## 4 Numerical Solution to the Viscous Free Interface Model
### Scales in the Transient Regime  
We assume that the source term \(S\) is constant, and that the viscosity \(\
u\) is also constant. Then, the solution to the viscous model (3)-(5) depends on the constant \(S\), the total thickness \(h\), the viscosity \(\
u\), the initial thickness of the static layer \(b^{0}\), and the initial velocity profile \(U^{0}(Z)\). We introduce dimensionless quantities, denoted by carets (hats),  
\[t=\tau\widehat{t},\quad Z=l\widehat{Z},\quad h=l\widehat{h},\quad b=l\widehat {b},\quad U=u\widehat{U}, \tag{33}\]  
where \(\tau\) is a time scale, \(l\) a space scale, and \(u=la_{1}\) with \(\alpha_{1}\) being the order of magnitude of the initial shear rate. In order to write Equation (3) in dimensionless form, we take  
\[l=\
u\frac{\alpha_{1}}{S},\qquad\tau=\frac{l^{2}}{\
u}=\
u\frac{(\alpha_{1})^ {2}}{S^{2}}. \tag{34}\]  
The dimensionless equation is then  
\[\partial_{\widehat{t}}\widehat{U}+1-\partial_{\widehat{Z}Z}^{2}\widehat{U}=0 \quad\text{ for all }\widehat{Z}\in(\widehat{b},\widehat{h}),\]  
with boundary conditions \(\widehat{U}=\partial_{\widehat{Z}}\widehat{U}=0\) at \(\widehat{Z}=\widehat{b}\) and \(\partial_{\widehat{Z}}\widehat{U}=0\) at \(\widehat{Z}=\widehat{h}\). If we take a linear initial velocity profile, this dimensionless solution \(\widehat{U}\) depends only on \(\widehat{h}=h/l\) and \(\widehat{b}^{0}=b^{0}/l\). Actually, since the problem is invariant by translation in \(Z\), the solution depends only on \((h-b^{0})/l\).  
Numerical investigations for a constant source \(S\) using the scheme described in Section 4.1 show the behavior illustrated in Figure 9. The static/flowing interface position \(b(t)\) first decreases until a time \(t^{c}\) and reaches a minimal value \(b^{\text{min}}\) (starting phase). Then \(b(t)\) increases and (if \(h\) is sufficiently large) reaches an asymptotic regime with upward velocity \(\dot{b}^{\infty}\) (stopping phase), before fully stopping when it reaches \(h\).  
According to the above scaling analysis, if the velocity profile is initially linear with shear \(\alpha_{1}\), and if  
\[\frac{h-b^{0}}{l}=\frac{(h-b^{0})S}{\
u\alpha_{1}}\gg 1, \tag{35}\]  
Figure 9: Schematic evolution of the thickness of the static/flowing interface as a function of time.  
then the dimensionless problem has no parameter left and the velocity \(\widehat{U}\) is therefore a fixed profile (in terms of \(\widehat{Z}-\widehat{b}^{0}\)), as well as \(\widehat{b}-\widehat{b}^{0}\). We conclude with (33) that the quantities \(t^{c}\), \(b^{0}-b^{\min}\), \(\dot{b}^{\infty}\) are proportional to \(\tau\), \(l\), \(l/\tau\), respectively. We obtain the following proportionality factors from the numerical simulation  
\[t^{c}=0.15\
u\frac{\left(\alpha_{1}\right)^{2}}{S^{2}},\qquad b^{0}-b^{\min}= 0.43\
u\frac{\alpha_{1}}{S},\qquad\dot{b}^{\infty}=0.95\frac{S}{\alpha_{1}}. \tag{36}\]  
Note the dependency on the ratio \(\alpha_{1}/S\), which is due to the homogeneity of the problem (3) and (4) with respect to \((U,S)\) (a new solution can be obtained by multiplying \((U,S)\) by a positive constant, with \(b\) unmodified).