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In this study, dynamic and heat transfer equations of two-dimensional laminar plane and axisymmetric stagnation flow are solved by Optimal Homotopy Analysis Method, Boundary Knot-Homotopy analysis method and compared by numerical solution. The optimal convergence-control parameter value is calculated using Chebyshev points. These points are corresponding to the range of solutions to get the best answer for both flows. Boundary Knot Method gives the best initial guess that applies in terms of primary answer of homotopy analysis method. Results are reported by the 50th order approximation. Also, it is considered that the total numbers of knots on the domain and the boundary is 40. It is shown that results have a good agreement with the numerical solution. The stream function, the velocity function, the shear stress function and the temperature distribution for small Prandtl values is shown for plane and axisymmetric stagnation flows using BK-HAM compared with the numerical solution. It can be found that, with increasing vertical distance, because of decreasing the effects of wall, the fluid shear stress will be reduced. Also the temperature distribution in the boundary layer changes linearly with distance from the wall. Also, increasing the Prandtl number and decreasing the thermal boundary layer thickness is leading to increase temperature distribution.

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In this study, the effect of chaotic advection on laminar mixing is analyzed experimentally and numerically. Mixer includes two circular rotors and a circular stator in the way that rotational speed of each rotor could be controlled by time. This kind of mixer is widely used in the food and pharmaceutical industry. The performance of the mixer was investigated experimentally with dye injection and image recording. Moreover, the effect of chaotic flow on mixing numerically is studied qualitatively and quantitatively with Lagrangian fluid particle tracing, calculating the stretching rate of fluid elements and Poincare sections. The experimental results indicate that as a result of applying sinusoidal perturbation on rotational velocity of rotors, weak mixing zones will disappear with the passage of time, and fluid particles will be distributed uniformly in the surface of the mixer. Numerical simulation shows that the mixer flow is sensitive to initial condition when rotational velocity of rotors is variable, and this is one of important factors in chaotic flow. Further, calculation of stretching rate of fluid elements indicates that average exponential value of fluid elements stretching in the chaotic flow is two times more than non-chaotic one. This research, based on scientific concepts and theoretical application of chaos in fluid mechanics, denotes that the efficiency of low-speed mixers in highly viscous fluid mixing is increased without any increase in the consuming energy.

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With the development of computers, the application of numerical methods in solving engineering problems has increased considerably. Methods such as Finite Element Method, Finite Volume Method and Finite Difference Method can be mentioned as some. In this research a Boundary Element Method is applied for numerical simulation. The main difference among the Boundary Element method and other numerical methods is the governing mathematics. At first In this method the governing equation is integrated. This leads to a decrease in the dimensions of the problem and then the simulation is performed. In this research, by a change of variable, the Navier Stokes equation is transformed to Navier equation in Elastostatics at first. Subsequently the methods proposed for solving the problems in Elastostatics is utilized to solve the viscous fluid flow. In fact, the applied fundamental solution is the main difference among the proposed method and other Boundary Element Methods. In the proposed method, in contrast to previously proposed methods, the fundamental solution of the Navier equation is utilized for simulation. At last, by considering the governing mathematics a computer code is developed for viscous flow simulation. The code is applied to two different geometries, a lid-driven-cavity and a backward facing step. Convergent solutions is achieved up to Reynolsds numbers equal with 600 and 100 respectively.