Thermal visualization of Ostwald-de Waele liquid in wavy trapezoidal cavity: Effect of undulation and amplitude

The present study is concerned with the numerical simulations of Ostwald-de Waele fluid flow in a wavy trapezoidal cavity in the presence of a heated cylinder situated at the center of the cavity. The work consists in characterizing the mixed convection as a function of the intensity of heat flow. The flow behaviour and temperature distribution in a cavity are the main focus of this study. The lower wall of the cavity is fixed and heated while the wavy surface is insulated and moves with a constant speed. The sloping walls are kept cold and are subject to zero no-slip conditions for velocity components. The whole setup is modeled as a set of coupled partial differential equations and is solved by the Finite Element Method. For pressure and velocity approximations, we use the stable finite element pair P 2 (cid:0) P 1 , while for temperature approximation we use the space of linear polynomial as P 1 . The ranges of the parameters involved in the study are the Ostwald-de Wale index ( 0 . 5 ≤ n ≤ 1 . 5 ) , Prandlt number ( 1 ≤ Pr ≤ 10 ) , Grashof number ( 100 ≤ Gr ≤ 1000 ) , the number of undulation ( 0 ≤ λ ≤ 4 ) , and the non-dimensional amplitude of the wavy surface ( 0 . 02 ≤ A ≤ 0 . 06 ) . The major findings of the study are shown using velocity profile, streamlines, and isotherms. Moreover, the kinetic energy and average ( Nu avg ) Nusselt number is determined for various values of the parameters involved.


Introduction
Research into mixed convection has gained a lot of attention because of its wide range of applications including building ventilation, storage of grains, chemical removal of thin films, disposing of water material, the cooling of electrical parts, etc. respectively Conventional approaches were studied extensively in ventilated enclosures.F. Penot [1] has analyzed the computational study of free convection inside the square cavity in the presence of a large Grashof number.Moreover, he examined the variation in Nusselt numbers for different inclinations.Xin et al. [2] studied the numerical effects of the partially heated air-filled cavity with a direct interpretation of the two-dimensional equations.They found that significant thermal performance in high Ra.Khanafer et al. [3] considered a numerical calculation of mixed convection heat transfer in open-ended enclosures.Also studied the average Nusselt number for different parametric values.Manca et al. [4] have investigated experimentally mixed convection in a cavity with a heated inflow wall.Also discussed is the impact of the Nusselt number in the presence of the Richardson number.Heat transfer in a gasfilled enclosure with a

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partially heated horizontal wall and thermally insulated side walls was studied by Lin et al. [5].Laguerre et al. [6] examined both experimental and numerical analysis of heat transfer by natural convection in a cavity filled with the cylinder.Also observed are the effects of thermal stratification and circular airflow in the cavity.Mahmoudi et al. [7] studied the numerical influence of mixed convection flow and heat transfer in a vented square cavity.Saury et al. [8] studied the characterization of a large Rayleigh number in the air-filled cavity.Furthermore, analyzed the impact of Nusselt number along both hot and cold walls.Xu et al. [9] described the unsteady thermal flow adjacent heated to the sidewall of the cavity.Yousaf et al. [10] examined the thermal and hydrodynamic behaviors were studied in a sinusoidal wavy square cavity.Chorin et al. [11] have numerically simulated the natural convection flow with thermal disturbance in the partially heated cavity.Furthermore, the local temperature variation of the hot wall affects global heat transport.
The power-law fluid is the simplest and most frequent model for non-Newtonian fluids, which has attracted a lot of attention from researchers.The important factor of this model is power-law index ″n ″ when n = 1, then fluid goes to Newtonian behaviors and when n > 1 and n < 1, means that the shear thickening and thinning behaviors in the fluid.Ohta et al. [12] investigated a numerical study of the thermal effect of pseudo-plastic fluid in a square cavity.In addition, analyzing the behavior of shear-thinning on viscous fluid.Lamsaadi et al. [13] studied the heat transfer in a rectangular cavity containing power-law fluids.Moreover, analyzed to the uniform heat flux along the side walls.Barth et al. [14] analyzed the comparison of both experimental and numerical impact of benchmark problem for natural convection of Newtonian fluid in the cubical cavity.Kaddiri et al. [15] investigated thermo-dependent viscous fluid confined in a square cavity and uniformly heated lower wall by using a finite difference scheme.Turan et al. [16] studied the effect of two-dimensional laminar natural convection of non-Newtonian Ostwald-de Waele fluid in a rectangular cavity with partially heated sidewalls.Sheremet et al. [17] numerically investigated the heat transfer in right angle trapezoidal cavity filled by nano-fluid of porosity medium in the presence of Buongiorno's mathematical method.Mahmood et al. [18] studied comparison of Newtonian and non-Newtonian fluid and also for more computational results by using highly refined hybrid mesh.Mahmood et al. [19] examined the effect of shear-thickening and shear-thinning flow in a channel-driven cavity.In addition, analyzed the significant impact of drag and lift forces.Mahmood et al. [20] characterized numerically non-linear Ostwald-de Waele fluid in a channel with a square obstacle.Also studied the influence of power-law index on pressure drop.
In this study, the Ostwald de Waele fluid in a sinusoidal wavy trapezoidal cavity having circular obstacles is studied by considering the different diameters of the circle.The paper is organized as follows: In section 2 geometry of the problem and its mathematical modeling is defined.In section 3 solution to the problem is proposed.In section 4 outcomes of the problem are summed up.In the last section conclusion based on the results and discussion, a section is drawn.

