# Computational fluid dynamics simulations of suspensions

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## Non-Newtonian matrix fluids

For non-Newtonian fluids filled with spheroidal particles not so many theoretical works can be found, besides the work of Lee and Mear [32] that was already mentioned multiple times. For example one finds a proposal of Thomasset et al. [73] to replace the Newtonian viscosity in the TIF equation with a Carreau law (2.17) and to use Lipscomb’s coefficients. Another approach in a similar style can be found in Wang et al. [49] and Kagarise et al. [50]. In these two works the authors also assume the validity of the TIF equation for a non Newtonian fluid with Lipscomb’s coefficients computed for a Newtonian fluid and replace the Newtonian viscosity by a shear-thinning viscosity described by the Giesekus model. Considering the discussion in section 2.1.2, these approaches seem to be a little bit brute force. To understand this, let us consider the TIF equation (2.50) for spherical particles = *p1 + 2d + 2’Cd.

### Viscosity as function of stress

To obtain the viscosity as a function of the stress such that = ./ one can invert the relation for the stress: = .  /  (2.63) to obtain the shear rate as a function of the stress  =  ./ and insert this relation into .  /. The shear stress in the modified Carreau model given by (2.36) is = X0 􀀀 1 + .Y  /2n_2  : (2.64). The above relation cannot generally be analytically inverted for n f 0. The inversion is nevertheless possible by considering only the asymptotic cases: the Newtonian limit where Y  ~ 1 and the power law regime with Y  ¸ 1.
In the Newtonian case the expression for ./ can be directly written down as in this case (2.64) is simply constant and thus = X0: (2.65). In the power law regime (2.64) reduces to = X0.Y /n  n+1.

#### Moving particle in a shear flow

In case of a moving particle the mathematical description of the problem becomes more involved. Hu et al. [97] showed that it is superior to use a strong coupling of the governing equations by setting up a weak momentum balance for the whole suspension instead of an iterative coupling. In the iterative coupling one would first solve the fluid problem assuming that the particles are fixed and compute with the solution of the flow field the forces and torques exerted by the fluid onto the particle. Knowing the forces and torques one would update the particle translational and angular velocities and solve again the fluid problem assuming that the particles move with the previously computed velocities. This procedure is then repeated over and over, until convergence is reached. While the iterative coupling is essentially easier to implement, since one can use a standard solver for the flow problem, its biggest problem is that the particles will always start an oscillatory movement with the amplitude increasing to infinity [97]. Thus, the iterative coupling is unstable and is not a feasible method to simulate a moving particle.
Figure 3.4 shows a sketch of the fluid domain with one immersed particle. In the following only one particle is considered since this is sufficient for the purpose of this work. The particle in figure 3.4 is described by its mass mp, moment of inertia tensor Ip and the position of its center of mass Xp. The boundary of the particle is denoted by ) p and is assumed to be fully immersed in the fluid. The matrix fluid is described by its density % and its viscosity .u/ which is again assumed to obey the Carreau law (2.17). The fluid can be described by the previously introduced Cauchy momentum equation for an incompressible fluid (3.10). The motion of the fluid is coupled to the motion of the particle by requiring that the force and torque exerted by the fluid on the particle must be equal to the force and torque exerted by the particle on the fluid. For a rigid particle the translatory motion is given by Newton’s second law mp dvp dt = *Ê) p . * p1/ n d􀀀.

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Abstract
Résumé
Zusammenfassung
1. Motivation
2. An overview of the rheology of suspensions
2.1. Rigid spherical particles
2.1.1. Newtonian matrix fluids
2.1.2. Non-Newtonian matrix fluids
2.1.3. Superposition of viscosity-stress curves
2.2. Rigid non-spherical particles
2.2.1. Newtonian matrix fluids
2.2.2. Non-Newtonian matrix fluids
2.3. Summary
2.A. Viscosity as function of stress
3. Computational fluid dynamics simulations of suspensions
3.1. Introduction
3.2. Computational domain and mesh generation
3.3. Fixed particle in elongational flows
3.3.1. Problem description
3.3.2. Boundary conditions for a fixed particle
3.4. Moving particle in a shear flow
3.4.1. Problem description
3.4.2. Boundary conditions for a moving particle
3.5. Computation of effective macroscopic properties
3.5.1. Introduction
3.5.2. Computing the intrinsic viscosity
3.6. Summary
4. Spherical particles
4.1. Introduction
4.2. Simulation results
4.3. Taylor expansion
4.4. Fitting intrinsic viscosity
4.5. Discussion of strain amplification
4.6. Extension to higher volume fractions
4.7. Summary
5. Spheroidal particles
5.1. Introduction
5.2. Simulation Results
5.3. Ad hoc model for the rheological coefficients
5.4. Re-evaluating spherical particles
5.5. Summary
6. Conclusions
A. Notation
Bibliography

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