Stability of Two Dimensional Immiscible Flow to Viscous Fingering

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Stability of Two Dimensional Immiscible Flow to Viscous Fingering
Michael J King
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With J, j and n denoting respectively the x, y and t indices, the numerical scheme used for (4a) consists of advection terms discretized by a leap-frog stencil, the viscosity term by a Crank-Nicholson stencil: 2 Af Ax A:y = — ^ — Ip" a iu"*^ ■¥ uV^) + J?" A (u""^^ -t- u"~^n 4 Ax^ 1 ' •■' ' ' ••' ' •■/ '''i "^'^"rj "r, ]>j - — I?" A (u"^^ + i/""^) -^ b" a (m""^^ -t- m''~^)1, 4Ay^ (8) where 8, * J ■ 8 81- J - 8 2 "f. Y ^ "I- 1, J A, u'l^j - uf^^j - ulj, Equation (4b) has the form V • X(5) VP = V ...his) V5, and was approximated by a Crank-Nicholson stencil: (9) (10) -7- The difference scheme is second order accurate in space and time. The linear system of equations generated on the numerical grid for each equation at each timestep were inverted using a multigrid algorithm^ of relaxation and injection through a sequence of coarser grids which is based on a variational formulation of the equations (4). The calculations were performed in a square, length L, width L. The boundary condi- tions imposed were (5 is defined in (11)) 5=1, at^^O, 5 = 0, aXy = L, — = at X = 0, Z, , dx P = Po, at >- = 0, P = 0, aty = L, (11) — = &tx = 0, L, dx uix, y, f = 0) = UQix, y) .

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