Mathematical modelling
Considered 2-dimensional, laminar, incompressible fluid flow with uniform physical properties in a trapezoidal cavity.The obstacle and the bottom wall are considered heated while the sinusoidal upper wall is considered cold as well as insulted and the other

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two walls are kept cold (see Fig. 1).Introducing the set of non-dimensional variables where x and y are non-dimensional coordinates measure along horizontal and vertical axes, respectively.Also u, v and θ are non-dimensional velocity and temperature.Using these non-dimensional variables in Navier Stokes equations [15] to obtain the non-dimensional governing equations Where the subscript X and Y represent X and Y components of the stress tensor τ, respectively.The non-dimensionalized equations Where v be the kinematic viscosity, g be the gravitational acceleration and β be the volumetric coefficient of expansion.
The lower wall of the trapezoidal cavity is fixed and uniformly heated while the wavy surface is insulated and moves with a constant speed.The sloping walls are kept cold and are subject to zero no-slip conditions for velocity components.For a generalized Newtonian fluid, which obeys the Ostwald-de Waele fluid [15] in non-dimensional Cartesian coordinate we define: The kinetic energy is a very major quantity for flow in a driven cavity which evaluates the scale of momentum for the entire flow.Kinetic energy is defined as To evaluate the characteristic of heat transfer, which is in terms of local Nusselt number is shown as Nu local = − ∂θ ∂n , compute the average Nusselt numWber on the walls of the cavity, the heated cylinder is obtained by utilizing the following mathematical formulation where n and S are normal direction and the thermal region of the surface, it is reasonable to assume that the Nu avg depends on the Prandlt number.Downloaded from mostwiedzy.pl

Numerical scheme and grid convergence
The system of partial differential equations ( 2)-( 5) cannot be solved analytically as they are nonlinear and the presence of non-Newtonian fluid in the trapezoidal wavy cavity; we used the numerical scheme FEM to compute the optimal solution with various diameters of a cylinder.The Newton method was used to solve the original discrete non-linear system of equations, and the linearized inner system was solved with the PARDISO direct solver.In order to achieve the needed level of convergence with less iterations, the PARDISO solver is used along with LU matrix factorization [21,22].
A hybrid grid is generated in the present work to capture the rapid changes in the dependent variables near the boundaries.Fig. 2 reveals the computational grids at fine level (L 6 ) for different diameters of the cylinder.To see the adequacy of the grid chosen for simulations, a grid independence study was performed by computing the average Nusselt number on heated surfaces as a benchmark quantity.
No noteworthy variations in Nu avg were detected with grid refinement at the higher level in space are shown in Table 1.All results were simulated for parameters setting as Pr = 5, β 1 = 0.10, n = 1, λ = 4, and Gr = 100, respectively.The criteria of convergence for the non-linear iteration is set as to , in which ψ represent the general solution component representing velocity, pressure, and temperature.Table 2 shows the mesh statistics for the number of elements (#EL) and the related degrees of freedom (#DOF) at various refinement stages.

