ALBERT

Notes: The convective interaction between a pair of droplets coarsening during the demixing transition of a binary fluid is examined. The starting point is the model H equation for binary fluids, and the droplet sizes are considered to be large enough that thermal fluctuations are neglected. Droplet motion is induced by the convective coupling in the concentration equation, where there is a flux of concentration due to the fluid velocity, and a reciprocal effect in the momentum equation. The effect of the convective force density is separated into two parts—one due to the sharp concentration gradients at the droplet interface, and the other due to the variation in the matrix. It is shown that the dominant contribution to the fluid velocity field is due to the sharp concentration variation at the interface, and this is proportional to the square of the droplet flux at the surface. The surface flux is determined by solving the diffusion equation in the matrix between the droplets, and matching the solution to that in the interfacial region. The analysis indicates that there is an attractive interaction if the two droplets have radii larger or smaller than the critical radius, while the interaction is repulsive if the radius of one droplet is larger and the other smaller than the critical radius. The magnitude of the induced droplet velocity is estimated. © 2000 American Institute of Physics.

Notes: The effect of a linear shear flow of a Newtonian fluid in the region 0〈z〈∞ on the fluctuations at the surface of an elastic medium of thickness H in the region −H〈z〈0 is analyzed in the regime Re(very-much-greater-than)1 and Λ∼1, where Re=ργH2/η is the Reynolds number and Λ=(ργ2H2/E)1/2 is the ratio of the inertial stresses in the fluid and the elastic stresses in the solid. Here ρ and η are the fluid density and viscosity, E is the coefficient of elasticity of the solid, and γ is the mean strain rate in the fluid. A linear analysis is used to determine the effect of the flow on the fluctuations in the surface displacement, and an asymptotic expansion in the small parameter ε=(Λ/Re) is employed. The dynamics in the bulk of the fluid is inviscid in the leading approximation, and the leading order growth rate is imaginary because energy is conserved in the absence of viscous dissipation. There are multiple frequencies of oscillation, all of which satisfy the equations of motion. An increase in the fluid velocity increases the frequency of the downstream traveling waves, and decreases the frequency of the upstream traveling waves. The structure factor for the surface modes of the upstream traveling waves increases with an increase in the fluid velocity because the kinetic energy of the fluctuations decreases due to the lower frequency.An opposite effect is observed for the downstream traveling waves; in addition, it is observed that the structure factor has a double-peaked structure and reaches zero at an intermediate value at sufficiently high velocities. This is due to a divergence in the ratio of the tangential and normal displacements, and a consequent divergence in the energy required for the normal fluctuations at the surface. There is an O(ε1/2) correction to the growth rate due to the presence of a viscous boundary layer of thickness Hε1/2 in the fluid at the interface. The O(ε1/2) calculation shows that the real part of the growth rate is negative for all values of Λ and wave number k, except along certain lines in the Λ−k parameter space where the real part of the growth rate is zero, because the amplitude of the boundary layer velocity becomes zero along these lines. The real part of the O(ε) correction to the growth rate at these points is negative, indicating the presence of a small stabilizing effect due to the dissipation in the bulk of the fluid and the elastic medium. © 1995 American Institute of Physics.

Notes: The interaction between a pair of non-Brownian droplets in the spinodal decomposition of a binary fluid is examined. The interaction arises due to the convective term in the model H momentum equation, which is reciprocal to the convective term in the concentration equation. The dominant contribution to this convective term is due to the interface between the droplet and the matrix, where concentration gradients are large, and this contribution is determined in the limit where the distance between the droplets L is large compared to the radius of a droplet R. The force on the fluid due to the interfacial concentration gradient is first calculated, and it is found that there is a net force on the fluid only if there is a deviation of the interfacial concentration profile from the equilibrium profile. This deviation is related to the flux of solute at the interface, which is calculated correctly to (R/L)2 for the interacting droplets. The average velocity of the droplets is then calculated by solving the momentum equations for the system. It is found that the interaction between the droplets does cause a spontaneous motion of the droplets towards each other. © 1998 American Institute of Physics.

Notes: The growth of random interfaces during the late stage spinodal decomposition for a near symmetric quench of a binary fluid is analyzed. Inertial effects are neglected, and the motion of the interface is determined by a balance between the surface tension, which tends to reduce the curvature, and the viscous stresses in the fluid. The interface is described by an "area distribution function" A(K,t), defined so that A(K,t)dKdx is the area of the interface with curvature in the interval dK about K in the volume dx at time t. Here, K=(K12+K22)1/2 is the magnitude of the curvature, and K1 and K2 are the principal curvatures. There is a change in the area distribution function due to a change in the curvature, and due to the tangential compression of the interface. Phenomenological relations for the change in curvature and surface area are obtained using the assumption that the only length scale affecting the dynamics of the interface at a point is the radius of curvature at that point. These relations are inserted in the conservation equation for the interface, and a similarity solution is obtained for the area distribution function. This solution indicates that the area of the interface decreases proportional to t−1 in the late stages of coarsening, and the mean curvature also decreases proportional to t−1. The effect of the motion of the interface on the interfacial concentration profile and interfacial energy is analyzed using a perturbation analysis. The diffusion equation for the concentration in the interfacial region contains an additional source term due to the convective transport of material caused by the motion of the interface, and this causes a correction to the equilibrium concentration profile of the interface. The excess interfacial energy due to the nonequilibrium motion of the interface is calculated using the Cahn–Hilliard square gradient free energy for a near-critical quench. It is found that the variation in the concentration causes an increase in the interfacial energy which is proportional to the curvature K of the interface. © 1998 American Institute of Physics.

