Stability of Two Dimensional Immiscible Flow to Viscous Fingering
Stability of Two Dimensional Immiscible Flow to Viscous Fingering
Michael J King
The book Stability of Two Dimensional Immiscible Flow to Viscous Fingering was written by author Michael J King Here you can read free online of Stability of Two Dimensional Immiscible Flow to Viscous Fingering book, rate and share your impressions in comments. If you don't know what to write, just answer the question: Why is Stability of Two Dimensional Immiscible Flow to Viscous Fingering a good or bad book?
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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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