Shear Thinning and Shear Thickening Non- Newtonian Confined Fluid Flow Over Rotating Cylinder

In present work, numerical solution is performed to study the confined flow of power-law non Newtonian fluids over a rotating cylinder. The main purpose is to evaluate drag and thermal coefficients as functions of the related governing dimensionless parameters, namely, power-law index (0.5 ≤ n ≤ 1.4), d imensionless rotational velocity (0 ≤ α ≤ 6) and the Reynolds number (100 ≤ Re ≤ 500). Over the range of Reynolds number, the flow is known to be steady. Results denoted that the increment of power law index and rotational velocity increases the drag coefficient due to momentum d iffusivity improvement which is responsible for low rate of heat transfer, because the thicker the boundary layer, the lower the heat transfer is implemented.


Introduction
Most real fluids in industrial and academic applicat ions aren't considered as Newtonian fluids, they denote non-Newtonian behavior, in wh ich the shear stress is not linearly p roportional to the velocity gradient. Typical examples can be poly mer me lts and solutions, foams, emu lsions and suspensions, etc. [1]. Non-Newtonian flu ids are div ided into two categories: (1) shear-thinning fluids, and (2) shear-thicken ing flu ids. Behavior and response to shear stress of shear-thinning and shear-thickening fluids are shown in Fig. 1. Newtonian flu ids flow across cylinders is one of those typical problems have been investigated in many researches due to extensive applications [2][3][4][5][6]. A lso imposing rotation to cylinders in order to control boundary layer on cy linder or other kinds of bluff bodies has been devoted much attention [7,8]. Non Newtonian flu ids flow over rotating cylinders is included in many industrial applications such as coating operations, roller bearing applications, oil drilling operations, mixing vessels with novel impeller designs, etc.
As Newtonian fluids flow over cylinder is not steady for every Re, non Newtonian fluids have also crit ical Re showing unsteadiness at about Re = 46-47 which is reported in [9,10]. Finally, again the flow reverts to a steady flow regime at Re = 100.
Admittedly, extensive accurate results are now available on the mo mentum and heat transfer characteristics for a stationary cylinder in unconfined power-law fluids in the steady flow reg ime [11][12][13][14][15] for both shear-thinning and shear-thickening flu ids. It has shown in the literature that for low Reynolds numbers, shear-thinning behavior increases drag above its Newtonian value and shear-thickening has repercussive effect. Patnana et al [10,11] studied the flo w and heat transfer phenomena fro m a cylinder swa mped in power law fluids in the laminar vortex shedding regime. A lso researcher's results are available for elliptic cylinders [16,17] and square [18,19]submerged in confined and unconfined power-law fluids using steady state assumption. But Sahu [20] reported on the unsteady flow over a square cylinder in the laminar vortex shedding regime.  [21]; At low Re, viscoelasticity decreases the drag on a cylinder beneath its Newtonian value and it increments the drag at high Re.
According to the authors knowledge, there have been only three reports on the flow of non-Newtonian flu ids over a rotating cylinder. Christiansen [22,23] has reported experimental data on the local velocity field for a cylinder rotating in Newtonian and polymer solutions of varying levels of shear-thinning and viscoelasticity using laser-Doppler anemo met ry. Also, Panda and Chhabra [24] has studied the low Re non Newtonian shear thinning over rotating cylinder. They co mpared their results with [22,23] and showed only the behavior of shear thinning fluids for unconfined flow. But it is well known that the fluids flow should be implemented through a pipe or other confined channels. So investigation of non-Newotonian confined flu id flow over rotating Cylinder is still in rare and the present work will fill this gap. Also [24] has not reported any result for shear thickening flu ids and as we know the industrial applications include the both kinds of non Newtonian fluids. It is thus safe to conclude that very little is known about the flow of h igh Re (in laminar range) non-Newtonian flu ids past a rotating cylinder. This work is thus concerned with the flow of power-law fluids past a rotating cylinder over the range of conditions power-law index (0.5 ≤ n ≤ 1.4), d imensionless rotational velocity (0 ≤ α ≤ 6) and the Reynolds number (100 ≤ Re ≤ 500) over which the flow is expected to be steady and two-dimensional. After validation of results with [1] and [24], the both drag and thermal coefficients will be developed for the above range of variations.

Problem Statement and Governing Equations
We assume that the fluid is inco mpressible with uniform velocity U 0 flowing over a cy linder of d iameter D (infin ite length in z-direction) rotating with an angular velocity of Ω in the counter-clockwise direct ion, as shown in Fig. 2. The cylinder is located at the center of a square of size H. The continuity and momentu m equations written as follows [24]: • Continuity: (2) where ρ, V, P, τ and g, respectively, are the fluid density, velocity vector, isotropic pressure, extra stress tensor and gravity. The co mponents of the extra stress tensor τ ij are emerged as: where the rate of deformation tensor and the scalar viscosity function are introduced by: n is the power-law index. Ad mittedly, n = 1 denotes the Newtonian flu id, n < 1 denotes shear-thinning and n > 1 shows shear-thickening treat ment. (4) Outlet: Fully developed condition is imposed on the exit plane. Th is is similar to the homogeneous Neumann condition, that is, ∂U x /∂x = 0 and ∂U y /∂x = 0 The governing Eqs. (1) and (2) are solved numerically together with the mentioned boundary conditions. Then Ux, Uy and p are post processed to calculate the derived quantities such as stream function, drag and thermal coefficients as functions Reynolds number (Re), power-law index (n) and the non-dimensional rotational velocity of the cylinder (α). It is appropriate to introduce the definitions of some of these parameters as follows: •

