The effect of Nusselt number on the bi-viscosity fluid subjected to the discrete heating effect

Discrete heating action has been established as an energy efficient method. Nusselt number plays a significant role in the heat transfer rate. Local Nusselt number is useful in analyzing the heat transfer rate along the sections of the side walls with different or same temperature in a heated enclosed cavity. Whereas the average Nusselt number quantifies the total heat change in the closed cavity. While the path of the trajectory of the heat flow is being analyzed by heat function. In the current work, multiple heaters of length l 1 , l 2 and l 3 are assembled at each side of the isosceles triangle. The numerical simulations of the modeled system along with boundary conditions are carried out by using the standard Galerkin Finite Element Method. The nonlinear algebraic system is solved by Newton method. The behavior of fluid inside the closed triangle with ( Pr = 0.7,10) is observed. The results are reported in terms of the graphs of the temperature contours, streamlines, average and local Nusselt numbers. The maximum heat transfer rate is observed in convection dominant case. By increasing the magnitude of a bi-viscosity parameter, secondary circulation is observed for Pr = 10. The rate of heat transfer has a direct relation with the bi-viscosity parameter. The maximum values of average and local Nusselt numbers are achieved by using the heat generation parameter against the Grashof number. Cite this article as: M. Adnan A, Mudassar R. The effect of Nusselt number on the bi-viscosity fluid subjected to the discrete heating effect. J Ther Eng 2021;7(7):1797–1814.


INTRODUCTION
In many phenomena of practical approaches, the flow and heat transfer inside a closed cavity have dragged the attention of many authors.As flow and heat transfer expounds the pivotal applications.Convection can be carried out in two different ways: one of them is natural convection and the other is mixed convection.Natural convection is also known as free convection in which the fluid motion inside the cavity is generated only by density differences in the fluid occurring due to temperature gradients.Whereas in mixed convection, convection is carried out by the motion of the boundary wall of the cavity along with gravity and density gradient.Both mechanisms of convection are constructed by using the fundamental laws of mass, momentum, and energy for the flows of fluids.The behavior of heat transfer inside the cavity depends mainly on the type of convection, fundamental laws, appropriate boundary conditions, and moderate heat transfer rate.The actual solution of governing equations of heat transfer in the cavity is a quite uphill task, so the results are computed via numerical approaches.Eventually, mixed convection can be quantified by unitless parameters: Reynolds, Grashof, and the Prandtl number.Some simulations require the combination of Grashof and Reynolds number.The literature survey of this topic is quite pervasive.
Bondareva in [1] numerically examined the convection melting inside a 2D and 3D enclosure with a heat source.Sivakumar in [2] performed numerical study to analyzed mixed convection heat transfer in a lid driven cavity with discrete heaters of different length and concluded that heat rate enhanced on reducing heating proportion area.Mohamad and Viskanta in [3] reported the three-dimensional flow and heat transfer in a shallow cavity filled with a fluid whose density increase with depth.The top wall was heated and lower was kept cold.Mixed convection heat transfer in an inclined enclosure whose one side was heated, and one side was cool under the impact of the magnetic field.The effect of thermal radiation and heat generation on two-dimensional heat transfer was also analyzed in [4].Heat transfer in a square cavity with linearly heated side walls was analyzed by Basak,et al. in [5].Numerical results were computed by using the finite element method.Mixed Convection heat transfer behavior in a square cavity containing porous medium with a movable lid was studied by Basak, et al. in [6], in which all the four walls were kept at different temperatures.
Madhuchhanda Bhattachary, et al. in [7] elaborated on the isothermal and non-isothermal cases of mixed convection heat transfer in the trapezoidal cavity with cold top wall and the hot bottom wall.Ching, et al. see [8] reported the mass and heat transfer with mixed convection in a right triangular enclosure and discussed two cases concerning the direction of motion of the left wall.Basak, et al. in [9] investigated the mixed convection heat flow in a square cavity, whose lower wall heated uniformly and non-uniformly and a top wall was moving at a constant speed.In the case of uniform heating plates, the local Nusselt number showed that the heat transfer at the edges is higher than the rate of heat transfer in the center of the bottom wall.In the non-uniform case, the rate of heat transfer at the edges was observed higher than at the center.A laminar steady state double diffusive mixed convection heat transfer inside a right triangular cavity analyzed by the M. Hasanuzzaman, et al. in [10].They also expounded on the impact of the Lewis number on the heat and mass transfer.Simulations were carried out by Galerkin weighted residual finite element method.M. Turkyilmazoglu in [11] examined the highly non-linear magnetohydrodynamic viscous fluid flow phenomenon analytically.
In the above-indicated literature survey, Newtonian fluids were taken into consideration.It is natural to think that what will be the effects on thermal efficiency when non-Newtonian fluids are taken inside the cavity.To give the lucid answer of this question, the current study is performed by considering the constitutive equations of the bi-viscosity model.A penalty finite element method on the constitutive equations of the bi-viscosity model is performed to investigate the impact of the discrete heater at the center of walls on the forced convection heat transfer in a triangular cavity.Moreover, the heater at the bottom wall is moving along the positive x-axis direction with uniform velocity.The motivation of the current work is also, due to diverse applications of non-Newtonian fluids.S. Siva Sankaran, et al. in [12] reported the numerical solution of the mixed convection heat transfer in a square cavity with different positions of heaters by using the finite volume method.Magnetohydrodynamics mixed convection heat transfer in a partly heated cavity filled with non-Newtonian fluid was investigated by Fatih, et al. in [13].A revolving cylinder was subjected at the center of the cavity.The simulations were carried out by using the Galerkin weighted residual finite element method.The mixed convection heat flow behavior in a square cavity subjected to the heater at the bottom wall with a top cold wall, and movable sidewall, filled with nanofluid was expounded by the Sebdani, et al. in [14].The effect of shear force was examined by keeping the buoyancy force constant and vice versa.The numerical approximation was made to elaborate the combined convection impact on the heat transfer in an open square cavity.The left wall of the cavity is heated, fully or partly with a uniform heating source.The observation was also made between the relationship of the length of the heater and Hartman number [15].Ali,et al. in [16] examined the mixed convection heat transfer in a triangular cavity with a moving heated lower wall at constant speed filled with a non-Newtonian fluid.The simulation was carried out by using a Galerkin finite element method.Debayan Das, et al. in [17] analyzed the relationship between the number of heaters and thermal efficiency.Double heaters were subjected to each sidewall of the cavities (square and triangular) yields the higher thermal efficiency in contrast to the single heater.Simulations were carried out by a Galerkin finite element method in [18].In a further study, they examined the impact of the position of multiple heaters on the local Nusselt number.Heat transfer in porous triangular and square enclosures with discrete double heaters along the vertical or inclined walls were elaborated in [19].

