A potential alternative CFD simulation for steady Carreau–Bird law-based shear thickening model: Part-I

  • Khalil Ur RehmanEmail author
  • M. Y. Malik
  • R. Mahmood
  • N. Kousar
  • Iffat Zehra
Technical Paper


The current article provides the numerical investigation into an infinite-length circular cylinder placed as an obstacle in the flow of non-Newtonian fluid. To be more specific, a channel of length 2.2 m and height 0.41 m is considered. The Power law fluid model is carried out with Carreau–Bird law as a non-Newtonian fluid model, and both the Power law linear (constant) and parabolic velocity profiles are initiated simultaneously at an inlet of the channel. The right wall as an outlet is carried with Neumann condition. The relative velocity of Power law fluid particles with both the lower and upper walls is taken zero. A mathematical model is structured in terms of nonlinear differential equations. A well-trusted numerical technique named finite element method is adopted commercially. The LBB-stable element pair is utilized to approximate the velocity and pressure. The nonlinear iterations are stopped when the residual is dropped by \(10^{ - 6}\). The impact of Power law index and Reynolds number on the primitive variables is inspected. The obtained observations in this direction are provided by means of both the contour plots and line graphs. Due to a circular obstacle, both the drag and lift coefficients are evaluated around the outer surface of an obstacle towards the higher values of the Power law index. The numerical values of drag and lift coefficients up to various refinement mesh levels of domain are provided by way of tables. It is noticed that the parabolic velocity profile at an inlet of channel is compatible as compared to the linear velocity profile. Further, both the drag and lift coefficients are increasing function of Power law index.


Power law fluid model Linear and parabolic profiles Finite element method Drag and lift coefficients 

List of symbols


Space variables

\(\vec{V} = (u,v,w)\)

Velocity field


Fluid density



\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {\tau }\)

Stress tensor


Body force


Fluid viscosity


Apparent viscosity


Power law index


Consistency coefficient


Mean velocity


Characteristic length (diameter of circular cylinder)


Reynolds number

\(U_{\hbox{max} }\)

Maximum velocity


Drag force


Lift force


Drag coefficient


Lift coefficient



The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha 61413, Saudi Arabia, for funding this work through research groups programme under grant number R.G.P-1/77/40.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Khalil Ur Rehman
    • 2
    • 3
    Email author
  • M. Y. Malik
    • 1
  • R. Mahmood
    • 2
  • N. Kousar
    • 2
  • Iffat Zehra
    • 2
  1. 1.Department of Mathematics, College of SciencesKing Khalid UniversityAbhaSaudi Arabia
  2. 2.Department of MathematicsAir UniversityIslamabadPakistan
  3. 3.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan

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