latter is an augmented Herschel-Bulkley model with viscoelastic regularization, whereas their framework has no explicit viscoelastic stress (Chemetov et al., 2024). Likewise, the compressible shear-density-coupling model for a single-component yield-stress fluid is presented as distinct from classic SHB because it includes compressibility and density-dependent rheology but does not include an explicit elastic memory term (Gross et al., 2018).

A further practical boundary concerns when SHB is actually needed. For nonthixotropic dense suspensions of soft particles studied in wide-gap geometries under steady conditions, MRI-based work concludes that a local monotonic Herschel-Bulkley law can account for the flow without steady-state shear banding, and explicitly suggests that more elaborate models become necessary when nonlocal effects, large normal stresses, or elastic response are important (Ovarlez et al., 2012). In combined shear and elongational experiments on soft glassy materials, the classic Herschel-Bulkley model qualitatively explained the observed coupling of orthogonal deformations, and the SHB model was not explicitly considered (Shaukat et al., 2012).

Within that broader landscape, the SHB model occupies a specific niche: it is the standard constitutive choice when the flow must simultaneously resolve yield, shear-dependent viscosity, and elastic stress evolution. Its characteristic signatures across the literature are unyielded elastic regions with unique stress histories, slow or oscillatory stress relaxation, negative wakes, elasticity-modified drag and entrainment, and transitions from steady to oscillatory or complex flow even under low-Reynolds-number conditions.