Results and discussion
The results are obtained from computations carried out in three diameters of the cylinder are present in the sinusoidal wavy trapezoidal cavity, for the several values of power-law index n, the non-dimensional parameter Reynolds number Re = 20 and Grashof number Gr = 1000 are fixed, respectively.The velocity field, streamline, velocity profile, and isotherms resulting from the calculations carried out are revealed in Fig. 3 to Fig. 7, respectively.
The impact of fluid flow in the sinusoidal wavy trapezoidal enclosure by the influence of cylinder diameter β 1 from left to right and characteristic of the power-law index from top to bottom n are examined in Fig. 3 at Re = 20, Gr = 1000 and Pr = 5.The streamlines are presented in Fig. 3 for different values of flow behavior index (n) and various diameter size of the cylinder.Enhancement in the size

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of diameter for shear-thinning regime, three small eddies occur near the upper region of the obstacle while at the higher values of n, reduces the numbers of eddy on that region.
In Fig. 4, the effects of undulation on velocity profile have been revealed against different diameters of the cylinder.From these plots, it is examined that the undulation of the upper surface effect the boundary layer thickness.The velocity profile for the different amplitude of the wavy surface and various values of the Ostwald -de Waele index (n) are shown in Fig. 5.It can be noticed from Fig.

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that the vortices clustered near the upper corners of the cavity.Moreover, the strong vortices are occurring in case of increasing n at the minimum amplitude of the wavy surface while enhancement of amplitude, the weak vortices appear at the corners of the wavy surface.
The corresponding isotherm contours for variations of number Pr from top to bottom and several diameter sizes of the cylinder from left to right are shown in Fig. 6.Further, it was observed that the heat transfer rapidly increases for both values of Pr and β 1 .Fig. 7 shows that the effects of insulated wavy surface on the isotherms for various Pr and β 1 .The adiabatic wavy surface gives rise to an increased heat transfer mechanism as compared with the cold one.
All these observations are due to the strengthening of the mixed convection with the variation of Prandtl number Pr and several lengths of cylinder diameter are illustrated in Fig. 5 with constant Grashof number Gr = 1000.
For a better description, we have evaluated multiple line graphs at different x locations as shown in Fig. 8.They reveal that the variation in velocity is very mild on the left side of the wavy region whose line graphs are drawn at x = 0.1, 0.2, 0.3, and 0.4.The same behavior is observed after the wavy region at x = 1.7, 1.8, and 1.9.However, the strength of backflow increases in the wavy region of the cavity as is evident from the line graphs at 0.6 ≤ x ≤ 1.4.
The effect of the size of the obstacle on the average Nusselt number against Pr is displayed in Fig. 9.This demonstrates that when we increase the Pr and β 1 that results in enhancement of the average Nusselt number showing more heat transfer due to an increase of the convection phenomenon.In order to determine a functional dependence of Nu avg versus Pr, the method of least squares has been implemented for the case β 1 = 0.10.The relation turned out to be Nu avg = 14.7421 + 0.5128 Pr where 14.7421turns out to be the yintercept and 0.5128 is the slope of linear trend.
We tabulate the kinetic energy at the various power-law exponent n for different diameters of the cylinder shown in Table 3.We observe that the kinetic energy is increased for the increment of both power-law exponents as well as the diameter of cylinders in a sinusoidal wavy trapezoidal cavity which implies both the parameters provide a direct relation with the kinetic energy.

Conclusions
In this pagination, the characteristics of heat and fluid flow inside a wavy trapezoidal cavity containing a heated cylinder have been explored.Finite Element Method (FEM) is employed using mixed finite elements utilized on a hybrid-refined grid to yield highly accurate results in terms of velocity profile, streamline patterns, and isotherms.The numerical simulations have been performed against restricted ranges of parameters n, λ, A, Pr, Re and Gr to generate different convection regimes.The flow rate enhances significantly for different diameter of cylinders and uniformly heating wall.The main findings of this study include the following: • Both the parameters n and β 1 provide a direct relation with the kinetic energy.
• The higher values of n, reduce the numbers of eddies near the wavy region.
• The strong vortices appear for increasing n at the minimum amplitude of the wavy surface.
• Adiabatic wavy surface gives rise to increased heat transfer mechanism as compared with the cold one.
• Number of eddies in the cavity is strongly influenced by the diameter of the obstacle as is evident from the stream function plots.
• A functional dependence of Nu avg on Pr has been calculated using the method of least squares that suggested a linear trend.

Fig. 1 .
Fig. 1.The schematic diagram for the physical problem.

Fig. 2 .
Fig. 2.A computational grid at fine level mesh for various diameters of the cylinder.

Fig. 8 .
Fig. 8. Evaluation of velocity profile at different positions for various n.

Table 1
Grid convergence test for various refinement levels.

Table 2
Mesh statistics versus different refinement levels for various diameters of the cylinder β 1 .

Table 3
Influence on kinetic energy for different power-law exponent n with various diameters of cylinders β 1 .
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