Cahn–Hilliard–Brinkman systems for tumour growth

Figure 1.  Typical situation for the formal asymptotic analysis

Figure 2.  The tumour and healthy regions $ \Omega_T $ and $ \Omega_H $

Figure 3.  Schematic sketch of the inner region close to $ \Sigma(0) $

Figure 4.  Initial tumour size for initial data $ r $: A slightly perturbed sphere

Figure 5.  Comparison of Cahn--Hilliard--Darcy and Cahn--Hilliard--Brinkman models: Tumour at time $ t = 12 $ for $ \beta = 0.1 $, left side for the CHD model, right side for the CHB model with $ \eta = 10^{-5} $

Figure 6.  Influence of different mobilities: Tumour at time $ t = 9 $ for $ \eta = 10^{-5} $, $ \beta = 0.1 $ and $ \alpha = \rho_S = 2 $, but with different mobilities, left $ m(\varphi) = \frac{1}{2}(1+\varphi)^2 $, middle $ m(\varphi) = \epsilon $, right $ m(\varphi) = 10^{-3}\epsilon $

Figure 7.  Influence of the adhesion parameter $ \beta $: Evolution of the tumour with $ m(\varphi) = \tfrac{1}{2}(1+\varphi)^2 $ and $ \eta = 0.1 $, above for $ \beta = 0.1 $ at time $ t = 1, 3, 6, 10 $, below for $ \beta = 0.01 $ at time $ t = 1, 1.5, 2, 2.5 $

Figure 8.  Influence of viscosiy I: Tumour and velocity for $ \beta = 0.01 $ at time $ t = 2.5 $, left for $ \eta = 0.1 $, right for $ \eta = 100 $, on top the tumour and below the velocity magnitude

Figure 9.  Influence of viscosity II: Evolution of the tumour at time $ t = 1, 3, 6, 10 $ with $ \beta = 0.1 $, $ \nu = 0 $ and a no-slip boundary condition on the left boundary, on top for $ \eta = 0.1 $ and below for $ \eta = 10 $

Figure 10.  Velocity profiles for different viscosities: The velocity magnitude at time $ t = 10 $ with $ \beta = 0.1 $, $ \nu = 0 $ and a no-slip boundary condition on the left boundary, left for $ \eta = 0.1 $, right for $ \eta = 10 $

Figure 11.  Influence of viscosity contrast: Tumour at time $ t = 10 $ with $ \beta = 0.1 $, $ \nu = 0 $ and a no-slip b. c. on the left boundary, with $ \eta_- = 0.01 $, $ \eta_+ = 1 $; $ \eta_- = 1 $, $ \eta_+ = 0.01 $; $ \eta_- = 0.01 $, $ \eta_+ = 10 $; $ \eta_- = 10 $, $ \eta_+ = 0.01 $

Figure 12.  Influence of initial profile I: Tumour at time $ t = 0, 0.3, 1, 1.6 $ with $ \eta = 100 $ and with the initial profile corresponding to $ r_1 $

Figure 13.  Influence of initial profile II: Tumour at time $ t = 0, 0.7, 1.2, 2.6 $ with $ \eta = 100 $ and with the initial profile corresponding to $ r_2 $

Figure 14.  Influence of initial profile III: Tumour at time $ t = 0, 0.5, 1.2, 2.4 $ with $ \eta = 100 $ and with the initial profile corresponding to $ r_3 $

Figure 15.  Influence of initial profile IV: Tumour at time $ t = 0, 0.3, 1.3, 2.3 $ with $ \eta = 0.01 $ and with the initial profile corresponding to $ r_4 $