It is well-known that viscoelasticity is a material property that exhibits both viscous and elastic characteristics with deformation. In particular, an elastic fluid strains when it is stretched and quickly returns to its original state once the stress is removed. In this review, we first introduce some mathematical results, which exhibit the stabilizing effect of elasticity on the motion of viscoelastic fluids. Then we further briefly introduce similar stabilizing effect in the elastic fluids.
Citation: Fei Jiang. Stabilizing effect of elasticity on the motion of viscoelastic/elastic fluids[J]. Electronic Research Archive, 2021, 29(6): 4051-4074. doi: 10.3934/era.2021071
[1] | Fei Jiang . Stabilizing effect of elasticity on the motion of viscoelastic/elastic fluids. Electronic Research Archive, 2021, 29(6): 4051-4074. doi: 10.3934/era.2021071 |
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It is well-known that viscoelasticity is a material property that exhibits both viscous and elastic characteristics with deformation. In particular, an elastic fluid strains when it is stretched and quickly returns to its original state once the stress is removed. In this review, we first introduce some mathematical results, which exhibit the stabilizing effect of elasticity on the motion of viscoelastic fluids. Then we further briefly introduce similar stabilizing effect in the elastic fluids.
Viscoelastic materials include a wide range of fluids with elastic properties, as well as solids with fluid properties. The models of viscoelastic fluids formulated by Oldroyd [51,52] (see [56,53,5] for alternative derivations and perspectives), in particular, the classical Oldroyd-B model, have been studied by many authors (see [10,79,69] and the references cited therein), since the pioneering work of Renardy [58] and Guillopé–Saut [15].
It is well-known that viscoelasticity is a material property that exhibits both viscous and elastic characteristics with deformation. In particular, an elastic fluid strains when it is stretched and quickly returns to its original state once the stress is removed. This means that the elasticity will have a stabilizing effect in the motion of viscoelastic fluids. Moreover, the larger elasticity is, the stronger this stabilizing effect will be. This stabilizing effect has been widely investigated by many authors. In particular, recently the stabilizing effect in viscoelastic fluids was mathematically verified by Jiang et.al. based on the following three-dimensional (3D) incompressible Oldroyd model, which includes a viscous stress component and a stress component for a neo-Hookean solid:
{ρvt+ρv⋅∇v+∇p−μΔv=κdiv(UUT),Ut+v⋅∇U=∇vU,div v=0, | (1.1) |
where the unknowns
For strong solutions of the both Cauchy and initial-boundary value problems for (1.1), the authors in [49,50,9,45] have established the global(-in-time) existence of solutions in various functional spaces whenever the initial data is a small perturbation around the rest state
In the investigation of the well-posedness problem of (1.1) with initial value condition and the following non-slip boundary value condition of velocity
v|∂Ω=0, |
where
12ddt∫Ω(ρ|v|2+κ|U|2)dx+μ∫Ω|∇v|2dx=0. |
However, it is not easy to see from the above basic energy identity how elasticity effects the motion of viscoelastic fluids. To clearly see the effect, we shall rewrite (1.1) in Lagrangian coordinates.
To this purpose, let the flow map
{ζt(y,t)=v(ζ(y,t),t)in Ω×(0,∞),ζ(y,0)=ζ0(y)in Ω. | (1.2) |
Here and in what follows, the notations
˜U(y,t):=∇ζ(y,t), i.e., ˜Uij:=∂jζi(y,t) for 1⩽i, j⩽3. |
When we study this deformation tensor in Eulerian coordinates, it is defined by
U(x,t):=∇ζ(ζ−1(x,t),t), |
where we have assumed that
U0=∇ζ0(ζ−10,0). | (1.3) |
Next, we proceed to rewrite the elasticity in Lagrangian coordinates. For this purpose, we should define the matrix
∇Aw:=(∇Aw1,∇Aw2,∇Aw3)T,∇Awi:=(A1k∂kwi,A2k∂kwi,A3k∂kwi)T, | (1.4) |
divA(f1,f2,f3)T:=(divAf1,divAf2,div Af3)T, divAfi:=Alk∂kfil,ΔAw:=(ΔAw1,ΔAw2,ΔAw3)TandΔAwi:=divA∇Awi | (1.5) |
for vector functions
It is well-known that
∂k(JAik)=0. | (1.6) |
By (1.6) and the relation
div(UUT/detU)|x=ζ=divA(∇ζ∇ζT/J)=div(AT(∇ζ∇ζT))/J=Δη/J. | (1.7) |
Now we assume
div(UUT)|x=ζ=Δη. |
Let
(u,q)(y,t)=(v,p)(ζ(y,t),t) for (y,t)∈Ω×(0,∞). |
By virtue of (1.1)
{ηt=u,ρut−μΔAu+∇Aq=κΔη,div Au=0, | (1.8) |
which couples with the boundary-value condition:
(η,u)|∂Ω=(0,0). | (1.9) |
It is easy to derive from (1.8) and (1.9) the following basic energy identity:
12ddt∫Ω(ρ|u|2+κ|∇η|2)dy+μ∫Ω|∇Au|2dy=0. |
Let
12∫Ω(ρ|u|2+κ|∇η|2)dy+μ∫t0∫Ω|∇Au|2dydτ=12∫Ω(ρ|u0|2+κ|∇η0|2)dy=:I0, |
which implies that
∫Ω|∇η|2dy⩽2I0/κ. | (1.10) |
In particular, for given the initial (perturbation) mechanical energy
∫Ω|∇η|2dy→0 as κ→∞. |
Noting that
Based on the above analysis result, Jiang et.al. mathematically proved that, under the proper large elasticity coefficient, the elasticity can inhibit flow instabilities such as Rayleigh–Taylor (RT) instability and thermal (or convective) instability. In addition, they further established the existence of large solutions of some class for the system of equations (1.1). We will review their results one by one.
