Definition 4.1. Viscosity sub/super/solution for the non-local eikonal equation)
4. The thin-rim (fast-nutrient-consumption or large-tumour) limit,
In the converse limit to the one we now discuss, i.e., asβ →0, we havec∼1 and the nutrient-rich formulations of [4] thus apply. The limit asβ→ ∞ also leads to significant simplifications, as we now describe.
(a) Darcy flow
Again denoting by−ν the inward normal distance from Γ(t), the boundary-layer scalings
t=
βT, ν =ζ/
β, p=P/β, v=V/
β, qν=Qν/
β, (27)
Mathematical Modelling of Nutrient-limited Tissue Growth 279 apply for Darcy flow, giving at leading order the one-dimensional ‘travelling-wave’
problem (in whichQν is of course independent ofζ)
∂
∂ζ ((Vν−Qν)n) =km(c)n−σkd(c)n,
∂
∂ζ((Vν−Qν)m) =σkd(c)n, n+m= 1,
∂
∂ζ
Dc(n)∂c
∂ζ
=kc(c)n, Vν =− 1
à(n)
∂P
∂ζ,
c= 1, P = 0, Vν=Qν onζ= 0, n→0, m→1, ∂c
∂ζ →0 asζ→ −∞,
(28)
from which P decouples, with the solution then depending on Vν and Qν only throughWν ≡Vν−Qν. The leading-order solutionsWν,n,mandcdepend only onζ and not onT or on the tangential components ofx (Qν depends on these, but is independent ofζ, being determined subsequently via the moving boundary problem formulated below). We note that the components ofV in the directions tangential to Γ are ofO(1/√
β). Since dWν
dζ =km(c)n, −∞< ζ <0, Wν = 0 onζ= 0 (29) it follows from (26) that
n= 1−σ, m=σ onζ= 0,
as is to be expected. As in the description of the large-time behaviour in [5], (26) determines a part of its solution the value of the constantsU andc∞, defined by
U=− lim
ζ→−∞Wν(ζ), c∞= lim
ζ→−∞c(ζ). (30)
We note that the above formulation (apart of course from the initial value prob- lem forP) holds irrespective of the choice of constitutive law (the leading-order problem being locally one-dimensional), so the quantities U and c∞ defined by (30) are of more general relevance.
The quantities in (30) represent the information we need from the inner (boundary-layer) analysis. In the outer (ν =O(1)) problem, i.e., in the necrotic core, we haven$1 (indeed, nis exponentially small), m∼1 and by matching with (30) we obtainc∼c∞and, to leading order,
∇ ãV= 0, V=− 1
àm∇p¯ in Ω(T),
¯
p= 0, Qν =Vãν+U on Γ(T),
(31) wherep= ¯p/√
β is the appropriate pressure scaling whenàm=O(1). The second condition on Γ(T) is a novel feature of this formulation, accounting for theO(1)
change in Vν between the matching region and the true interface which results from cell division in the viable rim (i.e., inζ=O(1)). The solution to the problem (31) is of course the trivial one
¯
p= 0, V=0, Qν =U, (32)
the last of which implies that the outward normal velocity of the interface is con- stant; nevertheless, we record the full formulation (31) here in order to generalise it subsequently, in particular to the two-phase case in which the properties of the surrounding material cannot be ignored. We remark that there is also a dis- tinguished limit àm = O(1/√
β) in which a Hele-Shaw type formulation results which contains a ‘kinetic undercooling’ regularisation.
