A Study Of The Solution For Mathematical Model On The Flow Between Rotating Frustrum And The Analysis Of The Solution For Combustion Model

The main work of this paper includes two parts:(1) A Study of the Solution of the Mathematical Model on the Flow between Two Coaxial Rotating Truncated Cones (Frustrum)Precipitation is one of the most effective methods used to prepare nano-sized particles. The method is easy to implement at large-scale in the chemical industry. Precipitation process is very fast and difficult to control. Experiment showed that rotating liquid film reactor (RLFR) is an ideal reactor for carrying out the precipitation. It has the advantage of both narrow reaction spaces and high shear forces. It’s found that the particles obtain using the RLFR were smaller in size and narrower in size distribution.In chapter 2 and chapter 3, the flow of the coaxial rotating truncated cone is studied, both theoretically and numerically. The gap between two cones is filled with viscous incompressible fluid.In chapter 2, the truncated cones with the inner one rotating and the outer one stationary is considered, where the radius at the top is larger than the bottom one. Numerical simulation method has been used to investigate the flow state of the gap between the truncated cones, and the transition from basic flow to Taylor vortices, as well as the distribution of the extreme values of pressure and velocity. It’s shown that with the increasing of Reynolds number above a critical number, the basic flow becomes instable, further increasing Re to 300, the first Taylor vortex is generated in the neighborhood of top area of the cones. The gap between the two cones is filled with six pairs of vortices when the Re is about 800, which is confirmed by experimental observation. The local extreme values of pressure and velocity are attained at adjacent lines between neighboring vortices or medium lines of large vortices. The local minimum values of velocity and the local maximum values of pressure are obtained at the same point, whereas the local maximum values of velocity of the flow is located at the point of inflection of pressure.In chapter 3, the case with the radius at the top smaller than the bottom one is discussed, where both the inner cone and the outer one rotate independently. We will prove the nonexistence of two-dimensional steady solution of the forms u= uθ (r, z)eθ+uz (r, z)ez, p= p(r, z) and u= ur(r,z)er+ue(r,z)eθ,p= p(r,z) for arbitrary angular velocities of both the inner and outer cone. Moreover numerical strategy has been used to investigate the flow state between truncated cones. The simulation results suggested that a three-dimensional steady solution exists, which confirmed Wimmer’s observation in the experiments.(2) A new Approach to the Mathematical Property of One-dimensional Steady Solution of Reverse Smolder WavesSmoldering is a kind of combustion reaction which occurs between the solid porous medium and oxygen. Heat is released by oxidation of the solid porous medium. Usually, smoldering combustion occurs at relatively low temperatures and small propagation velocity, which differs from the flaming combustion. If the material is sufficiently porous, the smolder wave can propagate through the interior of the porous sample.There are two kinds of smolder waves, the forward smolder wave and the reverse smolder wave, they are different at the directions between the flow of the oxidizer and the propagation of the smolder wave. The reverse smolder wave is initiated at one of the end and propagates in the direction opposite to the flow of the oxidizer.The mathematical property of one-dimensional steady solution of reverse smolder waves at infinity is studied in chapter 4 using analytic techniques. The asymptotic behavior of the solution is considered.Theorem 4.1 proves that both the solution and its derivative of the first and the second order have the limit at infinity. In particular, under the assumption of this theorem, it is proved that either the value of oxygen or the mass of fuel tend to zero at the positive infinity, which correspond to the fuel-rich case and fuel-lean case, respectively. This result is confirmed by Schult et al. using the method of asymptotic analysis. Moreover, a relational expression between the temperature and the value of oxygen is derived when the Levis number and the value of oxygen are reciprocal. Furthermore, an upper bound of the temperature is derived.