Notes: The coarsening of a random interface in a fluid of surface tension γ and viscosity μ is analyzed using a curvature distribution function A(Km,Kg,t) which gives the distribution of the mean curvature Km and Gaussian curvature Kg on the interface. There is a variation in the area distribution function due to the rate of change of Km, Kg and the compression of the interface due to tangential motion. The rates of change of mean and Gaussian curvature at a point are related to the rate of change of the normal velocity in the tangential directions along the interface. The fluid velocity is governed by the Stokes equation for a viscous flow, and the velocity field at a point is determined as an integral of the product of the Oseen tensor and the normal force at other points on the interface. Using a general form for this integral, it is shown that there is a characteristic variable K*=Kg/(Km2−4Kg)1/2 which is independent of time even as Km and Kg decrease proportional to t−1 and t−2, respectively. In the late stages, analytical forms for the distribution function are determined in the limit Km(very-much-less-than)K* using a similarity variable η=(γKmt/μ). Two reasonable approximations are used for the characteristic length for the correlation of the curvature and normal along the interface, and the results for these two approximations are quadratic polynomials in |η| which are nonzero for a finite interval about η=0. It is expected that the actual distribution function is in between these two limiting cases. © 1998 American Institute of Physics.

Notes: The effect of convective transport on the late stage growth of droplets in the presence of sedimentation and shear flow is analyzed. The high Peclet number limit (UR/D)(very-much-greater-than)1 is considered, where U is the characteristic velocity, R is the radius of the droplet, and D is the diffusion coefficient. The growth of the droplet depends on the boundary condition for the fluid velocity at the droplet interface, and two types of boundary conditions are considered. For a rigid interface, which corresponds to the interface between a solid and a fluid, the tangential velocity is zero and the normal velocity is equal to the velocity of the surface. For a mobile interface, which corresponds to an interface between two fluids, the tangential and normal velocities are continuous. These results indicate that the scaling relations for the critical radius are Rc(t)∝t(1/2) for a sedimenting droplet with a rigid interface, Rc(t)∝t(2/3) for a sedimenting droplet with a mobile interface, Rc(t)∝t(3/7) for a droplet with a rigid interface in a simple shear flow, and Rc(t)∝t(1/2) for a droplet with a mobile interface in a simple shear flow. The rate of droplet growth is enhanced by a factor of Pe(1/3) for rigid interfaces and Pe(1/2) for mobile interfaces. © 1998 American Institute of Physics.

Notes: The variation of the viscosity as a function of the sequence distribution in an A–B random copolymer melt is determined. The parameters that characterize the random copolymer are the fraction of A monomers f, the parameter λ which determines the correlation in the monomer identities along a chain and the Flory chi parameter χF which determines the strength of the enthalpic repulsion between monomers of type A and B. For λ(approximately-greater-than)0, there is a greater probability of finding like monomers at adjacent positions along the chain, and for λ〈0 unlike monomers are more likely to be adjacent to each other. The traditional Markov model for the random copolymer melt is altered to remove ultraviolet divergences in the equations for the renormalized viscosity, and the phase diagram for the modified model has a binary fluid type transition for λ(approximately-greater-than)0 and does not exhibit a phase transition for λ〈0. A mode coupling analysis is used to determine the renormalization of the viscosity due to the dependence of the bare viscosity on the local concentration field. Due to the dissipative nature of the coupling, there are nonlinearities both in the transport equation and in the noise correlation.The concentration dependence of the transport coefficient presents additional difficulties in the formulation due to the Ito–Stratonovich dilemma, and there is some ambiguity about the choice of the concentration to be used while calculating the noise correlation. In the Appendix, it is shown using a diagrammatic perturbation analysis that the Ito prescription for the calculation of the transport coefficient, when coupled with a causal discretization scheme, provides a consistent formulation that satisfies stationarity and the fluctuation dissipation theorem. This functional integral formalism is used in the present analysis, and consistency is verified for the present problem as well. The upper critical dimension for this type of renormalization is 2, and so there is no divergence in the viscosity in the vicinity of a critical point. The results indicate that there is a systematic dependence of the viscosity on λ and χF. The fluctuations tend to increase the viscosity for λ〈0, and decrease the viscosity for λ(approximately-greater-than)0, and an increase in χF tends to decrease the viscosity. © 1996 American Institute of Physics.

Notes: The effect of viscoelasticity on the early stages of spinodal decomposition is examined. In addition to the concentration and momentum equations for the fluid, the effect of viscoelasticity is included using a linear Maxwell equation for the stress tensor. The growth in the amplitude of the fluctuations depends on the transport coefficient, the viscosity of the fluid, and the relaxation time in the Maxwell model. For simplicity, the nonlinearity due to the quartic term in the expression for the Landau–Ginzburg expression for the free energy is neglected, as are the inertial terms in the momentum conservation equation. The momentum and Maxwell equations are solved exactly to obtain the velocity as a function of concentration, which is then inserted into the concentration equation. There are two types of nonlinearities in the conservation equation—one proportional to the cube of the concentration which leads to a four point vertex, and one proportional to the product of the concentration and the random noise in the stress equation which leads to a three point vertex. In the leading approximation, the renormalization of the transport coefficient due to these vertices is determined using the Hartree approximation, and the renormalization of the noise correlation due to the three point vertex is determined using a one-loop expansion. The renormalized transport coefficient and noise correlation are inserted into the concentration equation to determine the effect of the nonlinearities on the growth of the structure factor. It is found that an increase in the relaxation time tends to increase the rate of growth of the structure factor, and tends to decrease the wave number of the peak in the structure factor. © 1996 American Institute of Physics.