Correction to: Coupled Self-Organized Hydrodynamics and Stokes Models for Suspensions of Active Particles
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Journal of Mathematical Fluid Mechanics
Correction
Correction to: Coupled Self-Organized Hydrodynamics and Stokes Models for Suspensions of Active Particles Pierre Degond , Sara Merino-Aceituno, Fabien Vergnet and Hui Yu
Correction to: J. Math. Fluid Mech. (2019) 21:6 https://doi.org/10.1007/s00021-019-0406-9 This note provides a list of errata and their correction for Reference [1]. ˜ − 1) should come with a negative sign. This expression • There is a typo in Eq. (5.12): the term in (λ becomes: ˜ − 1)˜b ρ2 k 2 (λ 0 0 2 ¯ Ω iρ0 (−α + V0 · k) + γ|k| ρ0 − 2 |k|2 i 1 ρ0 ˜ ˜ ˜ ¯ = 2λk0 p¯ + b(λ + 1)(ρ0 k + ρ¯k0 ) − ρ¯ k ⊥ . (0.1) 2 |k|2 κ • In the statement of Th. 5.1 (Linear Stability Analysis), we have that Eq. (5.5) is not true. As a consequence equation (5.6) is slightly modified into
˜bρ0 k02 ˜ ˜ −4λ 2 + λ + 1 (−α + U0 · k) D(α, k) = 2|k|2 |k|
2 i k 0 ˜+1 ˜ −c1 k0 −2λ +λ + c1 (|k|2 − k02 ) |k|2 κ ˜ − 1)˜bρ0 k 2 (λ 0 −(−α + U0 · k) i(−α + V0 · k) − + γ|k|2 , (0.2) 2 |k|2 when k¯ = 0. And Eq. (5.7) is valid when k¯ = 0. When k¯ = 0 we have a different scenario where ¯ = 0 arbitrary and ρ¯ = 0 and Ω
˜ − 1)˜b ρ0 k 2 (λ 0 2 α = V0 · k + i − γ|k| , (0.3) 2 |k|2 ˜ ∈ [−1, 1] and so Im(α) ≤ 0. (Finally, notice that, the parameter which gives only stable modes since λ η in the original article does not play a role any more in this corrected version.) The original article can be found online at https://doi.org/10.1007/s00021-019-0406-9. 0123456789().: V,-vol
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To obtain these results we modify the proof of Th. 5.1 Case B, which we rewrite fully next (notice that the number of the equations that do not start by 0 refer to the number of the equations as they appear in the original article). Proof. Case (B) Suppose that k ⊥ = 0. Doing the inner product of Eq. (0.1) with k and using that k ⊥ · k = |k|2 − k02 , the dispersion relation is given by (using Eq. (5.11)):
2 ˜ 2 ρ i 2 λk k 0 0 0 ˜b ˜ + 1 + ρ¯k0 − ˜+1 ˜ +λ +λ − ρ¯ (|k|2 − k02 ) ρ0 k¯ −4λ 2|k|2 |k|2 |k|2 κ ˜ 2 2 ˜ − 1) bρ0 k0 + γ|k|2 ρ0 k, ¯ = iρ0 (−α + V0 · k) − (λ (0.4) 2 |k|2 and from Eq. (5.10b) we have the relation ρ + ρ0 c1 k¯ = 0. (−α + U0 · k)¯
(0.5)
Next we distinguish between the cases k¯ = 0 and k¯ = 0. Suppose that k¯ = 0, then from Eq. (0.5) we have that −α + U0 · k = 0 and multiplying Eq. (0.4) by −α + U0 · k = 0 and using Eq. (0.5), we get the ¯ dispersion relation in Eq. (0.2), after simplifying k. ¯ ¯ = 0 (thanks to Eq. (5.10a)) and, therefore, If, on the contrary, k = 0, then one can check that k ⊥ · Ω ⊥ they are normal. Since by assumption k = 0 from Eq. (0.1) we conclude that the coefficient in front of k ⊥ on the right hand side must be zero. Rewriting this term using that k¯ = 0 and Eq. (5.11) we have that ρ¯ I = 0, (0.6) with ρ0 k0 I := ˜b 2|k|2
˜ 2 i 2λk 0 ˜ − 2 +λ+1 − . |k| κ
From here we deduce that ρ¯ = 0 because otherwise we should have that I = 0 but the imaginary part of ¯ = 0 (otherwise we I is non-zero, so ρ¯ = 0. In particular this implie
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