International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 11 Issue: 03 | Mar 2024
p-ISSN: 2395-0072
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Numerical solution for two dimensional Non Newtonian boundary layer flow over a flat plate with suction/injection through porous media by using Collocation method Mihirbhai Prajapati 1, Jyoti Chaudhari2 ,Dilip Joshi3 1 Head& Lecturer, S. & H. Department, Bhulabhai VanmaliBhai Patel Inst. of Tech.,Umrakh, Bardoli, Gujarat, India 2 Head & Lecturer, S. & H. Department, Shri K J.Polytechnic , Bholav, Bharuch, Gujarat, India
3 Head & professor, Departmen of Mathematicst, Veer Narmad South Gujarat University, Surat, Gujarat, India
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Abstract - In this paper, we are given a graphical
Griffiths (2017) results show that the effects of shearthinning areto stabilize the boundary-layer flow.
presentation of the Falkner-Skan equation for the study of two-dimensional permeable steady boundary-layer viscous over a flat plate in the presence of Non-Newtonian power-law fluid and it is presented by a power-law model. Similarity transformation techniques are used to convert the boundary layer equations into a third order nonlinear differential equation. An equation containing three flow parameters like m is power-law relation parameter, omega is the porous parameter and beta is the Stream-wise pressure gradient. We converted the third order nonlinear differential equation into third order linear differential equation by using Quasi linearization techniques. Results are obtained for the velocity profile, viscosity profile, and skin friction for the value of physical parameters is discussed in brief.
In this paper, we consider third order nonlinear ordinary differential equations with three boundary conditions. This third order nonlinear ordinary differential equation converted into a linear ordinary differential equation by using Quasilinearization techniques. Apply the Collocation Methodmethod (D F Griffiths-1978) to solve the third order linear differential equation. In this method, we have to consider the linear combination of the trail function with a constant coefficient. Then apply this method to find the constant coefficient. These constants put in the assumed solution and from the solution we identify the behavior of the boundary-layer flow of the Ostwald-de Waele fluid.
2. FORMULATION OF THE PROBLEM
Key Words: Boundary-layer equation, Falkner-Skan equation, Quasilinearization Techniques, Similarity transforms, Collocation Method, Non-Newtonian fluid, power-law fluid.
Consider
.q 0
(1)
q. q p . kp t
1.INTRODUCTION
Applications of Non-Newtonian and Newtonian fluids are very useful in an industrial and technologically. Air or water which is Newtonian fluids serves as a benchmark for the fluid flow behavior. However, the behavior of NonNewtonian fluids is more important in the industry rather than Newtonian fluids. Non-Newtonian fluids like oil-water emulsions, foams, gas-liquid dispersions. Acrivosetal (1960) shows the thickness of boundary –layer for shear-thinning fluids is large compared to the shear-thickening fluids. Wu and Thomson (1996) that for moderate values of the Reynolds number, the boundary-layer equation for shearthinning fluids provides an accurate solution. Andersson and Irgens (1998) show that the boundary layer equation predicts the finite-width of the boundary-layer for shearthickening fluids. Results are obtained from Andersson and Irgens (1998) to support Filipuss et all (2001) gave rigorously mathematical analysis predicts the same finitewidth of the boundary-layer. Denier and Dabrowiski (2004) have shown that these are double solution for the boundarylayer equations when a self-similar form is assumed.
Which is the two-dimensional laminar boundarylayer flow of a viscous and incompressible fluid over flate through porous media with a Non-Newtonian power-law fluid. The above equation is express in the absence of body forces.
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(2)
In equation (2) the parameter is defined as is the fluid density, p is the pressure, k is the permeability of the porous medium and is the deviatoric stress tensor and is given by
= (q )
(3)
Where q is the second invariant of the strain-rate tensor and the shear rate q is given by q with
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1 1 (q : q ) 2 2
(4)
q (u u T )
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