# American Institute of Mathematical Sciences

March  2017, 10(1): 193-213. doi: 10.3934/krm.2017008

## Self-organized hydrodynamics with density-dependent velocity

 1 Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom 2 Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen, AB24 3UE, United Kingdom 3 Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, 52062, Germany

* Corresponding author: Pierre Degond

Received  February 2016 Revised  May 2016 Published  November 2016

Motivated by recent experimental and computational results that show a motility-induced clustering transition in self-propelled particle systems, we study an individual model and its corresponding Self-Organized Hydrodynamic model for collective behaviour that incorporates a density-dependent velocity, as well as inter-particle alignment. The modal analysis of the hydrodynamic model elucidates the relationship between the stability of the equilibria and the changing velocity, and the formation of clusters. We find, in agreement with earlier results for non-aligning particles, that the key criterion for stability is $(ρ v(ρ))'≥q 0$, i.e. a nondecreasing mass flux $ρ v(ρ)$ with respect to the density. Numerical simulation for both the individual and hydrodynamic models with a velocity function inspired by experiment demonstrates the validity of the theoretical results.

Citation: Pierre Degond, Silke Henkes, Hui Yu. Self-organized hydrodynamics with density-dependent velocity. Kinetic & Related Models, 2017, 10 (1) : 193-213. doi: 10.3934/krm.2017008
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The accuracy test at $t=1$ shows that the splitting scheme is of first order. The initial data is given by $\rho_0 = \rho_s(1+0.1\sin(\pi x)), \theta_0 = \theta_s(1+0.1\sin(\pi x))$ with $(\rho_s, \theta_s) = (0.01, \frac{\pi}{4})$ and the mesh sizes are iteratively $N_x = N_y = 32, 64,128,256$
Comparison between the particle (left) and SOH models (right). For the particle model, $N = 10^5, \nu = 100, D = 10, R_1 = R_2 = 0.1$. The result is the average of 40 simulations. For the SOH model, $N_x = N_y = 100$
Stability test of the SOH model with $\gamma = 0$ and $\sigma = 0.1$. The profiles show the RMSF of $\rho$ and $\theta$ in $\log$-scale with respect to time $t$. The numerical solutions evolves from the steady states, and locally high concentrations develop
Stability test of the SOH model with $d = 0.5$, $\sigma = 0.01$ and $\rho^* = 0.02$. RMSF of the density and angle are decreasing with time
Stability test of the SOH model with $d = 0.5$, $\sigma = 0.01$ and $\rho^* = 0.005$. The RMSF of the numerical solutions grows gradually and high local concentrations develop. The linear scaling of the $\log$ of RMSF implies an exponential growth of the perturbation as a function of time $t$
Growth rate of the perturbation $\rho_\sigma$. The parameters are $c_1 = 0.975, c_2 = 0.925, d = 0.05$ and $k_0 = 0.125$. The three parameters for $v(\rho)$ are chosen as $(\rho^*, \alpha, \beta) = (0.005, 2, 5)$. The steady state for the density is fixed at $\rho_s = 0.01$ and the final time is $t = 1$. (a) is computed using the fomula (11). In order to obtain (b), we compute the numerical solutions of the SOH model and perform a simple linear regression on the Discrete Fourier transform of the perturbed part, i.e. $\rho_\sigma = \rho - \rho_s$. The growth rate is interpreted as the slope of the function $t \to \hat\rho_\sigma(\xi, t)$
Growth rate of the perturbation $\rho_\sigma$ given by the particle model with $N = 10^5$. The parameters are $\nu = 100, D = 5, R_1 = R_2 = 0.1$. The three parameters for $v(\rho)$ are chosen as $(\rho^*, \alpha, \beta) = (0.005, 2, 5)$. For each $\theta_s$, the growth rate is computed using the average of $10$ simulations, in order to reduce the effects of noise
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