Analytic theory for the selection of Saffman-Taylor fingers in the presence of thin film effects

An analytic theory is presented for the width selection of Saffman-Taylor fingers in the presence of thin film effect. In the limit of small capillary number Ca and small gap to width ratio ϵ, such that ϵ ⪡Ca ⪡1, it is found that fingers with relative width λ < ½ are possible such that λ2(1-λ)(1-2λ) = k(ϵ/Ca3/2), where the positive constant k depends on the branch of solution and equals 2.776 for the first branch. A fully nonlinear analysis is necessary in this problem even to obtain the correct scaling law. It is also shown how in principle, the selection rule for arbitrary Ca can be obtained.