Viscoelastic-pizza

Background:

The charateristic scales: $R_0$, $G$, $\rho$: initial radius of the blob, elastic (pizza) modulus, and (pizza) density.

The characteristic frequency is $\Omega_c = \sqrt{G/(\rho R_0^2)}$: frequency required to significantly stretch the blob.

For Oldroyd-B, the parameters are:

• Dimensionless retardation time: $t_{\eta s} = (\eta_s/G)*\Omega_c = \eta_s/\sqrt{\rho G R_0^2}$. $t_{\eta s}$ is the time it takes to show an elastic response

  • For a purely elastic solid, $t_{\eta s}$ is 0
  • $t_{\eta s} \to \infty$ is liquid.

• Dimensionless relaxation time: $t_\lambda = \lambda * \Omega_c = \lambda * \sqrt{G/(\rho R_0^2)}$. $t_\lambda$ is the time it takes to show a viscous response

  • For a viscous liquid, $t_\lambda$ is 0 and for solids, $t_\lambda \to \infty$.
  • A purely elastic solid has $t_{\eta s} = 0$ and $t_\lambda \to \infty$.

Additionally, there are two other dimensionless numbers:

Pizza number: $\Pi = \rho\Omega^2R_0^2/G$.

This is the ratio of inertial to elastic forces. This is the square of the ratio of two characteristic frequencies: $(\Omega/\Omega_c)^2$.
$\Pi \gg 1$ is needed to show significant deformation of the blob. This is similar to the Bond number.

Elasto-capillary number: $Ec = \gamma/(G R_0)$.

This compares the elastic to capillary forces. To start with, we can assume $G \gg \gamma/R$.

• Another way to think about it is that $Ec = \Omega_\gamma^2/\Omega_c^2 = \frac{\gamma/(\rho R_0^3)}{G/(\rho R_0^2)} = \gamma/(G R_0)$.

Oldroyd-B model implementation

For details of the constitutive model Oldroyd-B model, see: https://github.com/comphy-lab/Viscoelastic3D.

How to run the code:

One core:

qcc -O2 -Wall -disable-dimensions pizza.c -o pizza -lm 
./pizza

OpenMP parallelization:

qcc -O2 -Wall -disable-dimensions pizza.c -o pizza -lm -fopenmp 
export OMP_NUM_THREADS=16
./pizza

Here, change 16 to the number of available threads.

OpenMPI parallelization:

CC99='mpicc -std=c99' qcc -Wall -O2 -D_MPI=1 -disable-dimensions pizza.c -o pizza -lm
mpirun -np 16 ./pizza

Here, change 16 to the number of available cores.

Note: In the pizza.c file, you can uncomment and edit ``argv" parts to pass parameters from terminal.