Consider two completely plane-parallel layers of stratified (immiscible) pure fluids, the heavier one is on top of the lighter one and both are subject to the earth's gravity. It is well-known that such an equilibrium state is unstable to sustain small disturbances, which will grow with time and lead to a release of potential energy, as the heavier fluid moves down under the gravitational force, and the lighter one is displaced upwards. This phenomenon was first studied by Rayleigh [57] and then Taylor [68] and is called therefore the RT instability. In the last decades, this phenomenon has been extensively investigated in mathematical, physical and numerical communities, see [7,71,17] for examples. It has been also widely investigated how other physical factors, such as rotation [7], internal surface tension [18,75,28], magnetic fields [35,29,30,34,73,74,32] and so on, influence the dynamics of the RT instability.
The RT instability was also often investigated in various models of viscoelastic fluids from the physical point of view, see [4] and [63] for examples. Later Jiang et.al. used the model (1.1) to investigate the effect of elasticity on the RT instability in viscoelastic fluids, see [36] and [39] for the stratified and non-homogeneous cases, resp.. Now we only introduce the RT instability in the stratified viscoelastic fluids.
To begin with, let us recall the RT problem of stratified viscoelastic fluids [36]:
{ρ±∂tv±+ρ±v±⋅∇v±+divS±(pg±,v±,U±)=0in Ω±(t),∂tU±+v±⋅∇U±=∇v±U±in Ω±(t),divv±=0in Ω±(t),dt+v1∂1d+v2∂2d=v3on Σ(t),[[v±]]=0,[[S±(pg±,v±,U±)−gdρ±I]]ν=ϑCνon Σ(t),v±=0on Σ±,(v±,U±)|t=0=(v0±,U0±)in Ω±(0),d|t=0=d0on Σ(0). | (2.1) |
Next, we should explain the notations in the above (stratified) VRT problem (2.1).
For each given
Σ(t):={(xh,x3)|xh∈T, x3:=d(xh,t)}, |
where
T:=T1×T2, | (2.2) |
The subscripts
Ω+(t):={(xh,x3)|xh∈T, d(xh,t)<x3<h} |
and
Ω−(t):={(xh,x3)|xh∈T, −l<x3<d(xh,t)}, |
are the domains of upper and lower fluids, resp..
