When a concentrated mixture of two immiscible fluid is sheared, a rather complex interface is formed due to the coagulation, rupture and deformation of droplets. Instead of formulating a droplet problem, Doi(1989) just forcused on the interface itself and derived a phenomenological constitutive equations which can describe the time evolution of the interfacial area(Q) and orientation($q_{ij}$) of the interface in the flow field. He restricted himself to a mixture of two immiscible fluids having same viscosity and density, mixed with the volume ratio 1:1. Here we are going to raise other questions on steady state structure of the interface with a finite but small flow field. Physically we can imagine that there should be a kind of steady structure of interface due to the competetion between flow and surface tension. But this theory failed to give nontrivial Q and $q_{ij}$ in the limit of dilute situation. It can be easily found that there is a certain term is missing in his phenomenological relaxation about the interfacial area when he wrote down.
$(\partial/\partial t)Q\mid_{relax} = -C_1(\Gamma Q/\mu_o)Q$
Here $\Gamma$ is the surface tension and $\mu_0$ is the viscosity. We proposed $-C_3(\Gamma/\mu_0)q_{ij}q_{ij}$ term from the contribution of surface tension on relaxation of the interfacial area in the first place. $C_1$term is related to the coagulation of droplets in concentrated system, which is going to vanish at the limit of dilution. Therefore $C_1$ can be treated as a function of volume fraction phenomenologically. On the other hand, $C_3$ is the basic term which describes the deformation to the spherical form due to the surface tension and also it contains some contribution from rupture of an elongated droplet. When compared with calculation and with experiments in polymer blending system (PS/LLDPE) our theoretical analysis generally was valid although, in our cases, the agreement is only qualitative.
