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Herschel Bulkley Model in Openfoam

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Old   July 28, 2021, 04:26
Default Herschel Bulkley Model in Openfoam
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mohammad
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Hi everybody,
I am trying to simulate the particle settling in a Non-Newtonian fluid, using Herschel-Bulkley model. As I know the HB model has been implemented in OF using the following equation for viscosity:
Code:
return
    (
        min
        (
            nu0_,
            (tau0_ + k_*rtone*pow(tone*sr(), n_))
           /(max(sr(), dimensionedScalar ("VSMALL", dimless/dimTime, VSMALL)))
        )
    );
Which is simply equivalent to the following equation:


\eta=min\left(\eta_0,\frac{\tau_0+k\dot{\gamma}^n}{\dot{\gamma}}\right)


where \eta is the final calculated viscosity, \dot{\gamma} as strain rate, n power law index, k consistency index, and \tau_0 as yield stress in viscoplastic fluids. Here my question is regarding \eta_0 which can only be found in OF and describes the largest possible viscosity in the domain for plastic phase.

Considering a settling particle, which starts to settle from zero velocity, I think it's an important factor, because my calculated viscosity based on the shear rate is always comparing with this parameter. So what you think about this parameter? Should it be kept as the highest possible value? If yes, for a settling particle, it fails to exceed the \eta_0 in initial iterations! So what is the highest value?!!!
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Old   December 20, 2021, 17:01
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Thanks for asking this question Mohammad.

I also came across this same question and spent few hours looking for it . Hope the following info might help someone .

The equations shown by you refer to the regularization approach (one of the ways to tackle the discontinuous behavior of non -Newtonian fluids).

The parameter \eta_{0} refers to the Creeping viscosity or Yielding viscosity, and it represents the viscosity when the shear stress is smaller than yield stress. Hence, it should be as high as possible to accurately model the Non-Newtonian behavior. Although, there is no standard rule to set this, I came across few papers that provided some rough approximations

1. Here ( https://www.researchgate.net/publica...zing_OpenfoamR) authors advised to take it somewhere from 10k -100k Pa.S (dont forget to convert it to kinematic viscosity)

2. In another paper (Eq -4 of Numerical study of the Bingham squeeze film problem https://www.sciencedirect.com/scienc...77025784800294), it is suggested to take its value 1000 times of dynamic viscosity. (again make sure to divide it by density b4 giving as input in FOAM)

P.S. - Curious fellows can also refer to this awesome piece of work for a nice overview (Progress in numerical simulation of yield stress fluid flows https://link.springer.com/article/10...397-016-0985-9)
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Old   December 20, 2021, 19:30
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Quote:
Originally Posted by SHUBHAM9595 View Post
Thanks for asking this question Mohammad.

I also came across this same question and spent few hours looking for it . Hope the following info might help someone .

The equations shown by you refer to the regularization approach (one of the ways to tackle the discontinuous behavior of non -Newtonian fluids).

The parameter \eta_{0} refers to the Creeping viscosity or Yielding viscosity, and it represents the viscosity when the shear stress is smaller than yield stress. Hence, it should be as high as possible to accurately model the Non-Newtonian behavior. Although, there is no standard rule to set this, I came across few papers that provided some rough approximations

1. Here ( https://www.researchgate.net/publica...zing_OpenfoamR) authors advised to take it somewhere from 10k -100k Pa.S (dont forget to convert it to kinematic viscosity)

2. In another paper (Eq -4 of Numerical study of the Bingham squeeze film problem https://www.sciencedirect.com/scienc...77025784800294), it is suggested to take its value 1000 times of dynamic viscosity. (again make sure to divide it by density b4 giving as input in FOAM)

P.S. - Curious fellows can also refer to this awesome piece of work for a nice overview (Progress in numerical simulation of yield stress fluid flows https://link.springer.com/article/10...397-016-0985-9)

Thanks SHUBHAM for your comprehensive explanation.

BTW, I have seen in OF, whenever we consider a higher \eta_0 in the simulation, the time step should be lower, as the viscosity in the unyielded will be so high, and there will be a problem in terms of stability in this kind of simulation. So we need to have an optimum viscosity value for the simulation. The papers you mentioned are so valuable. However, I think \eta_0 will be a case sensitive parameter, that can't be set to a very high value, as we see in the literature.
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Old   October 1, 2022, 11:28
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Quote:
Originally Posted by mostanad View Post
Hi everybody,
I am trying to simulate the particle settling in a Non-Newtonian fluid, using Herschel-Bulkley model. As I know the HB model has been implemented in OF using the following equation for viscosity:
Code:
return
    (
        min
        (
            nu0_,
            (tau0_ + k_*rtone*pow(tone*sr(), n_))
           /(max(sr(), dimensionedScalar ("VSMALL", dimless/dimTime, VSMALL)))
        )
    );
Which is simply equivalent to the following equation:


\eta=min\left(\eta_0,\frac{\tau_0+k\dot{\gamma}^n}{\dot{\gamma}}\right)


where \eta is the final calculated viscosity, \dot{\gamma} as strain rate, n power law index, k consistency index, and \tau_0 as yield stress in viscoplastic fluids. Here my question is regarding \eta_0 which can only be found in OF and describes the largest possible viscosity in the domain for plastic phase.

Considering a settling particle, which starts to settle from zero velocity, I think it's an important factor, because my calculated viscosity based on the shear rate is always comparing with this parameter. So what you think about this parameter? Should it be kept as the highest possible value? If yes, for a settling particle, it fails to exceed the \eta_0 in initial iterations! So what is the highest value?!!!
Hello,I have a question here, in FOAM code the HB model is about kinematic viscosity, but in ur equation it is about dynamic viscosity,and i searched about HB model and it seems yours is correct.
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