We present a renormalization-group (RG) approach to the nonlinear diffusion process BtU^DdxU, with Z) =* J for diu > 0 and Z) = (1 +6)/2 for dxU < 0, which describes the pressure during the filtration of an elastic fluid in an elastoplastic porous medium. Our approach recovers Barenblatt's long-time result that, for a localized initial pressure distribution, w(x,r)~/ ~^°^^'^^f{x/^,€), where / is a scaling function and a'=6/(2;r^)'^^ + 0(6^) is an anomalous dimension, which we compute perturbatively using the RG. This is the first application of the RG to a nonlinear partial differential equation in the absence of noise. PACS numbers: 47.55.Mh, 47.25.Cg, 64.60.Ak, 64.60.HtBuckingham's FI theorem' states that the dependence of a physical quantity on a set of dimensionful parameters may be expressed as the dependence of a dimensionless quantity fl on dimensionless combinations nclli, .. . ,n" of the governing parameters:
We consider the partial differential equation which describes phase separation in a block copolymer melt. We construct numerically the periodic solution which minimizes the free energy. The lamellar thickness of the final equilibrium pattern is found to scale with the molecular weight as a power law XL -N . The exponent 9 takes the value -, ' in the weak-segregation regime and -, in the strong-segregation regime. We propose a scaling theory of the dynamics, from which we obtain 8=2/, where P is the scaling exponent in spinodal decomposition, in agreement with a conjecture by Oono and Bahiana [Phys. Rev. Lett. 61, 1109]. Lastly, we also study the pattern formed by propagating fronts. The selection of the unique velocity of the front and of the wavelength of the pattern behind the front agrees well with the marginal-stability theory.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.