Low viscosity ratios in Oldroyd-B model, log-conform approach

[Anjishnu Choudhury]

Hello all,

I am writing to know if anyone has implemented the Elastic Viscous Stress Splitting (EVSS) technique in the log-conform approach [Viezel, Caroline, et al., Journal of Non-Newtonian Fluid Mechanics 285 (2020).]. This apparently helps one to obtain stable numerical solutions of viscoelastic free surface flows by alleviating the singularity in the interfacial stress conditions as \beta --> 0. 

I have tried using simple transformations of viscosities in the existing definitions. However, there is an upper-convective derivative of deformation tensor term which needs to be implemented in the formulations. Any tips on how to proceed ?

P.S: I am new to Basilisk and still not very confident in changing source files.

-Anjishnu

[Anjishnu Choudhury]

The above problem boils down to changing f_s(A) in log-conform approach, from f_s(A) = [A - I] to f_s(A) = [A - I - 2*lambda*D] where D is the deformation tensor. I have tried changing the part of the code (log-conform.h) to:

double fa = (mup[] != 0 ? lambda[]/(mup[]*eta) : 0.);

      pseudo_t D;
      D.x.y = lambda[]*(u.y[1,0] - u.y[-1,0] +
         u.x[0,1] - u.x[0,-1])/(2.*Delta);
      foreach_dimension()
        D.x.x = lambda[]*(u.x[1,0] - u.x[-1,0])/(Delta);

      pseudo_t A;
      A.x.y = (fa*tau_p.x.y[] + D.x.y)/nu;
      foreach_dimension()
      A.x.x = (fa*tau_p.x.x[] + 1. + D.x.x)/nu;
   
      /** Old code
      pseudo_t A;
      A.x.y = fa*tau_p.x.y[]/nu;
      foreach_dimension()
      A.x.x = (fa*tau_p.x.x[] + 1.)/nu;
       */

However, this gives floating point errors while compiling. 

Is there something trivial I'm missing ? 

Thanks a lot in advance !

[Stefan Bošković]

Hello Anjishnu,

I'm not sure whether this is important, but in the first row you used:
"mup[] != 0 ?"
and latter you used:
": 0.);"
So maybe you can change the first zero from I guess int to double:
"mup[] != 0. ?"

Best regards,
Stefan Bošković

>> [Viezel, Caroline, et al., _Journal of Non-Newtonian Fluid Mechanics
>> _285 (2020).]. This apparently helps one to obtain stable numerical

>> solutions of viscoelastic free surface flows by alleviating the
>> singularity in the interfacial stress conditions as \beta --> 0.
>>
>> I have tried using simple transformations of viscosities in the
>> existing definitions. However, there is an upper-convective
>> derivative of deformation tensor term which needs to be implemented
>> in the formulations. Any tips on how to proceed ?
>>
>> P.S: I am new to Basilisk and still not very confident in changing
>> source files.
>>
>> -Anjishnu
>

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[Anjishnu Choudhury]

Follow up:

With f_s(A)= A - I - 2*lam*D, the modified log-conform.h file seems to work for slowly starting BCs like u.t[bottom] = Dirichlet (U*sin(t)) or a Navier slip condition. However,  for suddenly started plate, u.t[bottom] = Dirichlet(U), it gives a floating point error.

My initial conditions are u.x[]=0. and u.y[]=0. Any ideas on how to fix it?

Thanks in advance!!

[Shyam Sunder Yadav]

Dear Anjishnu

To calculate Psi = log(A), we need to diagonalize A, for which A must be positive definite...

With f_s(A)= A - I - 2*lam*D,     the conformation tensor  A may be loosing its positive definiteness property...

Hence the divergence...

In Basilisk approach, the extra stress is decomposed as Tau = Tau_p + 2*mu_s*D where Tau_p is calculated by log-conform.h and the 2*mu_s*D part is handled by viscosity.h

In  EVSS, the extra stress is decomposed as follows:   Tau = (Tau_p - 2*mu_p*D)  + 2*mu_total*D, so viscosity one has to provide to the centered.h solver is the total viscosity, i.e, polymer plus the solvent viscosity.

So viscosity.h will have to handle a higher viscosity which may be problematic in spite of the fact that it is an implicit formulation...

Also, as I mentioned above, the (Tau_p - 2*mu_p*D) part may cause A to loose its positive definiteness, hence the divergence...

I may be wrong here, others please correct me!

What approach is used for solving the polymer stresses in the paper you mentioned?

Thanks, Regards

[Anjishnu Choudhury]

Dear Shyam,

Thanks a lot for your reply and opinion on this matter.

I have had this doubt of A losing its positive-definiteness property before. However, I thought if [\tau_p+I] and 2*lam*D are both positive definite, then the sum might be positive definite too. Also, the code works when I prescribe a slowly starting BC: u.t[bottom]=sin(omega*t) with a free surface at the top and periodic BCs for left-right. How is the positive-definiteness retained for such a condition then?

Regarding viscosity, the \mu_s can be interpreted as the total viscosity following the EVSS scheme. \tau_p is also interpreted as the 'purely elastic' tensor S. I tried using very low viscosities as the total viscosity as well, but it doesn't make a difference.

The polymer stresses in Viezel20 are calculated from simple finite difference upwind schemes and the log-conform approach is not used (so their simulations are limited to low Wi number). 

Perhaps, I should focus more on the positive definiteness of A, as you suggested. It might only be satisfied for slowly starting BCs and not for other situations.

Best regards,

-Anjishnu

[Shyam Sunder Yadav]

Dear Anjishnu

To calculate Tau_p - 2*mu_p*D, can you subtract the 2*mu_p*D part from the polymer stresses calculated using log-conform.h just before the acceleration event?

I doubt that will work but you can try...

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Dr. Shyam Sunder Yadav
Assistant Professor
Mechanical Engineering
BITS Pilani
09902346342
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[Giancarlo Esposito]

Hello everyone, did someone manage to implement the EVSS (or the Both sides diffusion, analogously) technique?