Σ+:=T×{h}, Σ−:=T×{−l}, Σ:=T×{0}. |
For given
[[ρ±]]>0. |
S±(pg±,v±,U±):=pg±I−μ±Dv±−κ±ρ±(U±UT±−I), |
where
For functions
C:=Δhd+(∂1d)2∂22d+(∂2d)2∂21d−2∂1d∂2d∂1∂2d(1+(∂1d)2+(∂2d)2)3/2. |
Finally, we briefly explain the physical meaning of each identity in (2.1). The equations (2.1)
[[S±(p±,v±,U±)]]ν=ϑCνon Σ(t), |
which represents that the jump in the normal stress is proportional to the mean curvature of the free surface multiplied by the normal to the surface [43,76]. The non-slip boundary condition of the velocities on both upper and lower fixed flat boundaries are described by (2.1)
The problem (2.1) enjoys an equilibrium state (or rest) solution:
∂3ˉpg=0 and [[ˉpg−gˉdρ±]]=0. |
Without loss of generality, we assume that
To simply the representation of the problem (2.1), we introduce the indicator function
ρ=ρ+χΩ+(t)+ρ−χΩ−(t), μ=μ+χΩ+(t)+μ−χΩ−(t), κ=κ+χΩ+(t)+κ−χΩ−(t),v=v+χΩ+(t)+v−χΩ−(t), U=U+χΩ+(t)+U−χΩ−(t), p=p+χΩ+(t)+p−χΩ−(t),v0=v0+χΩ+(0)+v0−χΩ−(0), U0=U0+χΩ+(0)+U0−χΩ−(0),S(pg,v,U):=pgI−μDv−κ(UUT−I). | (2.3) |
To focus on the elasticity effect upon the RT instability, from now on we only consider the case
d=d−0, v=v−0, V=U−I and σ=pg−ˉpg, |
one has a VRT problem in the perturbation form (with zero surface tension):
{ρvt+ρv⋅∇v+divS(σ,v,V+I)=0in Ω(t),Vt+v⋅∇V=∇v(V+I)in Ω(t),divv=0in Ω(t),dt+v1∂1d+v2∂2d=v3on Σ(t),[[v]]=0, [[S(σ,v,V+I)−gρdI]]ν=0on Σ(t),v=0on Σ+−,(v,V)|t=0=(v0,V0)in Ω(0),d|t=0=d0on Σ(0), | (2.4) |
where
If ignoring the effect of elasticity (i.e.,
{ρvt+ρv⋅∇v+divS(σ,v,V+I)=0in Ω(t),divv=0in Ω(t),dt+v1∂1d+v2∂2d=v3on Σ(t),[[v]]=0, [[S(σ,v,0)−gρd]]ν=0on Σ(t),v=0on Σ+−,(v,V)|t=0=(v0,V0)in Ω(0),d|t=0=d0on Σ(0), |
It is well-known that the classical RT problem is unstable, see [37,55] and the references cited therein. However, we will see that the VRT problem (2.4) may be stable due to the presence of elasticity.
It is well-known that the movement of the free interface
Assume that there are invertible mappings
ζ0±:Ω±→Ω±(0), |
such that
Σ(0)=ζ0±(Σ), Σ±=ζ0±(Σ±) |
and
det∇ζ0±=1, | (2.5) |
where
It should be noted that we define that
{ζt(y,t)=v(ζ(y,t),t)in Ω,ζ(y,0)=ζ0(y)in Ω. | (2.6) |
Denote the Eulerian coordinates by
Since
SA(q,u,η):=qI−μDAu−κDη−κ∇η∇ηT,DAu:=∇Au+∇AuT, Dη:=∇η+∇ηT, →n:=Ae3/|Ae3|, |
and refer to (1.4) for the definition of the operator
Now we further assume that
(u,˜U,q)(y,t)=(v,U,σ)(ζ(y,t),t) for (y,t)∈Ω×(0,∞), |
then,
{ηt=uin Ω,ρut+divASA(q,u,η)=0in Ω,divAu=0in Ω,[[η]]=[[u]]=0, [[SA(q,u,η)−gρη3I]]→n=0on Σ,(η,u)=0on Σ+−,(η,u)|t=0=(η0,u0)in Ω | (2.7) |
with
Before stating the main results for the transformed VRT problem (2.7), we shall introduce some notations of Sobolev spaces: for
Hk(Ω):=Wk,2(Ω), H1(Ω−):=W1,2(Ω−), H10(Ω):={w∈H1(Ω−)|w|Σ+−=0},H1σ(Ω):={w∈H10(Ω)|divw=0}, Hk0(Ω):={w∈H10(Ω)|w∈Hk(Ω)}, |
where
Theorem 2.1. [36,Theorem 2.1] Under the stability condition
Cr<1, | (2.8) |
where
Cr:=sup0≠w∈H1σ(Ω)2g[[ρ]]‖w3‖2L2(T)‖√κD(w)‖2L2(Ω), |
there is a sufficiently small constant
(1) the incompressible condition
(2) the volume-preserving condition
(3)
(4) the initial data
[[SA0(0,u0,η0)→n0−→n0⋅(SA0(0,u0,η0)→n0)→n0]]=0, |
there exists a unique global solution
eωt(‖u(t)‖2H2(Ω)+‖η(t)‖2H3(Ω)+‖(ut,∇q)(t)‖2L2(Ω)+‖[[q(t)]]‖2H1/2(T))+∫t0eωτ(‖(u,η)(t)‖2H3(Ω)+‖(ut,∇q)(t)‖2H1(Ω)+|[[q(t)]]|2H3/2(T))dτ⩽c(‖u0‖2H2(Ω)+‖η0‖2H3(Ω)). | (2.9) |
Here
There are some remarks of Theorem 2.1.
