Brownian motion of particles in a viscous flow

Illmatic · August 23, 2016, 04:19

Hi everybody!

I am trying to simulate the motion of particles in the nano meter scale (20nm-300nm) in a viscous flow. CFX enables me to calculate the fluid field, but what would be right strategy to proceed from there? How can I for include Brownian dynamics and diffusion of the nano particles to the forces that already act on the particle?

Thanks for your help in advance!

ghorrocks · August 23, 2016, 06:57

CFX does not have models for nanoparticles or Brownian motion. So any model you do will have to be either using a physical model in a regime it is not intended for or developing your own model.

How are you modelling this? Lagrangian particles, Eularian particles or a multicomponent mixture? Please explain why you chose the model you are using.

Illmatic · August 23, 2016, 07:47

Hi ghorrocks,

I already read your answers to that thread which is a very similar question. I intended to use a Langrangian frame work for the disperse phase as the volume fraction is considerably lower than the limit between dilute and dense flows mentioned in this book of Crowe et al. I have areas in my flow field where the viscous flow velocity is equal 0. In this areas I need to simulate the particle transport by the Brownian dynamics and diffusion. As I'm not really familiar with flow situations like this I actually have no idea how to proceed. What is general approach and what tools are generally used to simulate nano particle dynamics?

Edit: I just read another one of your answers where you recommend to add the diffusion of the particles as an additional variable instead of (falsely) using a multiphase flow model. So I think I understand the misconception I had in the beginning: On nano meter scale there is no slip between particles and viscous flow, right? How would that variable addition look like in CFX?

Best regards,
Axel

ghorrocks · August 23, 2016, 07:55

Based on your description I would recommend using an additional variable where you use diffusivity to model the dispersion due to Brownian motion. This is a very simple model to use and is cheap to run. So I would not recommend the Lagrangian particle tracking model.

Illmatic · August 23, 2016, 08:15

Thanks for your reply!

So even the slip of particles with 300nm diameter is not significant enough to consider in a simulation?

ghorrocks · August 23, 2016, 08:21

Don't take my word for it. Check the stokes number and work it out for yourself: https://en.wikipedia.org/wiki/Stokes_number

Illmatic · August 23, 2016, 10:01

So I used a formulation that I found here for the Stokes number, that is
$Stk = \frac{\rho_d d_d v_{f}}{9\mu_f}$
For a fluid flow with 15 m/s, a particle diameter of 300nm and density of 1000 kg/m^3 and a fluid viscosity of 40e-6 Pa s I calculated a Stokes number of 12.5. That would be well above 1 and therefore I would need to consider the particle slip too, right?

Edit: I know that the assumption of that formula is a Stokes flow, but I don't really know how to calculate the Stokes number in another way. Thanks for your help again!

ghorrocks · August 23, 2016, 21:12

No, you are using the wrong velocity. The velocity in the stokes number is the relative velocity of the particle compared to the surrounding fluid. It is not the velocity of the surrounding fluid.

The relative slip will be a function of the accelerations of the flow, mainly caused by streamline curvature. So have a look at the tightest bend in your flow and see how closely your particles stay with the flow as they go around the bend.

Illmatic · August 24, 2016, 04:48

Oh ok. In the Wikipedia article it said "

is the fluid velocity of the flow well away from the obstacle" where the obstacle is the particle in my case. That's why I inserted my bulk velocity to the equation.

ghorrocks · August 24, 2016, 07:15

To get a feel for how much slip you are going to get:

* Work out what the smallest radius of curvature is for a significant streamline in your fluid.
* Assume this can be modelled as a circular streamline of that radius with the fluid going at the local velocity.
* This will result in a force on particles in the flow. Work out the relative motion of the particle compared to the fluid in 1 revolution of this flow field.
* This is a guide to the relative slip of the particles to the fluid. I bet in your case it is really small, and small enough to ignore.

Illmatic · September 20, 2016, 05:09

Hi ghorrocks,

I didn't find time to work on this problem the last few weeks, but just came back to the topic. I am now trying to get the simulation running with an added scalar variable that represents the presence of particles. However I am not getting really far. Is it possible, that you could explain the approach in a little more detail for me?

ghorrocks · September 20, 2016, 07:30

Post #10 explains it in some detail. What are you having problems with?

Illmatic · September 20, 2016, 07:55

Sorry for not making my problem clear. I did the estimation of the slip velocity and it is well below the limit where the slip needs to be modeled.

I am now trying to implement the particle diffusion as an additional scalar field like you suggested in another thread. However I'm kind of stuck on the question how the balance equation for that should look like.

ghorrocks · September 20, 2016, 08:09

It sounds like a transport equation additional variable with diffusion. I am no expert on this, but it appears the diffusivity can be derived from the Brownian motion (see https://en.wikipedia.org/wiki/Brownian_motion for some introduction).

And that defines the additional variable equation, does it not?

Illmatic · September 20, 2016, 08:17

Sorry maybe we're still not on the same page

I wanna do exactly what you were describing in post #3 in this thread as "modelling diffusion with an increased diffusivity to account for the Brownian motion".

ghorrocks · September 20, 2016, 08:59

OK, let's go right back to the beginning. I am missing something here.

