Viscous Fingering in a Hele–Shaw Cell

Abstract

In this chapter, we study another interfacial phenomenon: the formation of viscous fingers in a Hele–Shaw cell. This phenomenon occurs in an entirely different physical system from dendritic growth, but it raises similar issues and can thus be treated using the same approach established in the previous chapters. We are interested in the study of this phenomenon in this book, not only because the subject itself occupies an important position in the field of pattern formation, but also because the resolution for this problem provides a keystone for analytically studying further pattern formation problems in solidification, such as cellular growth and eutectic growth.

Appendix: The Forms of Some Operators in the System of the Curvilinear Coordinate System (ξ, η)

For the curvilinear coordinate system (ξ, η), we have

$$\displaystyle{ \mathbf{r} = X(\xi,\eta )\mathbf{i} + Y (\xi,\eta )\mathbf{j}, }$$

(9.200)

where i and j are the unit vectors along the x-axis and y-axis respectively. Let the vectors e ₁, e ₂ be the unit vectors along the ξ, η directions respectively. One can write

$$\displaystyle{ \begin{array}{lllllllllll} &&\mathbf{e}_{1} = \frac{\partial \mathbf{r}} {\partial \xi } \Big/\Big\vert \frac{\partial \mathbf{r}} {\partial \xi } \Big\vert = \frac{1} {\mathcal{G}}\left (X_{\xi }\mathbf{i} + Y _{\xi }\mathbf{j}\right ), \\ &&\mathbf{e}_{2} = \frac{\partial \mathbf{r}} {\partial \eta } \Big/\Big\vert \frac{\partial \mathbf{r}} {\partial \eta } \Big\vert = \frac{1} {\mathcal{G}}\left (X_{\eta }\mathbf{i} + Y _{\eta }\mathbf{j}\right )\;. \end{array} }$$

(9.201)

To derive the mathematical formulation of the problem in the coordinate system {ξ, η}, we denote the field function by $\Phi = \Phi (\xi,\eta )$ and the interface shape by η = η _s(ξ, t). We have the formulas

$$\displaystyle{ \begin{array}{lllllllllll} &&\quad \quad \nabla \Phi = \frac{\mathbf{e}_{1}} {\mathcal{G}} \frac{\partial \Phi } {\partial \xi } + \frac{\mathbf{e}_{2}} {\mathcal{G}} \frac{\partial \Phi } {\partial \eta }, \\ &&(\mathbf{j} \cdot \nabla )\Phi = \frac{1} {\mathcal{G}^{2}}\left (Y _{\xi }\frac{\partial \Phi } {\partial \xi } + X_{\xi }\frac{\partial \Phi } {\partial \eta } \right ), \end{array} }$$

(9.202)

as well as

$$\displaystyle{ \nabla ^{2}\Phi = \nabla \cdot (\nabla \Phi ) = \frac{1} {\mathcal{G}^{2}}\left (\frac{\partial ^{2}\Phi } {\partial \xi ^{2}} + \frac{\partial ^{2}\Phi } {\partial \eta ^{2}} \right ) }$$

(9.203)

and

$$\displaystyle{ \frac{\partial \Phi } {\partial n} = \mathbf{n} \cdot \nabla \Phi = \frac{1} {\big(1 + \eta _{\mathrm{s}}^{{\prime}}{}^{2}\big)^{1/2}\mathcal{G}}\left (\frac{\partial \Phi } {\partial \eta } -\eta _{\mathrm{s}}^{{\prime}}\frac{\partial \Phi } {\partial \xi } \right ). }$$

(9.204)

In the moving frame, the local growth speed at the interface is

$$\displaystyle\begin{array}{rcl} & & u_{\mathrm{I}} = - \frac{\frac{\partial s} {\partial t}} {\big\vert \nabla s\big\vert } = \mathcal{G} \frac{\frac{\partial \eta _{\mathrm{s}}} {\partial t}} {\big(1 + \eta _{\mathrm{s}}^{{\prime}}{}^{2}\big)^{1/2}},{}\end{array}$$

(9.205)

$$\displaystyle\begin{array}{rcl} & & \mathbf{j \cdot n} = \frac{Y _{\eta } - Y _{\xi }\eta _{\mathrm{s}}^{{\prime}}} {\mathcal{G}\big(1 + \eta _{\mathrm{s}}^{{\prime}}{}^{2}\big)^{1/2}}\;,{}\end{array}$$