Remark 1. Theorem 2.1 still holds when
κC:=sup0≠w∈H1σ(Ω)2g[[ρ]]‖w3‖2L2(T)‖Dw‖2L2(Ω). |
Remark 2 The discriminant
g[[ρ]]˜c⩽Cr⩽g[[ρ]]min{hκ+,lκ−}, |
where the positive constant
(1)
(2)
Remark 3. For the current viscoelastic RT problem,
Remark 4. We remark on how to use the stability condition (2.8). Let us recall the basic energy identity of (2.7):
12ddtE1+μ2‖DAu‖2L2(Ω)=g[[ρ]]∫Ση3˜Ae3⋅udy1dy2−∫Ωκ(∇η∇ηT:∇AuT+(Dη):∇˜AuT)dy, |
where
E1(η):=g[[ρ]]‖η3‖2L2(T)−‖√κDη‖2L2(Ω)/2. |
We call
Obviously, to derive the a priori stability (2.9), we shall first pose a stability condition, which makes sure the mechanical energy to be positive definite. It is easy to see that the condition (2.8) is just the stability condition, for which we look. In particular, under the stability condition, we have the following stabilizing estimate: there exists a positive constant
‖w‖2H1(Ω)⩽−cE1(w) for any w∈H1σ(Ω). | (2.10) |
Thanks to (2.10), we can use an energy method to establish Theorem 2.1.
The failure of the stability condition (2.8) results in the following instability result, which exhibits that the elasticity can not inhibit RT instability for small elasticity coefficient.
Theorem 2.2. [37,Theorem 1.1] Under the instability condition
Cr>1, | (2.11) |
a zero solution to the transformed VRT problem is unstable in the Hadamard sense, that is, there are positive constants
(η0,u0,q0):=δ(˜η0,˜u0,˜q0)+δ2(ηr,ur,qr)∈H30(Ω)×H20(Ω)×H1(Ω), | (2.12) |
there is a unique strong solution
‖χ3(Tδ)‖L1(T), ‖χ3(Tδ)‖L1(Ω), ‖∂3χ3(Tδ)‖L1(Ω), ‖χh(Tδ)‖L1(Ω),‖∂3χh(Tδ)‖L1(Ω), ‖divhχh(Tδ)‖L1(Ω), ‖A3k∂kχ3(Tδ)‖L1(Ω), ‖A3k∂kχh(Tδ)‖L1(Ω), ‖(A1k∂kχ1+A2k∂kχ2)(Tδ)‖L1(Ω)⩾ϵ | (2.13) |
for some escape time
divA0u0=0inΩ,[[SA0(q0,u0,η0)−gρη03I]]→n0=0onΣ, |
where
We give some remarks for Theorem 2.2.
Remark 5. In this review, we only introduce the results of stability and instability for the transformed VRT problem (2.7) without surface tension. Interested readers can further refer to [78,37] for the case with surface tension. In addition, the corresponding version of compressible case can be found in [31].
Remark 6. It should be noted that, if
ζh(yh,0):R2→R2 is a C1-diffeomorphic mapping,ζ(y):¯Ω↦¯Ω is a C0-homeomorphism mapping,ζ±(y):Ω±↦ζ±(Ω±) are C1-diffeomorphic mappings, |
where
Remark 7. Finally we remark on how to use the instability condition (2.11). The eigenvalue problem of the linearized VRT problem reads as follows:
{λ˜η=˜u in Ω,λρ˜u+∇˜q=μΔ˜u+κρD˜η in Ω,div˜u=0 in Ω,[[˜u]]=0, [[(˜q−gρη3)I−D(μ˜u+κ˜η)]]e3=0 on Σ,(˜η,˜u)=0 on Σ+−. |
We expect to look for the largest positive eigenvalue
λ2=−infw∈A{12‖√λμDw‖2L2(Ω)−E1(w)}>0, | (2.14) |
where
E1(w)>0 for some w∈H1σ(Ω). |
This means that the variational problem (2.14) makes sense. Thus we can use the modified variational method in [18] to obtain the largest positive eigenvalue
u(y,t)=˜u(y)eλt, q(y,t)=˜q(y)eλt, η(y,t)=˜η(y)eλt, |
which is the solution with large exponent growth in time to the linearized transformed VRT problem. Thanks to this linear instability result, we can use a bootstrap instability method in [16] to establish Theorem 2.2.