What are you trying to model? Why are you trying to model it? What do you intend to learn from the model? What does the Brownian motion act on in your simulation and what effect does it cause?

Illmatic · September 20, 2016, 09:22

Sorry for the inconvience!

I want to model the motion of nano particles in a gas flow with several dead zones. What I am interested in is if the particles are able to leave the gas flow's motion and enter the dead zones and the average time they would remain in the dead zones. At the end I am interested on how long in average the particles need to pass the complete flow regime.

As the particles are in nano scale you pointed out I can't use a Langrangian approach. As far as I can see what I need to model are #1 diffusion of the particles and #2 thermophoresis because I will have a temperature gradient in the flow regime.

Thanks for help!

Edit: Basically my first step would be to do what you suggested in post #4. However I don't know how to do that in CFX.

ghorrocks · September 20, 2016, 20:42

If you need help setting up additional variables then have a look at the CFX tutorials. The Reacting Flow in a Mixing Tube tutorial uses additional variables.

Illmatic · September 22, 2016, 07:58

Hi ghorrocks,

Thanks for the link! However I'm not sure if I understood the idea of the whole procedere yet. Do I model the particles only as an scalar variable or do I use Langrangian particles and add the Brownian motion and thermophoresis as additional forces that act on the particle?

If the first is the case then how can I track the particles?

EDIT: I found this article by Albert Einstein (!) that provides the equation

$|\Delta x|(t)=\sqrt{t \cdot \frac{RT}{N}\frac{1}{3\pi \eta r_B}}$

for the displacement of a particle. Would it now be possible, to just add this displacement to the motion of the Langrangian particles or is that a wrong approach?

ghorrocks · September 22, 2016, 10:00

Based on what I know you are doing I would recommend using the scalar variable. Should be no need for Lagrangian particles.

If the flow is steady state then streamlines in the post-processor are particle tracks.

Yes, Einstein dabbled in Brownian motion.But I do not recommend using lagrangian particles so the equation you quote is not relevant.

September 20, 2016, 09:22		#17
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	Sorry for the inconvience! I want to model the motion of nano particles in a gas flow with several dead zones. What I am interested in is if the particles are able to leave the gas flow's motion and enter the dead zones and the average time they would remain in the dead zones. At the end I am interested on how long in average the particles need to pass the complete flow regime. As the particles are in nano scale you pointed out I can't use a Langrangian approach. As far as I can see what I need to model are #1 diffusion of the particles and #2 thermophoresis because I will have a temperature gradient in the flow regime. Thanks for help! Edit: Basically my first step would be to do what you suggested in post #4. However I don't know how to do that in CFX. Last edited by Illmatic; September 20, 2016 at 10:27.

September 22, 2016, 07:58		#19
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	Hi ghorrocks, Thanks for the link! However I'm not sure if I understood the idea of the whole procedere yet. Do I model the particles only as an scalar variable or do I use Langrangian particles and add the Brownian motion and thermophoresis as additional forces that act on the particle? If the first is the case then how can I track the particles? EDIT: I found this article by Albert Einstein (!) that provides the equation $\|\Delta x\|(t)=\sqrt{t \cdot \frac{RT}{N}\frac{1}{3\pi \eta r_B}}$ for the displacement of a particle. Would it now be possible, to just add this displacement to the motion of the Langrangian particles or is that a wrong approach? Last edited by Illmatic; September 22, 2016 at 09:37.

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
How to simulate dilute solid particles in gas flow?	chpjz0391	OpenFOAM Running, Solving & CFD	4	March 22, 2016 20:32
Compressible viscous flow newbie: when NOT to assume incompressible, non-gas EOS?	Srh	Main CFD Forum	0	April 19, 2013 03:12
How to set correct mass flow rate for lagrangian particles ?	sankarv	OpenFOAM	0	April 19, 2010 12:40
brownian motion of small particles	mazdak	Main CFD Forum	1	November 2, 2009 20:54
potential flow vs. Euler flow	curious ...	Main CFD Forum	23	July 21, 2006 08:40

August 23, 2016, 04:19	Brownian motion of particles in a viscous flow	#1
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	Hi everybody! I am trying to simulate the motion of particles in the nano meter scale (20nm-300nm) in a viscous flow. CFX enables me to calculate the fluid field, but what would be right strategy to proceed from there? How can I for include Brownian dynamics and diffusion of the nano particles to the forces that already act on the particle? Thanks for your help in advance!

August 23, 2016, 06:57		#2
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	CFX does not have models for nanoparticles or Brownian motion. So any model you do will have to be either using a physical model in a regime it is not intended for or developing your own model. How are you modelling this? Lagrangian particles, Eularian particles or a multicomponent mixture? Please explain why you chose the model you are using.