(b) Stokes flow
Other than forp, which is ofO(1), the scalings (27) again apply in the boundary layer, and Wν, n, m and c are determined, exactly as for Darcy flow, with the definitions (30) pertaining. The flow problem is rather more delicate, however, requiring the calculation ofO(1/√
β) terms in the momentum equations and re- quiring derivations of “viscous shell” equations, somewhat akin to those in [2] but with significant differences due to cell division. Denoting co-ordinates on Γ(T) by τ andυ, using (27) we have (in an obvious notation)
στ τ ∼ −
p+2
3à(n)dWν dζ
+ 1
√βσ(1)τ τ, συυ∼ −
p+2
3à(n)dWν
dζ
+ 1
√βσ(1)υυ, σνν∼ −
p−4
3à(n)dWν
dζ
+ 1
√βσ(1)νν,
(33)
στ υ∼ 1
√βσ(1)τ υ, στ ν∼à(n)∂Vτ
∂ζ + 1
√βσ(1)τ ν, συν∼à(n)∂Vυ
∂ζ + 1
√βσυν(1) (34) where p denotes the leading-order pressure; in conventional incompressible flow problems it follows from the continuity equation that Vν is independent of ζ, but here we instead have (29) and some of the resulting balances accordingly differ. Nevertheless, we can read off the relevant asymptotic formulations of the momentum equations directly from [2], which we essentially follow in parametrising the free surface in such a way that lines of constantτ andυare lines of curvature of Γ(T); we write Γ(T) asx=xΓ(τ, υ, T) and define
aτ = ∂xΓ
∂τ
, aυ= ∂xΓ
∂υ . At leading order the momentum equations then read
∂
∂ζ
aτaυσ(0)τ ν
= ∂
∂ζ
aτaυσυν(0)
= ∂
∂ζ
aτaυσ(0)νν
= 0,
Mathematical Modelling of Nutrient-limited Tissue Growth 281 where here and henceforthaτandaυdenote leading-order quantities. Sinceστ ν= συν =σνν= 0 on Γ(T), it follows that
στ ν(0)=συν(0)=σ(0)νν = 0 (35) and hence atO(1)
p= 4
3à(n)dWν
dζ , σ(0)τ τ =σ(0)υυ =−2à(n)dWν
dζ , ∂Vτ
∂ζ = ∂Vυ
∂ζ = 0. (36) AtO(1/√
β), we have (from [2], using (34) and (35))
∂
∂τ
aυστ τ(0)
+ ∂
∂ζ
aτaυσ(1)τ ν
−∂aυ
∂τ συυ(0)= 0,
∂
∂υ
aτσυυ(0)
+ ∂
∂ζ
aτaυσυν(1)
−∂aτ
∂υ στ τ(0)= 0,
∂
∂ζ
aτaυσ(1)νν
−aτaυ
κτστ τ(0)+κυσυυ(0)
= 0,
whereκτ andκυare the principal curvatures of Γ(T) (our sign convention differing from that of [2]). Using (36) (observing thataτ andaυare independent ofζ, while στ τ(0) andσυυ(0) are independent ofτ andυ) we thus obtain
∂
∂ζστ ν(1)= ∂
∂ζσυν(1)= 0, ∂
∂ζσνν(1)=−4κà(n)dWν dζ ,
whereκ= (κτ+κυ)/2 is the leading-order mean curvature of Γ(T). Hence σ(1)τ ν =σ(1)υν = 0, σνν(1)= 4κ
0 ζ
km(c(ζ))à(n(ζ))n(ζ)dζ, (37) We are now in a position to formulate the leading-order problem in the core, whereinnis again exponentially small andc∼c∞. Writing
σij = ¯σij/
β, p= ¯p/
β we thus obtain the standard Stokes flow problem
∇ ãV= 0, ∂σ¯ij
∂xj = 0, σ¯ij =−pδ¯ ij+àm ∂Vi
∂xj +∂Vj
∂xi
in Ω(T); (38) this is subject to the moving boundary conditions
¯
σijνj= 2γκνi, Qν=Vãν+U on Γ(T), (39) where we have matched with (37) and defined
γ= 2 0
−∞
km(c(ζ))à(n(ζ))n(ζ)dζ . (40) Cell division in the viable rim thus generates a surface-tension-like term, but with a negative coefficient of surface tension, −γ (determined by (40)). This is un- surprising in that cell division generates compressive stresses in the rim, leading to the likelihood of “viscous-buckling” instabilities; the formulation (38)–(39) is of course ill-posed, and in practice buckling will presumably occur on a length
scale comparable to the thickness of the viable rim (i.e., of O(1/√
β)), causing the thin-rim analysis above to cease to be valid. Were the U term absent from (39), the formulation would be equivalent to the time-reversal of the Stokes flow problem with positive surface tension, discussed in [1] for example, and for Γ(T) initially analytic a solution can be expected to exist for some finite time, despite the problems ill-posedness.