It is well-known that, in the development of the RT instability, gravity first drives the third component
Thermal instability often arises when a fluid is heated from below. The classic example of this is a horizontal layer of fluid with its lower side hotter than its upper. The basic state is then one of rests with light and hot fluid below heavy and cool fluid. When the temperature difference across the layer is great enough, the stabilizing effects of viscosity and thermal conductivity are overcome by the destabilizing buoyancy, and an overturning instability ensues as thermal convection: hotter part of fluid is lighter and tends to rise as colder part tends to sink according to the action of the gravity force [12]. The phenomenon of thermal convection itself had been recognized by Rumford [60] and Thomson [70]. However, the first quantitative experiment on thermal instability and the recognition of the role of viscosity in the phenomenon are due to Bénard [1], so the convection in a horizontal layer of a fluid heated from below is called Bénard convection. For many years, the question for understanding of convective flows has motivated numerous theoretical, numerical, and experimental studies [22,12].
Thermal convection in viscoelastic fluids is also a subject of considerable interest in contemporary fluid flow and heat transfer researches. The first work which deals directly with thermal instability of a viscoelastic fluid appears to be that of Herbert who studied plane Couette flow heated from below [21]. He found that a finite elastic stress in the undisturbed state is necessary for elasticity to affect stability. Since Herbert's pioneering work, many physicists have continued to develop the linear theory and nonlinear numerical method in the studies of thermal instability in viscoelastic fluids, see [11,14,47,40,59,61,65,66] and the references cited therein. Moreover, it has also been widely investigated how thermal convection in viscoelastic fluids evolves under the effects of other physical factors, such that rotation [13,42,67], magnetic fields [2,3,54], the porous media [77,62] and so on.
As Rosenblat pointed out in [59], the nature of (linear) convective solution depends strongly on the particular constitutive relation used to characterize the viscoelasticity. For certain models and certain parameter ranges the convection is supercritical and stable, while for other models and parameter ranges it can be subcritical and unstable. In other words, the influence of elasticity on the thermal convection is closely related to the choice of model describing the motion of a viscoelastic fluid.
Recently Jiang–Liu mathematically prove the phenomenon of inhibition of thermal instability by elasticity by the following (nonlinear) Boussinesq approximation equations of viscoelastic fluids [38]:
{vt+v⋅∇v+∇p/ρ=g(α(Θ−Θb)−1)e3+νΔv+κdiv(UUT)/ρ,Θt+v⋅∇Θ=kΔΘ,Ut+v⋅∇U=∇vU,divv=0. | (3.1) |
We shall explain the mathematical notations in (3.1). The unknowns
The rest state of the Boussinesq approximation equations (3.1) can be given by
∇ˉp=gρ(α(ˉΘ−Θb)−1)e3,ΔˉΘ=0. |
For the simplicity, we consider that
ˉΘ=Θb−ϖx3for0⩽x3⩽h, |
where
Ωh:=T×(0,h) and Ω:=T×(0,1), | (3.2) |
where
Denoting the perturbation around the equilibrium state by
v=v−0,θ=Θ−ˉΘ,V=U−I,β=p/ρ−ˉp/ρ, |
then,
{vt+v⋅∇v+∇β=gαθe3+νΔv+κdiv((V+I)(V+I)T)/ρ,θt+v⋅∇(θ+ˉΘ)=kΔθ,Vt+v⋅∇V=∇v(V+I),divv=0. | (3.3) |
We shall pose the following initial-boundary value conditions for the well-posedness of (3.3):
(v,θ)|t=0=(v0,θ0), | (3.4) |
(v,θ)|∂Ωh=0. | (3.5) |
We call the initial-boundary value problem (3.3)–(3.5) the viscoelastic Rayleigh–Bénard problem.
