Numerical Simulations of Convection Induced by Korteweg Stresses in a Miscible Polymer–Monomer System: Effects of Variable Transport Coefficients, Polymerization Rate and Volume Changes

Abstract

We modeled a miscible polymer-monomer system with a sharp transition zone separating the two fluids to determine if convection analogous to Marangoni convection in immiscible fluids could occur because of thermal and concentration gradients. We considered three cases: with a temperature gradient along the transition zone, with a variable transition zone width, and one with a gradient in the conversion of polymerization. Using the Navier-Stokes equations with an additional term, the Korteweg stress term arising from non-local interactions in the fluid, we demonstrated with realistic parameters that measurable fluid flow would result in the absence of buoyancy-driven convection for all three cases. We show that even if the Korteweg stress is not a function of temperature, the increase in the diffusion coefficient with temperature can result in convection because a gradient in the transition zone width develops. We also examine the effects of a polymer viscosity that is not only a function of concentration but also temperature. We demonstrate that a constant flux of heat, as would be realistic for a heating element in contact with the side of the reactor, would produce a greater flow than a linear thermal gradient parallel to the transition zone. We demonstrate that qualitatively different flow patterns can be realized by using unusual initial conditions that could be realized with different masks for the photopolymerization. We also demonstrate that the volume change during polymerization and caused by side heating could not cause significant fluid flow that would confound the observation of Korteweg-stress induced flows. To avoid buoyancy-driven convection, the experiment would have to be performed in microgravity.

Phase-field modelling of a miscible system in spinning droplet tensiometer

“…This necessary step has never been done. Only the capillary constant was previously estimated [20,30], and the previous estimations are close to the value obtained in the current work. non-dimensional radial and axial dimensions of the droplet (measured in units of R); radius of an initial (spherical) droplet; droplet's radial dimension in the equilibrium state (for an immiscible droplet) C 0 ; C in equilibrium concentration profile for a flat interface (21); initial concentration profile (24); δ; δ 0 ; δ in interface thickness; thickness of an equilibrium flat interface (21); initial thickness of an interface; r and z; ⃗ e r radial and axial coordinates; the unit vector in the radial direction; ⃗ u = (u r , u z ) vector of velocity; Π pressure field; ω and ψ vorticity and streamfunction (20); L * ; τ * ; v * ; p * ; µ * typical length and time, velocity, pressure and chemical potential (9); Pe; Ca; Re; Gr Peclet (10), capillary (11), Reynolds (12), and Grashof (13) numbers; A non-dimensional parameter determining the thermodynamic state of a mixture σ; σ 0 ; σ V ; σ CH surface tension coefficient; surface tension coefficient for a flat interface (23); surface tension coefficient calculated from the Vonnegut formula (25); surface tension coefficient calculated from the structure of the solute/solvent interface (22); m; m 0 rate of change of the droplet's radial dimension; a constant (26)…”

“…The numeric value of the surface tension coefficient, σ CH , can be approximately calculated from the formula for a flat interface σ 0 = for a monomer/polymer solution [20,30]. 5 Finally, in the experiments (e.g.…”

“…There are only some estimates of the capillary constant in e.g. [29,30]. The lack of knowledge of these new parameters hinders the wider usage of the phase-field method.…”

Nonlinear simulations of miscible viscous fingering with gradient stresses in porous media

“…Such stress, called the Korteweg stress (Korteweg, 1901), acts at the miscible diffusive interface against the growth of the instabilities (Pojman et al, , 2006(Pojman et al, , 2009Joseph, 1990;Hu and Joseph, 1992;Chen and Meiburg, 2002;Pramanik andMishra, 2013, 2014;Swernath et al, 2010). Experiments are conducted with the miscible fluids in microgravity to determine if they could exhibit transient interfacial phenomena seen with immiscible fluids .…”

On the phase-field modelling of a miscible liquid/liquid boundary

“…The consideration was however limited for a rather fast process at very high Peclet numbers, when the effect of diffusion was negligible. The similar approach was used to study the spreading of the interface boundary and generation of the 55 convective motion near the boundary by the action of the Korteweg force [16,17].…”