Q2. what is the Stokes approximation for the local velocity v?
The Stokes approximation is employed for the local velocity v and it takes the form0 = − ∇p − φ ∇ δF δφ − c ∇ δF δc + η0∇2 v, (4)where p is determined such that the velocity field satisfies the incompressibility condition ∇ · v = 0.
Q3. What is the relaxation rate of the droplet?
when the relaxation rate is finite, the droplet tends to shift further since the interfacial energy is an increasing function of c.
Q4. What is the reason why the concentration c tends to decrease at the front?
Since the first order correction to the concentration c is given by c( r) = −[−D∇2 + γ ]−1H ( r) and the operator [− D∇2 + γ ]−1 is positive definite, the concentration c tends to increase (decrease) at the front (rear).
Q5. What is the second assumption in the theory?
Since the interface velocity is arbitrarily small in the vicinity of the drift bifurcation threshold, the second assumption is consistently justified in the theory.
Q6. What is the theory of a droplet?
if the time-delayed effect τuα dominates the term −uα which corresponds to the Stokes drag force, the droplet undergoes migration.
Q7. What is the first order correction from the convective term?
The first order correction from the convective term gives us P = 31/56 as shown in Appendix D. Since migration of droplet occurs for τ ≥ 1, this indicates that the stronger Marangoni effect is necessary when the convection of the third component exists.
Q8. What is the chemical reaction of the droplet?
In this experiment, the molecules which constitute the droplet are produced by a chemical reaction which takes place at the droplet surface.
Q9. What is the concentration variation of a droplet?
Phys. 136, 074904 (2012)that the concentration variation is φ(x) = φe > 0 inside the droplet and φ(x) = −φe at the surrounding matrix.
Q10. What is the c-dependence of the Onsager coefficient?
21 The second term in Eq. (3) indicates consumption of c with the rate γ > 0 due to a chemical reaction and with c = c∞ for | r| → ∞ whereas the last term represents production of c, which occurs inside a droplet with radius R, whose center of mass is located at rG.
Q11. What is the support for this work?
This work was supported by the JSPS Core-to-Core Program “International research network for non-equilibrium dynamics of soft matter” and the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
Q12. What is the simplest equation for a fluid mixture?
The authors consider a fluid mixture where the free energy is given in terms of the local concentration difference φ = φA − φB byF {φ} = ∫ d r [ B(c)2 ( ∇φ)2 + fGL(φ) + f0(c)] , (1)where φA(φB) is the local concentration of the component A (B) and f0(c) = cln c.
Q13. What are the dimensions of the coefficients?
The dimensionless coefficients depend only on R̂ = Rβ and are given bym̂(R̂) = mDβ2τc, (48)τ̂ (R̂) = ττc, (49)ĝ(R̂) = g(Dβ)2τc. (50) These scaled coefficients have been evaluated numerically and plotted in Figs.
Q14. What is the normal component of the interface velocity?
From the left-hand side of Eq. (2), the authors note that the normal component V(a, t) of the interface velocity is given byV (a, t) = vα( r(a), t)nα(a).
Q15. What is the simplest way to derive the time-evolution equation for u?
(21) Equations (20) and (21) are manipulated in Appendix B asuα1 = − 8σ1R15 η0∫ da′nα(a′)cI (a′), (22)uα2 = σ1R25 η0∫ da′(δαδ − nα(a′)nδ(a′))(∇δc)I . (23)In Sec. III, the authors will derive the time-evolution equation for u from Eq. (19) with Eqs. (20) and (21) by solving Eq. (3) for the third component c.