Now we set the unknowns in Lagrangian coordinates by
(u,ϑ,q)(y,t)=(v,θ,β)(ζ(y,t),t) for (y,t)∈Ωh×(0,∞), |
where
{ζt=u,ut−νΔAu+∇Aq=gαϑe3+κΔη/ρ,ϑt=kΔAϑ+ϖu⋅∇Aζ3,divAu=0 | (3.6) |
with initial-boundary value conditions
(ζ,ϑ,u)|t=0=(ζ0,ϑ0,u0)and(u,ϑ,ζ−y)|∂Ωh=0. |
From now on, we still define that
Let
y∗=y/h,t∗=νt/h2η∗=η/h,u∗=hu/ν,θ∗=Rkϑ/hϖνandq∗=h2q/ν2 |
to rewrite (3.6) as the following non-dimensional form defined in
{ηt=u,ut−ΔAu+∇Aq=Rϑe3+QΔη,Pϑϑt−ΔAϑ=Ru⋅∇Aζ3,divAu=0, | (3.7) |
where
(η,u,ϑ)|t=0=(η0,u0,ϑ0)and(η,u,ϑ)|∂Ω=0. | (3.8) |
We call the initial-boundary value problem (3.7)–(3.8) the transformed VRB problem. Before stating the main results for the transformed VRB problem (2.7), we shall introduce some notations of Sobolev spaces in this section: for
Hk(Ω):=Wk,2(Ω), Hk0(Ω):={w∈Hk(Ω)|w|∂Ω=0},Hkσ(Ω):={w∈Hk0(Ω)|divw=0}, H_k(Ω):={w∈Hk(Ω)|∫Ωwdy=0}. |
We have the following stability result for the transformed VRB problem:
Theorem 3.1. [38,Theorem 2.1] Under the condition
Q>R2max{4P2ϑ+136π2,6Pϑπ2}, | (3.9) |
then there is a sufficiently small
‖η0‖2H3(Ω)+‖(u0,ϑ0)‖2H2(Ω)⩽δ, |
there exists a unique strong solution
eωt(‖η(t)‖2H3(Ω)+‖(u,ϑ)(t)‖2H2(Ω)+‖(∇q,ut,ϑt)‖2L2(Ω))+∫t0eωτ(‖(η,u,ϑ)(t)‖2H3(Ω)+‖(∇q,ut,ϑt)‖2H1(Ω))dτ⩽c(‖η0‖2H3(Ω)+‖(u0,ϑ0)‖2H2(Ω)). |
The above positive constants
The viscoelastic Rayleigh–Bénard problem (3.3)–(3.5) in the absence of deformation tensor reduces to the classical RB (i.e., Rayleigh–Bénard) problem
{vt+v⋅∇v+∇β=gαθe3+νΔv,θt+v⋅∇(θ+ˉΘ)=kΔθ,divv=0,(v,θ)|t=0=(v0,θ0),(v,θ)|∂Ω=0. |
We mention that there exists a threshold
1R0:=sup(ϖ,ϕ)∈H1σ(Ω)×H10(Ω)2∫Ωϖ3ϕdy‖∇(ϖ,ϕ)‖2L2(Ω), |
such that if the convection condition
Next we briefly explain why the stability condition is given by the form (3.9). Let us first recall the basic energy identity:
12ddt(Q‖∇η‖2L2(Ω)+‖u‖2L2(Ω)+Pϑ‖ϑ‖2L2(Ω))+‖∇(u,ϑ)‖2L2(Ω)=2R∫Ωu3ϑdy+ int. (i.e., integrals involving nonlinear terms). |
By the idea of the inhibition of instability by the elasticity, (3.7)
12ddt(E2(η,ϑ)+‖u‖2L2(Ω))+‖∇(u,ϑ)‖2L2(Ω)=2RPϑ∫Ω∇η3⋅∇ϑdy+int., | (3.10) |
where
E2(η,ϑ):=Q‖∇η‖2L2(Ω)+2R2Pϑ‖η3‖2L2(Ω)+Pϑ‖ϑ‖2L2(Ω)−4R∫η3ϑdy. |
To control the first integral on the right hand of (3.10), we shall derive from (3.7)
ddt(∫∂αhη⋅∂αhudy+12‖∇∂αhη‖2L2(Ω))+Q‖∇∂αhη‖2L2(Ω)=‖∂αhu‖2L2(Ω)+R∫Ω∂αhη3∂αhϑdy+int., |
Here and in what follows,
ddtE2(η,u,ϑ)+D2(η,u,ϑ)=int., |
where
E2(η,u,ϑ):=12(2(E2(η,ϑ)+‖u‖2L2(Ω))+∑0⩽α⩽1‖∂αh∇η‖2L2(Ω))+∑0⩽α⩽1∫∂αhη⋅∂αhudy,D2(η,u,ϑ)=2‖∇(u,ϑ)‖2L2(Ω)+Q∑0⩽α⩽1‖∂αh∇η‖2L2(Ω)−∑0⩽α⩽1‖∂αhu‖2L2(Ω)−4RPϑ∫Ω∇η3⋅∇ϑdy−R∑0⩽α⩽1∫∂αhη3∂αhϑdy. |
It is easy to see that, for sufficiently large
E2(η,u,ϑ) and D2(η,u,ϑ) are positive definite. |
Moreover precisely, they are equivalent to, for sufficiently large
‖(η,u,ϑ)‖2L2(Ω)+∑0⩽α⩽1‖∂αh∇η‖2L2(Ω) and ‖(u,ϑ)‖2H1(Ω)+∑0⩽α⩽1‖∂αh∇η‖2L2(Ω), |
resp.. Based on the above idea, we can easily find out the stability condition (3.9), which is relatively more complicated than the stability condition (2.8) in the transformed VRT problem (2.7). Moreover, we can establish Theorem 3.1 under the stability condition by an energy method.
