[atul1018]

Dear Community

I am trying to understand how pressure gradient force acts on particles. I have gone through several papers and i find it confuisng and unable to interprete different formulations of pressure gradient force acting on a particle.

In the review paper (Experimental methods in chemical engineering:
Unresolved CFD-DEM) published by Bérard et al., they defined pressure gradient force as volume of particle multiplied by pressure gradient

F_p=-V_p (∂p/∂x)=1/6 πd_p^3 ∇P_g

On the other hand many papers such as Point-Particle DNS and LES of Particle-Laden Turbulentflow - a state-of-the-art review by Kuerten, Particle-fluid interaction forces as the source of acceleration PDF invariance in particle size bu Meller and Liberzon habe defined it in terms of fluid acceleration term (total time derivative of fluid velocity)

F_p= m_f Du/Dt


I am not able to understand if the different formulations mean the same thing and can be derived from one form to another. Please help me to understand the things properly.

Best Regards
Atul Jaiswal

[FMDenaro]

Originally Posted by atul1018

Dear Community

I am trying to understand how pressure gradient force acts on particles. I have gone through several papers and i find it confuisng and unable to interprete different formulations of pressure gradient force acting on a particle.

In the review paper (Experimental methods in chemical engineering:
Unresolved CFD-DEM) published by Bérard et al., they defined pressure gradient force as volume of particle multiplied by pressure gradient

F_p=-V_p (∂p/∂x)=1/6 πd_p^3 ∇P_g

On the other hand many papers such as Point-Particle DNS and LES of Particle-Laden Turbulentflow - a state-of-the-art review by Kuerten, Particle-fluid interaction forces as the source of acceleration PDF invariance in particle size bu Meller and Liberzon habe defined it in terms of fluid acceleration term (total time derivative of fluid velocity)

F_p= m_f Du/Dt


I am not able to understand if the different formulations mean the same thing and can be derived from one form to another. Please help me to understand the things properly.

Best Regards
Atul Jaiswal

I don't follow the topic about particle-laden since my last meet with Hans Kuerten several years ago, I can just try to think about your question.


If I write



F_p= m_f Du/Dt = V_f*rho_f*Du/Dt = V_f*(-dp/dx + mu Lap u)


I could see a similar formula for vanishing viscosity.

[atul1018]

Hello

Thanks for your explainantion. I am still confused how you expended the total derivative term for velocity field. When you write:

F_p= m_f Du/Dt = V_f*rho_f*Du/Dt = V_f*(-dp/dx + mu Lap u)

I see your expansion for total derivative as:

Du/Dt = (1/rho)*-dp/dx + nu*Lap*u

I am unable to follow this expansion. As per the definition of total derivatiive the expansion should be:

Du/Dt = du/dt + u*du/dx

Another thing: following the paper of Maxey and Riley (equation of motion for a small rigid sphere in nonuniform flow), i see the similar expression for total derivative for fluid velocity. I cite the exact paragraph and expression from the paper here:

The pressure gradient term was written in this form on the assumption that the undisturbed flow in incompressible and satisfies

Du/Dt = (1/rho)*-dp/dx + nu*Lap*u + g

You see that there is extra gravity term compared to your expansion of total derivative. I am also not sure, how total derivative term can be written as above equation when flow is incompressible.

Looking farward to hear from you.

Best Regards
Atul

[FMDenaro]

Originally Posted by atul1018

Hello

Thanks for your explainantion. I am still confused how you expended the total derivative term for velocity field. When you write:

F_p= m_f Du/Dt = V_f*rho_f*Du/Dt = V_f*(-dp/dx + mu Lap u)

I see your expansion for total derivative as:

Du/Dt = (1/rho)*-dp/dx + nu*Lap*u

I am unable to follow this expansion. As per the definition of total derivatiive the expansion should be:

Du/Dt = du/dt + u*du/dx

Another thing: following the paper of Maxey and Riley (equation of motion for a small rigid sphere in nonuniform flow), i see the similar expression for total derivative for fluid velocity. I cite the exact paragraph and expression from the paper here:

The pressure gradient term was written in this form on the assumption that the undisturbed flow in incompressible and satisfies

Du/Dt = (1/rho)*-dp/dx + nu*Lap*u + g

You see that there is extra gravity term compared to your expansion of total derivative. I am also not sure, how total derivative term can be written as above equation when flow is incompressible.

Looking farward to hear from you.

Best Regards
Atul

I used the momentum equation for the fluid