Numerical Details
Govern ing equations have been solved using FLUENT (version 6.3.26). The cells (Fig. 3) were generated using the commercial grid tool GAM BIT (version 2.3.16). 2D laminar flow of inco mpressible non Newtonian flu id was used on the collocated grid arrangement. While the main purpose of this work is to solve the governing equations using steady state assumption, limited time-dependent simulations for extreme values of the governing parameters such as Re = 200, n = 3 and α = 6, etc were also conducted to make sure that the flow regime was steady in the study. Momentum equations were discritized using second order upwind scheme. The SIMPLE scheme was used for solving the pressure-velocity coupling equations. Relative convergence criteria of 10 −5 for the residuals of the continuity and x and y-co mponent of the mo mentu m equations were set in this work.

Gri d Inde pe nde ncy
Three different sized cells are used to investigate the independency of the results to the mesh size. Coarse, med iu m and fine meshes which are related to 117600, 151100, 218100 nu mber of cells, respectively, were put in use. Table 1 summa rizes the relative quality and performance of the three grids for the resulting values of drag coefficient. As seen in table 1 results for G2 and G3 are close to each other resulting for a mesh independent values. Due to larger CPU time and co mputational cost of G3, g rid G2 denotes a good compromise between the accuracy and computational efforts and hence all results reported herein are based on the use of grid G2.

Results and Discussion
The results are governed by 3 dimensionless parameters, namely, Re, n and α and the objective of the present study is to develop this functional relat ionship. Results fro m numerical solution are presented in parts containing the variation of the 3 dimensionless parameters. In each part, the effects on drag coefficient and heat transfer coefficient are studied.
However, before demonstration of new results, it is useful to prove and ascertain the accuracy of the new results reported herein. Table 2 shows comparison among the results of present work and [1,24]. A good agreement is verified through the simulation.

Variation of Reynol ds Number
In this part, Reynolds number increment was set from 100 to 500 while rotating velocity has been fixed in α = 2. As shown in Fig. 4, increasing Reynolds number increases the specified thermal coefficient. As the velocity of fluid increases, the thickness of the hydrodynamic boundary layer on the upper surface of cy linder increases as a result of high velocity gradient considering the cylinder rotational direction. So the net rate of heat transfer fro m the cylinder decreases. Although the boundary layer at the bottom surface of cylinder contains less velocity gradient, but increment of the difference among the flu id's velocity and the cylinder rotating velocity wh ich causes an increase in the boundary layer thickness plays an important role in reducing the rate of thermal coefficient increment. It can be infrared fro m Fig. 4 that shear thickening of non-Newtonian fluids, make th icker boundary layer wh ich reduces thermal treat ment efficiently.
As shown in Fig 5, the Hydrodynamic aspect of the results denotes that increasing Reynolds number increases the drag coefficient as a result of boundary layer thickness increment which is responsible for skin friction in the upper and lower sides of cylinder.

Variation of Non-Newtoni an Power
This section is devoted to non-Newtonian power index effects at various α. For the constant Re of 200, power index is varied fro m 0.5 to 1.4 for stationary cylinder and cylinder with α = 3 and 6. As shown in Fig 6, heat transfer decrease as non-Newtonian power increases. Velocity gradient increases faster for n > 1 in co mparison with n < 1which causes a great decrease in heat transfer phenomenon. Even though rotational cylinder reduces the viscosity of fluid due to reduction of velocity gradient on cylinder surface. Fig.7 shows that increasing power index will arise the drag coefficient due to large skin friction at cylinder surface. It can be deduced that the industrial consumption of shear -thickening non Newtonian flu ids at these conditions is not recommended, because the both increasing drag coefficient and descending thermal coefficient is encountered, which is not recommended.

Conclusions
In the present study non-Newtonian laminar fluid flow over a rotating cylinder is investigated. Non-Newtonian power law model is used for simulation of the problem. Variation of d ifferent parameters such as non-Newtonian power, rotating velocity of the cylinder and Reynolds number of the inlet fluid are studied. Results indicated a great dependence of the thermal and hydrodynamic aspect of the problem on these parameters. So me of the results are pointed below: 1) Increasing Reynolds number increases heat transfer amount and drag coefficient.
2) Increasing non-Newtonian power, make the hydrodynamic boundary layer develop faster, resulting in a great reduction of heat transfer amount and therefore a great increase in drag coefficient.
3) Increasing rotating velocity of the cylinder increases drag coefficient.
Further studies are needed to develop the Re range and other effective parameters such as reducing the distance of rotating cylinder to the walls. This may influence on the thermal and drag coefficients.