Effective Computational Methods for Solving the Jeffery-Hamel Flow Problem

في هذا البحث، تم تنفيذ الطريقة الحسابية الفعالة (ECM) المستندة إلى متعددة الحدود القياسية الأحادية لحل مشكلة تدفق جيفري-هامل غير الخطية. علاوة على ذلك، تم تطوير واقتراح الطرق الحسابية الفعالة الجديدة في هذه الدراسة من خلال وظائف أساسية مناسبة وهي متعددات الحدود تشيبشيف، بيرنشتاين، ليجندر، هيرمت. يؤدي استخدام الدوال الأساسية إلى تحويل المسألة غير الخطية إلى نظام جبري غير خطي من المعادلات، والذي يتم حله بعد ذلك باستخدام برنامج ماثماتيكا®١٢. تم تطبيق تطوير طرق حسابية فعالة (D-ECM) لحل مشكلة تدفق جيفري-هامل غير الخطية، ثم تم عرض مقارنة بين الطرق. علاوة على ذلك، تم حساب الحد الأقصى للخطأ المتبقي ( )، لإظهار موثوقية الطرق المقترحة. تثبت النتائج بشكل مقنع أن ECM و D-ECM دقيقة وفعالة وموثوقة للحصول على حلول تقريبية للمشكلة.In this paper, the effective computational method (ECM) based on the standard monomial polynomial has been implemented to solve the nonlinear Jeffery-Hamel flow problem. Moreover, novel effective computational methods have been developed and suggested in this study by suitable base functions, namely Chebyshev, Bernstein, Legendre, and Hermite polynomials. The utilization of the base functions converts the nonlinear problem to a nonlinear algebraic system of equations, which is then resolved using the Mathematica®12 program. The development of effective computational methods (D-ECM) has been applied to solve the nonlinear Jeffery-Hamel flow problem, then a comparison between the methods has been shown. Furthermore, the maximum error remainder ( ) has been calculated to exhibit the reliability of the suggested methods. The results persuasively prove that ECM and D-ECM are accurate, effective, and reliable in getting approximate solutions to the problem

Analytical investigation of laminar flow through expanding or contracting gaps with porous walls

AbstractLaminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions is investigated analytically using optimal homotopy asymptotic method (OHAM). OHAM is a powerful method for solving nonlinear problems without depending to the small parameter. The concept of this method is briefly introduced, and it׳s application for this problem is studied. Then, the results are compared with numerical results and the validity of these methods is shown. After this verification, we analyze the effects of some physical applicable parameters to show the efficiency of OHAM for this type of problems. Graphical results are presented to investigate the influence of the non-dimensional wall dilation rate (α) and permeation Reynolds number (Re) on the velocity, normal pressure distribution and wall shear stress. The present problem for slowly expanding or contracting walls with weak permeability is a simple model for the transport of biological fluids through contracting or expanding vessels

MHD nanofluid flow in a semi-porous channel

This research project paper, the problem of laminar nanofluid flow in a semi-porous chan- nel is investigated analytically using Homotopy Perturbation Method (HPM), Least Square Method (LSM) and Differential Transformation Method (DTM). This problem is in the presence of transverse magnetic field. Due to existence some shortcomings in each method, a novel and efficient method named LS-DTM is introduced which omitted those defects and has an excellent agreement with numerical solution. Here, it has been attempted to show the capabilities and wide-range applications of the Homotopy Perturbation Method in comparison with the numerical method used for solving problems. Then, we consider the influence of the three dimensionless numbers: the nanofluid volume friction, Hartmann number for the description of the magnetic forces and the Reynolds number for the dy- namic forces

Numerical Study of the Casson Non-Newtonian Fluid Flow over a Nonlinear Stretching Sheet

In this paper, numerical study of Casson non-Newtonian fluid over a nonlinear stretching sheet using shooting method will be presented. The governing nonlinear partial differential equation is converted to an ordinary differential equation  by using similarity transformations. This ordinary differential equation is handled numerically with the use of well-known shooting technique aided by Runge-Kutta method. For comparison of the results, MATLAB in-built solver BVP4C is used. The discussion of the results is provided in the forms of tables as well as graphically

Dual Solutions for MHD Jeffery–Hamel Nano-Fluid Flow in Non-parallel Walls Using Predictor Homotopy Analysis Method

The main purpose of this study is to present dual solutions for the problem of magneto-hydrodynamic Jeffery–Hamel nano-fluid flow in non-parallel walls. To do so, we employ a new analytical technique, Predictor Homotopy Analysis Method (PHAM). This effective method is capable to calculate all branches of the multiple solutions simultaneously. Moreover, comparison of the PHAM results with numerical results obtained by the shooting method coupled with a Runge-Kutta integration method illustrates the high accuracy for this technique. For the current problem, it is found that the multiple (dual) solutions exist for some values of governing parameters especially for the convergent channel cases (α = -1). The fluid in the non-parallel walls, divergent and convergent channels, is the drinking water containing different nanoparticles; Copper oxide (CuO), Copper (Cu) and Silver (Ag). The effects of nanoparticle volume fraction parameter (φ), Reynolds number (Re), magnetic parameter (Mn), and angle of the channel (α) as well as different types of nanoparticles on the flow characteristics are discussed