HI jessica
I will explain various hydrodynamic forces and/or the equations to work out the pressure and velocity of the Force.
Please forgive any spelling mistakes.



To correctly simulate the rheology of concentrated suspensions at high shear rates requires the inclusion of hydrodynamic forces between particles. The computation of such interactions is very intensive, and it presents a significant challenge: to produce algorithms that enable a study of systems with large numbers of particles.(I have tried many times Jessica.) The work of Brady and coworkers in the early eighties showed the way forward to such a goal. The technique, Stokesian Dynamics presents a formalism for studying such systems. We first write down the Langevin equation for a colloidal particle within a hydrodynamic medium:
Please note Jessica that temprature is very important hear.(IE the colder the area is, the less effective a hydrodynamic force would be)

FH + FC + FB = M · dU/dt

Here, F represents forces acting on an individual particle: Hydrodynamic, Colloidal (charge repulsion/attraction for example) and Brownian forces. M is the mass of the particle, and U its velocity.Within a colloidal system, the time scales of interest dictate that inertia may be neglected. Thus the right hand side of above equation can be neglected. We also work with the assumption that the fluid obeys the Stokes equations, i.e. we are within a low Reynolds number regime. Thus the hydrodynamic forces can be considered linear in the velocities, therefore FH = R·U, where R is a resistance tensor, the elements of which describe the hydrodynamic interactions between individual particles. To simulate colloid particles, we need to determine the particles velocities at each time step, thus the above equation becomes:

U = -R-1 · (FC + FB)

Thus solving for U requires the inversion of the resistance matrix R. This typically restricts the size of the systems that one can study to less than 100 particles. In order to simulate systems on a larger scale, Ball and Melrose have proposed an approximation that is applicable to colloids at high concentrations in shear flow. They argue that at such concentrations, where the typical interparticle spacing is small, the near-field terms in the resistance matrix (which are given by lubrication theory) will dominate in shear flow. Thus R can be made sparse; this greatly increases computational efficiency. The method is now typically of O(N2) compared to O(N3) for the full matrix, making the study of larger systems feasible (typically as much as 4,000 particles). Even though the resistance tensor is sparse and consists of near-field terms, the inversion of such a matrix results in a many-bodied far-field mobility tensor. Ref. gives full details on the technique, in particular how to correctly handle the correlated Brownian forces through use of a 2nd order difference scheme. The code has been implemented and validated. It has a suite of colloid interactions, including charge, depletion, and polymer coats. Particles may interact with just squeeze lubrication or with full modes of shear and rotation. Nearest neighbours are identified through the construction of a Delaunay mesh which is co-moving with the particle centres.
( i have included a diagram)

Jessica if you work out the equation you will find my theory proved right. If you need help please pm me and i will give you the eqaution format and answers.

as i have said please forgive my speling errors.

pleas enote that, i dont want the Titanic to be raised

I could also go into; Sphere-Sphere gravitational softening and point-Sphere-gravitational softening.

If you wish to see this then please tell me and i will right another report on the subject.

Your reprt on the wreck is very intresting but, Is lacking alot of crucial parameters, Equations Ect.
i do agree with you in part about the hydrodynamic forces, as i have explained above though. It all depends on temprature and depth.

Attached file:   hydro.jpg (0.00 KB)

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