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Re: [ESPResSo-users] Lubrication correction in Stokesian dynamics

From: Ulf Schiller
Subject: Re: [ESPResSo-users] Lubrication correction in Stokesian dynamics
Date: Thu, 12 Jan 2017 08:07:29 -0500
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You may wish to read these articles:

Claeys, T. L., & Brady, J. F. Lubrication singularities of the grand resistance tensor for 2 arbitrary particles. PHYSICOCHEMICAL HYDRODYNAMICS, 11(3), 261-293 (1989) Cox, R. G. The motion of suspended particles almost in contact. International Journal of Multiphase Flow 1, 343–371 (1974).

Ladd's paper is based on the expansions therein.


Dr. Ulf D. Schiller
Assistant Professor, Department of Materials Science and Engineering
Faculty Scholar, School of Health Research
Clemson University
161 Sirrine Hall
Clemson, SC 29634

Office: 299c Sirrine Hall
Phone: 1-864-656-2669
Fax: 1-864-656-5973

On 01/12/2017 07:39 AM, Lei Liu wrote:
Hello David,

thanks a lot for reading and answering my tedious questions.
Because I am a novice in this field, I try to understand more details.

As you said, r2bcorr_para_self is \alpha_s+\beta_s,
which equals to the first part on the right hand side of equ. (3.17),
and has corresponding implementation in Line 1410 in the source file

How about r2bcorr_perp_self ?
If this term is \beta_s, based on equ. (3.19),
I would expect that a line of code like
"r2bcorr_perp_self = 1/( 1 - 9/16/dr2 - 3/4/dr4 - 1/4/dr6 )".
But in Line 1412 it equals to
"r2bcorr_perp_self = 1/( 1 - 25/16/dr2 )"
This is the point I get confused.

Would you please explain a little bit more ?

Best regards

On Thu, Jan 12, 2017 at 7:24 PM, David Schwörer
<address@hidden <mailto:address@hidden>> wrote:

    Hi Lei,

    It's been a while since I looked at this, but I think:
    r2bcorr_para_self is alpha_s+beta_s
    r2bcorr_para_mix is alpha_m+beta_m
    r2bcorr_perp_self is beta_s
    r2bcorr_perp_mis is beta_m

    the confusion is that in the one basis set is in rr and one, the other
    in rr and one-rr, that is why in 3.17 b_s is substracted from alpha_s,
    so that after adding the one beta_s, only the first part 1/(1-alpha_s^2)

    I hope that helps.


    On 01/12/2017 08:50 AM, Lei Liu wrote:
    > Dear all,
    > by reading related documents, now I understand that the terms containing
    > log(s) come from R^{lub}.
    > The only left question is about the variables {r2bcorr_para_self,
    > r2bcorr_para_mix, r2bcorr_perp_self, r2bcorr_perp_self} in function
    > "sd_compute_resistance_matrix_sparse()".
    > The first two variables corresponds to \alpha_{s} and \alpha_{m} in
    > equations (3.17), (3.21) in David's thesis.
    > But how about the latter two ?
    > Would anyone like to do me a favour, and to explain a little bit where
    > they come from ?
    > I get confused because they are different from my expectation,
    > \beta_{s} or \beta_{m} in equations (3.19) and (3.23).
    > With my best wishes
    > Lei
    > On Wed, Jan 11, 2017 at 8:30 PM, Lei Liu <address@hidden 
    > <mailto:address@hidden <mailto:address@hidden>>> wrote:
    >     Dear all,
    >     I am trying to understand the lubrication correction
    >     in Stokesian dynamics implemented in current developing version of
    >     ESPResSo.
    >     According to David Schwoerer's thesis, the function
    >     "sd_compute_resistance_matrix_sparse()"
    >     computes the lubrication correction described in equation
    (3.24) as
    >     R^{lc} = R^{lub} - R^{2b,ff}.
    >     In addition, there is one comment in the code
    >     referring R^{lub} to 'N.-Q. Nguyen and A. J. C. Ladd, PHYSICAL
    >     REVIEW E 66, 046708 (2002) equation (34)'.
    >     But I still do not understand this function quite well, especially
    >     the terms containing the variable "ls = log(s) = log(|r|/a - 2)",
    >     which I cannot find neither in section 3.1.2 in the thesis nor in
    >     Ladd's paper.
    >     Would anyone like to give me more references about how ESPResSo
    >     calculates this correction?
    >     Many thanks in advance
    >     Lei

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