In addition, we also find that the thermal instability can occur if
Theorem 3.2. [37,Theorem 2.2] We define that
A:={(ϖ,ϕ)∈H1σ(Ω)×H10(Ω)|‖ϖ‖2L2(Ω)+Pϑ‖ϕ‖2L2(Ω)=1},B:={(ϖ,ϕ)∈A|1R0=2∫Ωϖ3ϕdy‖∇(ϖ,ϕ)‖2L2(Ω)}, ξ:=sup(ϖ,ϕ)∈B{∫Ωϖ3ϕdy}. | (3.11) |
Let
√Q⩽min{1,2(R−R0)ξ2H+3√H,2(R−R0)ξ1+√H}, | (3.12) |
where we have defined that
(˜η0,˜u0,˜ϑ0,ηr,ur)∈H3σ(Ω)×H2σ(Ω)×H20(Ω)×H30(Ω)×H20(Ω), |
such that, for any
(η0,u0,ϑ0):=δ(˜η0,˜u0,˜ϑ0)+δ2(ηr,ur,0)∈H30(Ω)×H20(Ω)×H20(Ω), |
there is a unique strong solution
Remark 8. Similarly to (6), if
ζ:=η+y:¯Ω↦¯Ω is a C0-homeomorphic mapping,ζ:R2×(0,1)↦R2×(0,1) are C1-diffeomorphic mappings. |
Thus we can also recover the exponential stability of the VRB problem (3.3)–(3.5) in Eulerian coordinates from Theorems 2.1 and 2.2 by an inverse transformation.
Similarly to Theorem 2.2 we can use a bootstrap instability method in [16] to establish Theorem 3.2. However, the construction of unstable linear solutions is relatively more complicated. In fact, following the argument of linear instability of the linearized VRT problem in Remark 7, we will face the following variational problem
λ=−inf(u,ϑ)∈A(‖∇u‖2L2(Ω)+‖∇ϑ‖2L2(Ω)+Q‖∇u‖2L2(Ω)λ−2R∫Ωu3ϑdy), |
where
The two stability results in the previous sections present that the elasticity has the stabilizing effect in the motion of viscoelastic fluids under the small perturbations. In this section we will consider that this stabilizing effect also be observed under the large perturbations.
From now on, we define
(u,q)(y,t)=(v,p)(ζ(y,t),t) for (y,t)∈T×R+. |
Then the evolution equations for
{ηt=u,ρut−μΔAu+∇Aq=κΔη,div Au=0, | (4.1) |
with initial data
(η,u)|t=0=(η0,u0)in T. | (4.2) |
Now we further define some simplified function spaces:
Hk(T):=Wk,2(T), H_k(T):={w∈Hk(T)|(w)T=0},H3∗,1(T):={w∈H3(T)|det(∇w+I)=1, ζ:=η+y:R3↦R3 is a C1-diffeomorphic mapping}. |
Here and in what follows,
Next we introduce the first result, which is concerned with the existence of strong solutions to the initial value problem (4.1)–(4.2) in some classes of large initial data:
Theorem 4.1. [33,Theorem 1.2] There are constants
κ⩾1c2max{2√c1Ih0(u0,η0),(4c1Ih0(u0,η0))2}, |
where
‖(u,√κ∇η)‖2H2(T)+∫t0‖∇(u,√κη)‖2H2(T)dτ⩽cIh0(u0,η0). | (4.3) |
Here and in what follows the positive constants
Theorem 4.1 exhibits the global existence of strong solutions to the initial value problem of (1.1) defined in a spatially periodic domain, when the initial deformation (i.e.,
We briefly sketch the proof idea of Theorem 4.1. Recalling (1.10), we easily conclude that the equation (4.1) may be approximated by the following linear system of equations for sufficiently large
{ηlt=ul,ρult−μΔul+∇ql=κΔηl,div ul=0. | (4.4) |
Since the linear system of equations has global solutions with large initial data, we could expect that the initial value problem (4.1)–(4.2) may also admit a global large solution for sufficiently large
In order to obtain Theorem 4.1, the key step is to derive the a priori estimate (4.3) under sufficiently large
sup0⩽t⩽T‖(u(t),√κ∇η(t))‖H2(T)⩽K/2, |
provided that
sup0⩽t⩽T‖(u(t),√κ∇η(t))‖H2(T)⩽K for any given T>0 |
and
max{K,K4}/κ∈(0,δ2]. |
Based on the above fact and the existence of a unique local solution, we can immediately obtain Theorem 4.1.
