Diffuse interface models in fluid mechanics

More documents

Recommendations

Info

$G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport ...$

Thus equation (27) readsP 0 (ρ v ) = P 0 (ρ l ) (29)This relation means that the liquid and vapor phases at equilibrium are equal. We recover theclassical condition of equilibrium of a planar interface.It is worth noting that the two conditions (23) and (29) are two equations of the two unknownsρ v and ρ l that can thus be determined. Moreover, these conditions have a simple graphical interpretationon the graph of the function F 0 (ρ). Indeed, the condition (23) means that the two slopesof the tangent to this graph at ρ v and ρ l are equal. The condition (29) means that the y-interceptsof these tangents are equal. Therefore, these two tangents are the same. Thus, ρ v and ρ l are definedby the bi-tangent to the graph of the function F 0 (ρ). Moreover, this allows to show that W (ρ) (definedby (25)) is actually the height between F 0 (ρ) and its bi-tangent.For the sake of simplicity (as will be shown hereafter), the function W (ρ) is often modeled asfollows:W (ρ) = A (ρ − ρ v ) 2 (ρ − ρ l ) 2 (30)where A is a parameter characteristic of the function F 0 (ρ) (cf. figure 2). This particular formallows to simplify the differential equation (24):This differential admits the following solutionwheredρdz = √2 Aλ (ρ − ρ v) (ρ l − ρ)ρ(z) = ρ l + ρ v2where h represents the interface thickness.+ ρ l − ρ v2h ˆ= 1ρ l − ρ v√λ2 A( z)tanh2 h(31)2.1.4 Surface excess energyThis analysis also allows to determine the energy “concentrated” at the interface. This energy,denote F ex is defined as follows:whereF ex ˆ=∫ zi−∞(F (ρ, ∇ρ) − F 0 (ρ v )) dz +ρ ex ˆ=∫ zi−∞∫ +∞z i(ρ(z) − ρ v ) dz +(F 0 (ρ l ) − F (ρ, ∇ρ)) dz − g eq ρ ex∫ +∞z i(ρ l − ρ(z)) dzwhere z i is any position. Actually it is straightforward to show that F ex does not depend on z i .Using the relations (25) and (24), it can be shown thatF ex =∫ +∞−∞λ( ) 2 ∫ dρρl√ √dz = 2 λ W (ρ) dρ (32)dzρ vThis energy concentrated at the interface is interpreted as the surface tension.It is worth noting that the above expression is general and is valid for any expression for thefunction F 0 (ρ) and therefore W (ρ). In the particular case where W (ρ) is given by (30), one findsthat the expression for the surface tension is the following:σ = (ρ l − ρ v ) 36√2 A λ (33)12
2.1.5 Equilibrium of a pherical inclusionIn section 1.2, we showed that surface tension has an effect on the outer bulk phases through theLaplace relation. Is it recovered by the van der Waals model?The equilibrium condition (21) is always valid and is in particular valid for a spherical inclusion(bubble or droplet) at equilibrium. In this case, we consider a spherical system of coordinateswhose origin is the center of the inclusion. The only variations that must accounted for are alongthe radial direction. The equilibrium condition thus readswhere r is the radial coordinate.This equation shows in particular thatg − λ d2 ρdr 2 − λ 2 rdρ= cste (34)drg(ρ s v) = g(ρ s l ) ˆ= g s eq (35)This relation shows that the bulk phase specific Gibbs free enthalpies are equal. It is importantto note that, at this point, the value of geq s is unknown and is a priori different from g eq (theequilibrium value of a planar interface). This implies in particular that the densities of the bulkphases (ρ s v and ρs l) are different from those of a planar interface.By multiplying equation (34) by dρ/dr and integrating from r = 0 to r yieldsF 0 (ρ) − F 0 (ρ s i ) − g s eq(ρ − ρ s i ) = λ 2( ) 2 ∫ dρr( ) 22 dρ+dr 0 η λ dη (36)dηwhere the subscript i denotes the interior phase (i.e. vapor for a bubble or liquid for a drop).This expression shows in particular thatF 0 (ρ s e ) − F 0 (ρ s i ) − gs eq (ρs e − ρs i ) = ∫ ∞where the subscript e denotes the exterior phase.Using the definition (28) of the pressure, this relation reads:P 0 si∫ ∞( ) 2− Pe 0 s 2 dρ=0 η λ dηdη0( ) 22 dρη λ dηdηThe variations of ρ are significant only in the vicinity of the radius R of the inclusion; thus∫ ∞0( ) 22 dρη λ dη ≃ 2 dη R∫ ∞0λ( ) 2 dρdηdηMoreover, if the density profile (determined by the integro-differential equation (36)) is onlyweakly influenced by the curvature effects the last integral term can be approximated by takingthe density profile of a planar interface, in which caseThereforeThis is the Laplace relation.∫ ∞0λP 0 si( ) 2 dρdη ≃ σdη− P 0 se≃ 2 σRThe graphical determination of the condition of equilibrium of a spherical inclusion is illustratedin figure 4.13
Page 8 and 9: is no longer the case and some of t
Page 10 and 11: where δ represents the variation,
Page 14: WW2σ/Rρρ2σ/RbubbledropFigure 4:
Page 17: q = −φ dρ − k ∇T (49)dtT =
Page 20 and 21: we show that particular boundary co
Page 22 and 23: 3.1.2 Equilibrium conditionsSince w
Page 24 and 25: φ ˆ= ∂F∂∇cThe following exp
Page 26 and 27: alance equation and a convection-di
Page 28 and 29: This system is thermodynamically co
Page 30 and 31: P 0modifiedP eqρ vρ lρFigure 10:
Page 32 and 33: Figure 11: Numerical simulation of
Page 34 and 35: in the Cahn-Hilliard model than in

Diffuse interface models in fluid mechanics

Create successful ePaper yourself

Delete template?

Save as template?