The transient deformation of liquid capsules enclosed by elastic membranes with bending rigidity in shear flow has been studied numerically, using an improved immersed boundary-lattice Boltzmann method. The purpose of the present study is to investigate the effect of interfacial bending stiffness on the deformation of such capsules. Bending moments, accompanied by transverse shear tensions, usually develop due to a preferred membrane configuration or its nonzero thickness. The present model can simulate flow induced deformation of capsules with arbitrary resting shapes (concerning the in-plane tension) and arbitrary configurations at which the bending energy has a global minimum (minimum bending-energy configurations). The deformation of capsules with initially circular, elliptical, and biconcave resting shapes was studied; the capsules' minimum bending-energy configurations were considered as either uniform-curvature shapes (like circle or flat plate) or their initially resting shapes. The results show that for capsules with minimum bending-energy configurations having uniform curvature (circle or flat plate), the membrane carries out tank-treading motion, and the steady deformed shapes become more rounded if the bending stiffness is increased. For elliptical and biconcave capsules with resting shapes as minimum bending-energy configurations, it is quite interesting to find that with the bending stiffness increasing, the capsules' motion changes from tank-treading mode to flipping mode, and resembles Jeffery's flipping mode at large bending stiffness.

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Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10 −1 to 10 −6 , porosity values from 0.2 to 0.8, Rayleigh numbers from 10 3 to 10 7 , and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy-Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β 1 and β 2 were varied from −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.

Forced convection after a backward-facing step, with a porous floor segment, is investigated numerically using the SIMPEC method. The Brinkman-Forcheimmer extended model is used to govern the flow in the porous-medium region. At the interface, the flow boundary condition imposed is a shear stress jump, which includes the inertial effect, together with a continuity of normal stress. The thermal interfacial condition is continuities of temperature and heat flux.Results are presented for Reynolds number up to 800 and Darcy number up to 10 À1 . Also varied are the length and depth of the porous segment. Compared with the case with no porous floor, the local heat transfer is augmented after the porous floor. Within the porous floor, the heat transfer is reduced, but this may be offset by using a porous medium of higher conductivity than the fluid. To obtain good heat enhancement after the porous segment, it should approximately match the length of the recirculation region. The porous segment should have large permeability (Darcy number around 10 À1 ), but it is not necessary that it be of great depth. The interfacial stress jump coefficients b and b 1 are varied from À5 to þ5, and some effects are observed on the local Nusselt numbers, velocity profile, and temperature distribution.