Hele-Shaw: Difference between revisions

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The Hele-Shaw model describes an incompressible flow lying between two nearby horizontal plates<ref name="MR0097227"/>. The following equations are given for a non-negative pressure $u$, supported in in a time dependent domain,
The Hele-Shaw model describes the evolution of an incompressible flow lying between two nearby horizontal plates<ref name="MR0097227"/>. The following equations are given for a non-negative pressure $u=u(x,t)$, supported in in a time dependent domain,
\begin{align*}
\begin{align*}
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
\Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \partial \{u>0\}\cap \Omega
\frac{\partial_t u}{|Du|} &= |Du| \text{ on } \Gamma = \partial \{u>0\}\cap \Omega
\end{align*}
\end{align*}
The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side). Particular solutions are given for instance by the planar profiles
The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side).  
 
Particular solutions are given for instance by the planar profiles
\[
\[
P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad  A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0
P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad  A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0
\]
\]


Non-local aspects of the equation can be appreciated by noticing that a given deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one.
The model has a non-local nature as any deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one.


Let $u = P + \varepsilon v$. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives
Let $u = P + \varepsilon v$ where $P$ is a planar profile. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives
\[
\[
\frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2
\frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2
Line 22: Line 24:
\partial_t w &= a\partial_n w \text{ on } \{x_n=0\}
\partial_t w &= a\partial_n w \text{ on } \{x_n=0\}
\end{align*}
\end{align*}
Or in terms of the half-laplacian in $\mathbb R^{n-1} = \{x_n=0\}$,
Or in terms of the half-laplacian we can say that $w$ when restricted to $\mathbb R^{n-1} = \{x_n=0\}$ satisfies the heat equation
\[
\[
\partial_t w = a\Delta_{\mathbb R^{n-1}} w
\partial_t w = a(t)\Delta_{\mathbb R^{n-1}}^{1/2} w.
\]
\]
Notice the space invariance of the linearized problem implies Holder estimates for any spatial derivative of $w$ meanwhile the time regularity depends on the regularity of the coefficient $a(t)$. This approach allows to prove interior Holder estimates in space and time for the normal vector of the free boundary from a flatness hypothesis<ref name="2016arXiv160507591C"/>.


== References ==
== References ==
Line 31: Line 35:


<ref name="MR0097227">{{Citation | last1=Saffman | first1= P. G. | last2=Taylor | first2= Geoffrey | title=The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid | journal=Proc. Roy. Soc. London. Ser. A | issn=0962-8444 | year=1958 | volume=245 | pages=312--329. (2 plates)}}</ref>
<ref name="MR0097227">{{Citation | last1=Saffman | first1= P. G. | last2=Taylor | first2= Geoffrey | title=The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid | journal=Proc. Roy. Soc. London. Ser. A | issn=0962-8444 | year=1958 | volume=245 | pages=312--329. (2 plates)}}</ref>
<ref name="2016arXiv160507591C">{{Citation | last1=Chang-Lara | first1= H.A. | last2=Guillen | first2= N. | title=From the free boundary condition for Hele-Shaw to a fractional parabolic equation | journal=ArXiv e-prints | year=2016}}</ref>


}}
}}

Latest revision as of 07:45, 1 August 2016

This article is a stub. You can help this nonlocal wiki by expanding it.

The Hele-Shaw model describes the evolution of an incompressible flow lying between two nearby horizontal plates[1]. The following equations are given for a non-negative pressure $u=u(x,t)$, supported in in a time dependent domain, \begin{align*} \Delta u &= 0 \text{ in } \Omega^+ = \{u>0\}\cap \Omega\\ \frac{\partial_t u}{|Du|} &= |Du| \text{ on } \Gamma = \partial \{u>0\}\cap \Omega \end{align*} The first equation expresses the incompressibility of the fluid. The second equation, also known as the free boundary condition, says that the normal speed of the inter-phase (left-hand side) is the velocity of the fluid (right-hand side).

Particular solutions are given for instance by the planar profiles \[ P(x,t) = a(t)(x_n-A(t))_+ \qquad\text{where}\qquad A(t) = \int_t^0 a(s)ds \qquad\text{and}\qquad a(t)>0 \]

The model has a non-local nature as any deformation of the domain $\Omega^+$ affects all the values of $|Du|$, at least in the corresponding connected component. To be more precise let us also formally show that the linearization about a planar profile leads to a fractional heat equation of order one.

Let $u = P + \varepsilon v$ where $P$ is a planar profile. Then $u$ and $P$ harmonic in their positivity sets imply $v$ harmonic in the intersection, notice that as $\varepsilon\searrow0$, $v$ becomes harmonic in $\{x_n>A(t)\}$. On the other hand, the free boundary relation over $\{x_n=A(t)\}$ gives \[ \frac{a^2+\varepsilon \partial_t v}{|ae_n+\varepsilon Dv|} = |ae_n+\varepsilon Dv| \qquad\Rightarrow\qquad \partial_t v = 2a\partial_n v+\varepsilon |Dv|^2 \] By taking the reparametrization $w(x,t) = v(x+Ae_n,t)$ and letting $\varepsilon\searrow0$ we get that $w$ satisfies \begin{align*} \Delta w &= 0 \text{ in } \{x_n>0\}\\ \partial_t w &= a\partial_n w \text{ on } \{x_n=0\} \end{align*} Or in terms of the half-laplacian we can say that $w$ when restricted to $\mathbb R^{n-1} = \{x_n=0\}$ satisfies the heat equation \[ \partial_t w = a(t)\Delta_{\mathbb R^{n-1}}^{1/2} w. \]

Notice the space invariance of the linearized problem implies Holder estimates for any spatial derivative of $w$ meanwhile the time regularity depends on the regularity of the coefficient $a(t)$. This approach allows to prove interior Holder estimates in space and time for the normal vector of the free boundary from a flatness hypothesis[2].

References

  1. Saffman, P. G.; Taylor, Geoffrey (1958), "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid", Proc. Roy. Soc. London. Ser. A 245: 312--329. (2 plates), ISSN 0962-8444 
  2. Chang-Lara, H.A.; Guillen, N. (2016), "From the free boundary condition for Hele-Shaw to a fractional parabolic equation", ArXiv e-prints