August 23, 2016, 07:47		#3
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	Hi ghorrocks, I already read your answers to that thread which is a very similar question. I intended to use a Langrangian frame work for the disperse phase as the volume fraction is considerably lower than the limit between dilute and dense flows mentioned in this book of Crowe et al. I have areas in my flow field where the viscous flow velocity is equal 0. In this areas I need to simulate the particle transport by the Brownian dynamics and diffusion. As I'm not really familiar with flow situations like this I actually have no idea how to proceed. What is general approach and what tools are generally used to simulate nano particle dynamics? Edit: I just read another one of your answers where you recommend to add the diffusion of the particles as an additional variable instead of (falsely) using a multiphase flow model. So I think I understand the misconception I had in the beginning: On nano meter scale there is no slip between particles and viscous flow, right? How would that variable addition look like in CFX? Best regards, Axel

August 23, 2016, 07:55		#4
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	Based on your description I would recommend using an additional variable where you use diffusivity to model the dispersion due to Brownian motion. This is a very simple model to use and is cheap to run. So I would not recommend the Lagrangian particle tracking model.

August 23, 2016, 08:15		#5
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	Thanks for your reply! So even the slip of particles with 300nm diameter is not significant enough to consider in a simulation?

August 23, 2016, 08:21		#6
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	Don't take my word for it. Check the stokes number and work it out for yourself: https://en.wikipedia.org/wiki/Stokes_number

August 23, 2016, 10:01		#7
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	So I used a formulation that I found here for the Stokes number, that is $Stk = \frac{\rho_d d_d v_{f}}{9\mu_f}$ For a fluid flow with 15 m/s, a particle diameter of 300nm and density of 1000 kg/m^3 and a fluid viscosity of 40e-6 Pa s I calculated a Stokes number of 12.5. That would be well above 1 and therefore I would need to consider the particle slip too, right? Edit: I know that the assumption of that formula is a Stokes flow, but I don't really know how to calculate the Stokes number in another way. Thanks for your help again!

August 23, 2016, 21:12		#8
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	No, you are using the wrong velocity. The velocity in the stokes number is the relative velocity of the particle compared to the surrounding fluid. It is not the velocity of the surrounding fluid. The relative slip will be a function of the accelerations of the flow, mainly caused by streamline curvature. So have a look at the tightest bend in your flow and see how closely your particles stay with the flow as they go around the bend.

August 24, 2016, 04:48		#9
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	Oh ok. In the Wikipedia article it said " is the fluid velocity of the flow well away from the obstacle" where the obstacle is the particle in my case. That's why I inserted my bulk velocity to the equation.

August 24, 2016, 07:15		#10
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	To get a feel for how much slip you are going to get: * Work out what the smallest radius of curvature is for a significant streamline in your fluid. * Assume this can be modelled as a circular streamline of that radius with the fluid going at the local velocity. * This will result in a force on particles in the flow. Work out the relative motion of the particle compared to the fluid in 1 revolution of this flow field. * This is a guide to the relative slip of the particles to the fluid. I bet in your case it is really small, and small enough to ignore.

September 20, 2016, 05:09		#11
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	Hi ghorrocks, I didn't find time to work on this problem the last few weeks, but just came back to the topic. I am now trying to get the simulation running with an added scalar variable that represents the presence of particles. However I am not getting really far. Is it possible, that you could explain the approach in a little more detail for me?

September 20, 2016, 07:30		#12
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	Post #10 explains it in some detail. What are you having problems with?

September 20, 2016, 07:55		#13
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	Sorry for not making my problem clear. I did the estimation of the slip velocity and it is well below the limit where the slip needs to be modeled. I am now trying to implement the particle diffusion as an additional scalar field like you suggested in another thread. However I'm kind of stuck on the question how the balance equation for that should look like.

September 20, 2016, 08:09		#14
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	It sounds like a transport equation additional variable with diffusion. I am no expert on this, but it appears the diffusivity can be derived from the Brownian motion (see https://en.wikipedia.org/wiki/Brownian_motion for some introduction). And that defines the additional variable equation, does it not?

September 20, 2016, 08:17		#15
Illmatic Member Axel Join Date: May 2016 Location: Augsburg, Germany Posts: 46 Rep Power: 10	Sorry maybe we're still not on the same page I wanna do exactly what you were describing in post #3 in this thread as "modelling diffusion with an increased diffusivity to account for the Brownian motion".

September 20, 2016, 08:59		#16
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	OK, let's go right back to the beginning. I am missing something here. What are you trying to model? Why are you trying to model it? What do you intend to learn from the model? What does the Brownian motion act on in your simulation and what effect does it cause?

September 20, 2016, 20:42		#18
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	If you need help setting up additional variables then have a look at the CFX tutorials. The Reacting Flow in a Mixing Tube tutorial uses additional variables.

September 22, 2016, 10:00		#20
ghorrocks Super Moderator Glenn Horrocks Join Date: Mar 2009 Location: Sydney, Australia Posts: 17,871 Rep Power: 144	Based on what I know you are doing I would recommend using the scalar variable. Should be no need for Lagrangian particles. If the flow is steady state then streamlines in the post-processor are particle tracks. Yes, Einstein dabbled in Brownian motion.But I do not recommend using lagrangian particles so the equation you quote is not relevant.