From:  "Heine, David R" <HeineDR@...233...> 
Date:  Fri, 30 Jun 2017 14:18:47 +0000 
Ranga, Yeah, it looks like you have done your homework. Tim submitted a GRM version of pair_lubricate based on Chapter 11, but maybe we are better off using the equations
you highlighted here. Do you have this implemented in LAMMPS? It would be interesting to compare the behavior of two nearcontact spheres with the three versions of pair_lubricate we now have. David
From: Ranga Radhakrishnan [mailto:r.radhakrishnan@...652...]
Hi David, Thanks for your reply. I have attached the pdf as per your request. I will try to address your email in a enumerated list to make my points clear.
1) \omega^\infty seems to have the wrong units. It is because "h_rate" has the units of length/time (Please correct me if I am wrong). Specifically, I am referring to lines: "omega[i][0] += 0.5*h_rate[3]; ..." which subtracts h_rate from omega. In case "h_rate"
has the right units of 1/time, I am confused about the units of Ef (E^\infty) which should be the rate of strain tensor. 2) The results given in Chapter 9 of Ref. (1) are slightly misleading, because they cite Jeffrey and Onishi's work (doi: 10.1017/S0022112084000355) before giving the final formulae for the forces, and torques. In the original work by Jefferey and Onishi,
the gap distance is nondimensionalised by (radi+radj)/2. 3) Eqs. 9.26, and 9.27 in Ref. (1) are the solutions for force and torques for shearing of two surfaces only due to rotation. Why does the pump term account only for the torque? I don't think the current formulation of the force considers the shearing of
two surfaces only due to rotation correctly. (Please see Sec. IV of the attached document). 4) a.The squeeze term in lubricate/poly is taken from the force given in Eq. 9. 33 of Ref. (1). According to the resistance matrix formulation, the first term in Eq. 9. 33 should be multiplied by a prefactor of "2/(1+\beta)" , and the second term by "\beta"
(apologies for the mistake in my previous email). One simple way to see that is that the magnitude of the leading order terms given in Sec. 11.2.2 should be twice of Eq. 9. 33, which is not the case in the textbook. 4) b. The squeeze or the shearing terms should be independent of the particle velocities and rotations, so Chapters 9 and 11 of Kim and Karilla should be consistent with each other. In case they are not, I have tried to refer to the original research articles
and verify the same. 5) I don't think we need to bring in volume fraction dependencies at the moment, because the issues that I have raised can be tracked down using just two particles of unequal sizes. The general solution of the problem of two unequal spheres in a fluid is given in Chapter 11 of Kim and Karilla, or originally in Jeffrey's research article (doi:10.1063/1.858494). In the attached pdf, I have
mainly relied on Kim and Karilla as the reference. The results that you are referring to in Chapter 9 of Kim and Karilla can be derived as cases of the general result in Chapter 11 (as shown in Section IV of the attached pdf). As you rightly mention the grand
(shear) resistance matrix formulation is slightly more involved to implement efficiently. However, one can get simplified expressions for forces and torques that are easier to implement in LAMMPS by considering only the first two leading order terms as shown
in the attached document (Eqs. 12, or 13, and 23, or 24). Regards, Ranga On 28/06/17 14:51, Heine, David R wrote:
