number is
P eI↔II =−(1 +M)/(f′(c0)c0). (6.15) In Fig. 6.3 the dispersion relation is shown for three different unstable cases:
• M = 0 (region O in Fig. 6.4): all modes are real (the case in [Bois et al., 2011]) and initial growth shows stationary patterns (typeIIs)
• M = 2 (region III): there are unstable real and imaginary modes
• M = 4 (region II) : all unstable modes are imaginary and initial growth is dominated by traveling or standing waves (typeIIo) To get the P´eclet number that marks the transition from imaginary modes to mixed unstable modes one has to find the wavenumber of the real-imaginary transition
q1,2tr = q
(√
M±p
−f′(c0)c0P e). (6.16) When both positive numbers qtr1,2 are real then there exists a region of wavenumbers with complex unstable eigenvalues. If the lower qtr1 gets com-plex there is no region with unstable real modes any more, since always Re(˜ω(q2tr))<0. For the transition one gets
P eII↔III =−(1 +√
M)2/(f′(c0)c0). (6.17) With this a phase diagram classifying the initial instability is drawn in Fig.
(6.4). From Eq. 6.15 it becomes obvious that the instability is only present if f′(c0)<0. This means that the regulator must be an inhibitor of tension generation. This is true for Physarum myosin, where the free intracellular Ca2+is an inhibitor. However, in most of the other cell types Ca2+is an activator of contraction.
6.4 Numerical simulations
The investigation is continued with numerical simulations. The dominant wavenumber arising from linear stability analysis is not always observed in the finite nonlinear regime. A good example for this is the coarsening process for M = 0. Structures from the initial instability grow, such that smaller wavenumbers dominate in the end when a stationary stable solu-tion is reached. In Fig. 6.5 b)-e) four exemplary numerical simulasolu-tion in one dimension obtained with the finite difference method are shown. The color encodes the concentration field c. A random perturbation around the steady state with c0 = 1 was used as initial condition. The sol fraction was chosen to be ρsol = 0.01 leading to small deformations only, which ensures numerical stability in both one and two dimensions. As boundary conditions
0.0 0.2 0.4 0.6 0.8 1.0 0
5 10 15 20
M
Pe
I II
III
O
Figure 6.4: Phase diagram in the M-P e-plane: There are four regions cor-responding to the different kind of dispersion relation in Fig. 6.3. Region O is line defined by M = 0 when all unstable modes have real eigenvalues (black line). In region I (violet) all modes are stable, in region II (blue) all modes are imaginary and in region III (white) the unstable modes are mixed real and imaginary. The black dots give the positions of the curves in Fig. 6.3. Here, f′(c0) =−1/4 and c0= 1.
no-flux Neumann for the concentrationcand zero Dirichlet for displacement uand flow fieldwwere imposed. In Fig. 6.5 a) the spatiotemporal dynamics is classified in the phase plane of P e and M. For M = 0 coarsening to a stationary domain takes place (see b).
For any small nonzeroM the coarsening changes over to a stable traveling domain that is reflected from the boundaries (c). These behavior is a con-sequence of the conservation of the regulating species with concentration c.
An inhomogeneous velocity profile leads to the accumulation of sol in some region an thus an expansion of the gel. IfM is very small it takes a long time until the deformation is large enough to cause significant elastic stresses. As consequence the domains start to move. Also the reflections can be under-stood in a very simple way: When a domain encounters a boundary, due to conservation, it can either travel backwards or stay there. But having a static velocity profile at the boundary would mean the deformations would grow infinitely. The only possibility that is left is that the wave is reflected.
At onset of pure complex modes (beginning of region II) turbulent waves are observed (d). Further increase of M finally leads the regular standing waves (e). For periodic boundary conditions there would be traveling waves
6.4. NUMERICAL SIMULATIONS 97
Figure 6.5: Numerical simulations for ρsol = 0.01 and c0 = 1.0 in 1D with Dirichlet boundary conditions for u and w: a) phase diagram with separa-tion lines (black) obtained from linear stability analysis and color coding that represents the spatiotemporal dynamics. Bottom: space-time plots for selected parameters. b) coarsening to stationary pattern, c) traveling do-mains, d) turbulence and e) standing waves (published in [Radszuweitet al., 2013]
instead. The onset of turbulence coincides very well with the separatrix 1 between regions II and III characterizing the linear growth behavior.