(9.206)

$$\displaystyle{ \frac{\partial \Phi } {\partial n} = \mathbf{n} \cdot \nabla \Phi = \frac{1} {\big(1 + \eta _{\mathrm{s}}^{{\prime}}{}^{2}\big)^{1/2}\mathcal{G}}\left (\frac{\partial \Phi } {\partial \eta } -\eta _{\mathrm{s}}^{{\prime}}\frac{\partial \Phi } {\partial \xi } \right ). }$$

(9.207)

The curvature operator can be derived as

$$\displaystyle\begin{array}{rcl} \mathcal{K}\Big\{\eta _{\mathrm{s}}(\xi,t)\Big\}& =& - \frac{1} {\mathcal{G}(\xi,\eta _{\mathrm{s}})}\Bigg\{ \frac{\eta _{\mathrm{s}}^{{\prime\prime}}} {\big(1 + \eta _{\mathrm{s}}^{{\prime}}{}^{2}\big)^{3/2}} + \frac{\Pi _{0}(\xi,\eta _{\mathrm{s}})} {\mathcal{G}^{2}(\xi,\eta _{\mathrm{s}})\big(1 + \eta _{\mathrm{s}}^{{\prime}}{}^{2}\big)^{1/2}} \\ & & \ + \frac{\Pi _{1}(\xi,\eta _{\mathrm{s}})} {\mathcal{G}^{2}(\xi,\eta _{\mathrm{s}})\big(1 + \eta _{\mathrm{s}}^{{\prime}}{}^{2}\big)^{1/2}}\eta _{\mathrm{s}}^{{\prime}}\Bigg\}\,, {}\end{array}$$

(9.208)

where

$$\displaystyle{ \begin{array}{lllllllllll} \Pi _{0}(\xi,\eta ) =\big (Y _{\xi \xi }X_{\xi } - X_{\xi \xi }Y _{\xi }\big), \\ \Pi _{1}(\xi,\eta ) =\big (X_{\xi \xi }X_{\xi } + Y _{\xi \xi }Y _{\xi }\big). \end{array} }$$

(9.209)

Assume that as ε → 0,

$$\displaystyle{\eta _{\mathrm{B}}(\xi ) =\epsilon h_{1}(\xi ) + \cdots \,.}$$

We derive that

$$\displaystyle{ \mathcal{K}\{\eta _{\mathrm{B}}(\xi )\} = \mathcal{K}_{0}(\xi ) +\epsilon \mathcal{K}_{1}(\xi ) + \cdots \,, }$$

(9.210)

where

$$\displaystyle{ \left \{\begin{array}{lllllllllll} \mathcal{K}_{0}(\xi )& =& -\frac{\Pi _{0,0}} {\mathcal{G}_{0}^{3}}, \\ \mathcal{K}_{1}(\xi )& =&\frac{2\pi ^{2}\lambda _{0}(1 -\lambda _{0})} {\mathcal{G}_{0}^{3}\mathcal{R}_{0}} \Bigg[1 -\frac{2(1 -\lambda _{0})} {\lambda _{0}\mathcal{R}_{0}} + \frac{3\Pi _{0,0}} {\pi \mathcal{G}_{0}^{2}} \Bigg]h_{1}(\xi )\quad \\ &- &\frac{2\pi (1 -\lambda _{0})^{2}\tan ( \frac{\pi \xi }{2})} {\mathcal{R}_{0}\mathcal{G}_{0}^{3}} h_{1}^{{\prime}}(\xi ) - \frac{1} {\mathcal{G}_{0}}h_{1}^{{\prime\prime}}(\xi ). \end{array} \right. }$$

(9.211)

At the root point T, ξ = ±1, η = η _T(t): since Y _ξ (±1, η _s) = −X _η (±1, η _s) = 0, the slope of the interface shape y _s can be calculated as

$$\displaystyle{ \frac{\text{d}y_{\mathrm{s}}} {\mathrm{d}x} \Big\vert _{x=\pm W} = \frac{Y _{\xi } + Y _{\eta }\eta _{\mathrm{s}}^{{\prime}}} {X_{\xi } + X_{\eta }\eta _{\mathrm{s}}^{{\prime}}}\Big\vert _{X=\pm 1} =\eta _{ \mathrm{s}}^{{\prime}}(\pm 1). }$$

(9.212)