## Hard-sphere model and collisions of cohesive particles

Fluid flows with solid particles occur in many important industrial applications and in nature. In this project we focus on particle-particle collisions.

Our main tool will be so-called Lagrangian approach: a technique where we track all the particles (as points) in the coordinate system. Their motion is due to Newton's 2nd law.

And now: how can we model/describe the collisions between particles? Imagine two colliding solid particles:

Our goal is simple: we know the velocities of the particles before the collision. And we want to find/calculate the velocities after the collision. Please note that "velocity" means both the linear and the angular velocity.

How to describe then the whole "collision process". There are two techniques: a so-called soft-sphere model and a hard-sphere model. In the former we simply analyse the particles as they touch, deform and bounce-off using our skills from the contact/solid mechanics. When we use the hard-sphere model we use a simpler technique: we write impulse equations so that we obtain the "new" velocities directly as a function the "old" velocities.

To illustrate it better: let's have a look at the following equations [1,2]:

They look complicated but in fact it is not really true. On the left-hand side we can see the "new" velocities of both particles (subscript 1 refers to the first particle while 2 refers to the second one).

On the right-hand side we can see the "old" velocities (see superscript (0) that means "old"). In addition we have some other stuff: particle masses (denoted as m), radii (denoted as r), so-called restitution coefficient e_m (it describes if the collision is ellastic - then it is equal to 1.0 - or inellastic - then it is equal to 0.0) and friction coefficient (denoted as f).

There are also some other parametres, but they are less important now.

Summarizing: these equations tell us how to calculate the velocities after the collision knowing the velocities before. In fact the above equations represent only an illustration. If you want to know more, see [1,2] (where Ref. [2] you can download, see below).

Nevertheless, these relations do not include cohesion. Therefore the objective of this research is to implement cohesion into the hard-sphere model, i.e. extend the above relations.

This was done in our paper [2] and it can be shortly described as:

These equations are almost identical as the previous ones. But the difference is one extra term: J_n,c. This term we call "cohesive impulse" that it accounts for cohesion. See [2] for details.

Thus: the main objective is to obtain a tool for modelling of flows of cohesive particles. In addition we look into issues such as agglomeration and deposition. In addition to Ref. [2] you can also have a look at Ref. [3] that introduces some modifications/improvements and at Ref. [4] that models particle-wall collisions.

Or you can have a look at Ref. [5] where the model was used for simulating of a shear flow with particles.