In two dimensions the dispersion relation has exactly the same form as in one dimension except for that the ”elastic P´eclet number” has the form
M =ρsol(2µ+λ)/(Dβ) with 2µ+λ=K+ 2d−1
d G (6.18)
For the simulations in 2D the same boundary conditions are assumed as in one dimension. The same set of equations on a disc was solved as in Chapter 5 but with only one concentration without any reaction term and
1In this context theseparatrix marks the boundary between regions in the phase dia-gram, where the dynamics of the system has qualitatively different properties.
with relation (6.11) for the active tension. A different mechanical parameter set as for Physarum is used that has the same orders of magnitude like in [Bois et al., 2011]. A small random perturbation from c0 = 0.78 is applied for all the following simulations. Zero-Dirichlet conditions for u and ware imposed, while no-flux for the concentration field c is used. As in one di-mension the sol fraction is ρsol= 0.01.
The diversity of pattern in two dimensions is richer, due to the additional spatial dimension. Though, the main characteristic equals the onedimen-sional case. For small elastic number M but a P´eclet number above the instability a formation of domains is observed that move after some time.
This timescale can be estimated from Eqs. (6.10):
τmove=ηeff/(ρsol(2µ+λ)). (6.19) This situation is shown in Fig. 6.6 as space-time plot of a cut through the disc and snapshots of the concentration field. A Larger P´eclet number leads to higher wavenumbers and thus smaller droplets. These domains travel and coarsen via collisions (see Fig. 6.7). When the coarsening process is finished the single domain is traveling and is reflected at the boundaries. This com-plies very well with the onedimensional case as Fig. (6.8) shows. In region II near the separatrix to region III in theP e−M plane a turbulent behavior in agreement with one dimensional simulations appears (see Fig. 6.9). Here, the domains form and break up again in an irregular way. Increasing further the elastic number M leads again to regular patterns. Fig. 6.10 shows a stable spherical standing wave (or breathing pattern). The parameters are closely above the threshold of instability, where only the first nonzero mode is unstable. For higher P´eclet number (see Fig. 6.11) the next modes eigen-values cross the zero axes leading to a rotational symmetry breaking and a more complex standing wave. This solution is unstable and turns into a stable rotating pattern. The shape of the rotating wave could misleadingly lead to the impression that this is a spiral wave. But in this case there are no phase waves (no chemical oscillator) and hence, no phase singularity.
These five cases are summarized in Fig. 6.12 in theM−P ephase diagram.
Finally, to present the flow field of a moving domain it is depicted in Fig.
6.13 by arrows on top of the concentration field.
6.4. NUMERICAL SIMULATIONS 99
Figure 6.6: Domain formation: A space-time plot of the concentration field c on a line through the center is shown on the left side for time duration 50τ. To the right two snapshots are presented with mesh on top of the color coding. The P´eclet number is P e= 10 and M = 0.08.
Figure 6.7: Coarsening: A space-time plot of the concentration field con a line through the center is shown on the left side for time duration 50τ. To the right two snapshots are presented with mesh on top of the color coding.
The P´eclet number isP e= 20 and M = 0.08.
6.4. NUMERICAL SIMULATIONS 101
Figure 6.8: Coarsening: A space-time plot of the concentration field c on a line through the center is shown on the left side for time duration 500τ. To the right two snapshots are presented with mesh on top of the color coding.
The P´eclet number is P e= 15 andM = 0.375.
Figure 6.9: Turbulence: A space-time plot of the concentration field con a line through the center is shown on the left side for time duration 500τ. To the right two snapshots are presented with mesh on top of the color coding.
The P´eclet number isP e= 20 and M = 2.5.
6.4. NUMERICAL SIMULATIONS 103
Figure 6.10: Spherical standing wave: A space-time plot of the concentration fieldcon a line through the center is shown on the left side for time duration 500τ. To the right two snapshots are presented with mesh on top of the color coding. The P´eclet number is P e= 25 and M = 5.
Figure 6.11: Standing wave and rotating pattern: A space-time plot of the concentration field con the left is given on a circle depicted by the dashed line. The time duration is 500τ. To the right two snapshots are presented with mesh on top of the color coding. The P´eclet number is P e= 27.5 and M = 5.
6.4. NUMERICAL SIMULATIONS 105
æ æ
æ
à à
ì ì
0 1 2 3 4 5 6
0 5 10 15 20 25 30
M
Pe
I II
III
Figure 6.12: Summary of the five presented patterns on the disc: Coars-ening and traveling domains (blue dots), turbulence (purple square) and standing/rotating waves (olive diamonds).
Figure 6.13: Flow field w of a moving domain represented by arrows: the length of the arrows correspond to the absolute of w. It is shown on top of the concentration field c. The data is the same as in Fig. 6.8 with P e= 15 and M = 0.375.