Next we further introduce the second result, which is concerned with the properties of the solution
Theorem 4.2. [33,Theorem 1.2] Let
(1) Exponential stability of
ec3t‖(ˉu,√κ∇η)‖2H2(T)+∫t0(‖ˉu‖2H3(T)+κ‖∇η‖2H2(T))ec3τdτ⩽cIh0(ˉu0,η0), | (4.5) |
where
(2) Large-time behavior of
‖ˉη‖2H2(T)+∫t0‖ˉη‖2H2(T)dτ⩽ce−c3tIh0(ˉu0,η0)/κ, | (4.6) |
ec3t‖η−(u0)Tt−ϖ‖2H3(T)+∫t0‖η−(u0)Tt−ϖ‖2H3(T)ec3τdτ⩽cIh0(ˉu0,η0)/κ, | (4.7) |
where
(3) Stability of
ec3t(‖ud‖2H2(T)+κ‖ηd‖2H3(T))+∫t0‖(ud,√κηd)‖2H3(T)ec3τdτ⩽c√κ−1max{1,√κ−1}×(√Ih0(u0,η0)+Ih0(u0,η0))(‖∇η0‖2H2(T)Ih0(u0,η0)+Ih0(ˉu0,η0)). |
Here
(ηl,ul)|t=0=(η0+ηr,u0+ur), | (4.8) |
where
(a)
(b)
where the constants
We end this section by listing some remarks for Theorem 4.2.
Remark 9. We explain on why the initial data
(1) Since the initial data for
(2) The initial data
Remark 10. Let us try to give a physical meaning hidden in (4.6). Consider the viscoelastic fluid in a periodic cell
ln:xh=ηh(yh,0,t)+yh, x3=η3(yh,0,t)+y3, 0⩽y3⩽2πL3. |
At time
By (4.6) and the interpolation inequality, we see that
|ˉη(y,t)|⩽ce−c3tIh0(ˉu0,η0)/κ. |
In particular, for the case
from which and the geometric meaning of
Remark 11. Similarly to Remark 10, we can also give a physical meaning hidden in (4.7). Namely, consider a straight line segment
Remark 12. The corresponding versions of Theorems 4.1 and 4.2 can be found in Theorems 1.4 and 1.5 in [33].
In previous three sections, we have seen the stabilizing effect of elasticity on the motion of viscoelastic fluids. In fact, this stabilizing effect has been also mathematically verified in the elasticity fluids. Next we further briefly introduce the relevant results in elastic fluids.
The system without viscosity reduces to the following system, which describes the motion of elasticity fluids:
(5.1) |
In the absence of
The global regularity of solutions to the two-dimensional Euler equations has been known for a long time. It is also known that the gradient of the vorticity given by
However Lei proved the existence of global solutions with small initial data to the 2D Cauchy problem of (5.1) in Lagrangian coordinates, where the highest-order energy solutions at most algebraically grows in time [44]. Recently the uniform bound of the highest-order energy of global solutions to the 2D case was further proved by Cai [6], which exhibits that the elasticity can inhibit the growth of solutions. We mention that Wang also gave the existence of global solutions to the 2D Cauchy problem of (5.1) by using space-time resonance method and a normal form transformation [72], and the relevant result of the 3D system of (5.1) can be found in [64].
In addition, the stabilizing effect of elasticity on the local-in-time motion of elasticity fluids can be found in the free-boundary case, interested readers can refer to [8] for the vortex sheet problem and [46] for the RT problem.
In this review, we have introduced the mathematical results concerning the stabilizing effect of elasticity on the motion of viscoelastic/elastic fluids. However these results were verified by the Oldroyd model, which includes a viscous stress component and a stress component for a neo-Hookean solid. We except that similar stabilizing results can be extended to the other Oldroyd-B models in future. In addition, there are still many relevant interesting stabilizing problems, which should be further investigated, for examples,
(1)
(2) Theorem 3.1 provides the stability condition of
(3) Theorem 4.1 proves the existence of large solutions of some class in the spatially periodic domain case. It is not clear that whether a similar result can be found in a general bounded domain.
The author is grateful to Dr. Binqiang Xie in South China Normal University for communication in the vortex sheet problem of elasticity fluids.
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