VISCOSITY_MODEL_Bingham

Question

Dear Altair Support Team, I hope this email finds you well. We are currently using the bingham type within the VISCOSITY_MODEL command in AcuSolve to simulate a non-Newtonian fluid and have some questions regarding its parameterization. Upon reviewing the model's governing equation in the AcuSolve documentation, we understand that it is a generalized model incorporating several physical effects, which differs from the conventional Herschel-Bulkley-Papanastasiou (HBP) model we are accustomed to. Specifically, we are uncertain about the best practice for determining the following three parameters from our experimental rheological data: * bingham_viscosity (μ₀) * bingham_time_constant (η) * bingham_infinite_shear_viscosity (μ∞) Our attempts to calibrate these parameters using our rheology data have led to simulation results that deviate significantly from those reported in our reference literature. To resolve this discrepancy, we would be very grateful if you could provide some clarification on the following: * What is the theoretical source or reference literature for this generalized Bingham model as implemented in AcuSolve? * Has this model been officially validated against any benchmark cases? If so, would it be possible to access the corresponding validation reports or examples for our reference? A precise understanding and correct implementation of these parameters are critical to the success of our research. Thank you for your time and assistance. We look forward to your guidance. Best regards, Yu Yin

yinyu · Answer

Dear AcuSolve Support Team,

Thank you for your follow-up.

Yes, we are fully aware that the standard Herschel-Bulkley (HB) model is discontinuous; as the shear rate approaches zero, the viscosity tends toward infinity, which inevitably causes numerical divergence. However, we noted in the AcuSolve documentation that the viscosity equation has been regularized to ensure continuity at zero shear rate, effectively transforming it into the Herschel-Bulkley-Papanastasiou (HBP) model:
μ = k(γ̇)^n−1 + τ_y / (γ̇(1 − e^(−Mγ̇)))

Furthermore, we observed that AcuSolve implements a further modification to the standard HBP model by imposing an upper viscosity limit at zero shear rate and a lower limit at infinite shear rate:
μ = μ_∞ + (μ_0 − μ_∞)(ηγ̇)^n−1 + τ_y / (γ̇(1 − e^(−Mγ̇)))

Additionally, an extra time constant term is introduced. This formulation differs from the HBP models commonly found in published literature. We are uncertain whether this time constant carries a specific physical meaning or what its recommended value range should be. We would greatly appreciate it if you could share the theoretical background or reference materials for this specific modified HBP implementation in AcuSolve.

Given this regularization approach, we believe the modified HBP model should run and converge robustly within AcuSolve.

Moreover, in contemporary debris flow research, the HBP model is predominantly used to capture non-Newtonian rheology. The use of the Carreau model for debris flows is rarely reported in peer-reviewed literature. For this reason, we strongly prefer to maintain the HBP model for our simulations.

However, based on our available rheological data, we are unsure how to correctly assign values to these parameters, particularly the time constant. We are currently extracting parameters by fitting AcuSolve’s modified HBP equation to our shear stress vs. shear rate data. Should we instead be fitting against shear viscosity vs. shear rate data? We would greatly appreciate your guidance on the recommended fitting workflow.

Assuming shear stress vs. shear rate fitting is correct, our curve-fitting equation is:
y = 16.0937*x^0.149 + 5.031*(1 − exp(−190.8756*x))
(where y = shear stress, x = shear rate)

Based on this fit, we assigned the following parameters in AcuSolve:

* Viscosity (μ0) = 16.0937
* Infinite_shear_viscosity（μ∞）=0
* Yield Stress (τ_y) = 5.031
* Regularization/Stress Growth Parameter (M) = 190.876
* Time Constant = 1
* Power Law/Flow Index (n) = 0.149

Using these values results in a significant discrepancy compared to the reference study. Specifically, at identical simulation times, the debris flow runout distance is markedly shorter, indicating a much slower migration speed (see figure below):

Interestingly, when we artificially scale down the reference viscosity (consistency index) by a factor of 1,000, our results align closely with the published study (see figure below):

Could this discrepancy stem from our parameter extraction methodology, or might there be a unit conversion issue (e.g., Pa·s vs. mPa·s or kPa) during the fitting process? We would be grateful for your insight on this.

For your reference, the literature we are benchmarking against is listed below:

Investigating the effects of claysand content.pdf

Best regards,
Yu Yin

acupro · Answer

I'm pasting here a comment/discussion from a well-versed user several years ago.  He recommends using the Carreau model rather than the Herschel-Bulkley model (which isn't actually supported in AcuSolve).  From what I've found, the software that comes with the rheometer should be able to generate the coefficients for the Carreau model - consistent with what AcuSolve requires.  I've attached another 'instance' of your data - with viscosity vs shear-rate in Log-Log scale.  While there may be some issues with the lower shear-rate values (same shear-rate with multiple viscosity values) - maybe zero_shear_viscosity is something like 700 and the infinite_shear_viscosity something like 0.6.  The other parameters for the Carreau model (time_constant, index or power-law index, and transition_index or transition parameter) probably come from the rheometer software.

Rheometer_Data_Mod.xlsx

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Herschel Bulkley is probably not a good viscosity model for most solvers due to it not being differentially continuous.  In the end, you need to express the rheological model in terms of apparent viscosity versus shear rate.  Knowing the shear stress is insufficient, you must also define a critical shear rate (gamma 0) for HB.  Ultimately, you also need to put limits on the viscosity at zero and infinite shear rate (many forget to do this).  Most of the models will be Z-shaped as shown below.  The HB model is really a power law model with upper and lower viscosity limits.

The Carreau model is continuous over the entire shear rate range. HB is not. I discovered convergence problems with HB early on.  A guy said they had been struggling with power law models for years in Fluent and had much better results with Carreau in Fluent.

My advice is to convert the HB to Carreau if you want stability.  The HB model is from a time when rheometers had a shear rate range from 1-10 s^-1.  You would use the 2 points to estimate a K and n value and another device to estimate yield stress.  Now, rheometers can easily go from 0.01-1000 s^-1 so the need for HB is low since 0.01 s^-1 is pretty stagnant.  The real difficulty comes in trying to estimate the infinite shear rate asymptote.  You might need to get to 1,000,000 s^-1 in a capillary rheometer to observe it.  Fortunately, there are physical limits that say a fluid will stop thinning above the base viscosity (e.g., a water based formula will not thin below 4 cP).

Rheometer_Data_Mod.xlsx

yinyu · Answer

Dear Acupro,

Thank you for your follow-up. Certainly, please find our raw experimental data in the attached file.

The file contains two sets of data:

* The original raw data presented as Shear Stress vs. Shear Rate.
* A calculated table of Apparent Viscosity vs. Shear Rate, which we derived by dividing the shear stress by the corresponding shear rate.

To provide some context for our simulation, we are studying submarine debris flows (a type of gravity flow). Our literature review indicates that the Herschel-Bulkley (HB) model is widely used by researchers to characterize the non-Newtonian behavior of these flows. For this reason, we are strongly inclined to use the HB model for our simulation.

Given this, we would be very grateful for your expert guidance. Specifically, we hope you could help us with:

* Confirming if the HB model is indeed the most appropriate choice for our application within AcuSolve.
* Assisting us in selecting the correct model parameters based on the data provided.

We appreciate your time and expertise and look forward to your reply.

Thank you again for your help.

Best regards,

Yu Yin

the actual data.xlsx