MATHEMATICAL MODEL
In the present study, a mixed convection problem is considered in a triangular geometry.Figure 1 illustrates the coordinate systems and geometry of given problems.The flow is assumed to be steady two dimensional in a triangular shaped cavity filled with viscous fluid.Three heaters l 1 , l 2 and l 3 are placed at the center of each wall as shown in Figure 1.These heaters have uniform temperature θ h .While temperature at w 1 , w 2 , w 3 , w 4 , w 5 and w 6 is θ c (cold walls).The bottom heater l 1 moves in positive x direction.A uniform magnetic field is also applied in positive x direction.
So, there are three factors that caused the fluid motion inside triangle.Which are moving heater, density difference and gravity force.Due to different temperature along the walls, density difference arises.These factors (gravity and density) produce buoyancy force.The constitutive relation related to the bi-viscosity fluid, see in [20], can be written as: where, It is remarkable that the matter will act as solid for π < π c and it performs as fluid for π > π c .Considering above deliberation, the governing system of this problem becomes: ) .
( ) The Eq. ( 2) is the continuity equation, also called incompressible condition, Eqs. ( 3) and ( 4) are momentum equations, Eq. ( 5) is the heat equation.The radiative heat flux is defined as: Assume that, the temperature difference within the flow fluid is small.So expending θ 4 by using Taylor series about θ c , gives θ 4 = 4θ c 3 θ -3θ c

4
. Establish the dimensionless parameters: The Eqs. ( 2) -( 5) and boundary conditions defined in Eq. 6, takes the following form: The velocity and temperature boundary conditions within the cavity can be expressed as follow: .
The velocity components u and v are subjected to the no-slip boundary conditions.To analyze the energy efficiency researchers suggested the study of thermal boundary conditions on the cavity.

SOLUTION PROCEDURE
The governing equations ( 9)-( 12) are approximated numerically by using Galerkin Finite Element Method with boundary conditions stated in (13).In order to penalize the pressure term in flow model the Penalty finite element method [21], [22] is applied by using Penalty parameter γ given below: Eqs. ( 10) and ( 11) can be written as follows by using ( 14) The weak formulations of the Eqs.( 12) and ( 15) -( 16), on the element of Ω with the weight functions η 1 , η 2 and η 3 are: ) .
Where the subscript Ω ∈ on the integral demonstrates that the integral is being calculated over the given triangular cavity i.e. the notion Ω ∈ the limit of integration.Where η 1 , η 2 and η 3 are trial functions.
Here, u(x,y), v(x,y) and T(x,y) are approximated by the finite element interpolations u e , v e and T e over the triangular elements ψ e , where, ψ i e trial function defined on the six nodal triangular elements.By Substitution of Eq. ( 18) into the weak formulation, the j th equation of the finite element model becomes: from above equation, we can write as follows: The positive value of Ψ refers to the anticlockwise circulation, likewise the negative value of Ψ means clockwise circulation.Since we are using a no-slip boundary condition, so the value of stream function near the wall is zero.
Nusselt Number is a leading dimensionless number for heat transfer studies.It plays a vital role in both forced and natural convection.Nusselt number is also called a dimensionless heat transfer coefficient for convection.
The mathematical expression of local Nusselt number Nu is given below: Local Nusselt number for the bottom and side heater is given as under:  The relation between average and local Nusselt number can be defined as:

CODE VALIDATION
To fortify the algorithm, we have tested and validated our code for the previous study of T. Basak, et al. [9], in which uniform and nonuniform heating effect of bottom wall in a lid driven cavity was investigated by keeping the vertical wall at constant temperature.It is noted that our results agree well with the corresponding results reported in [9].

NUMERICAL RESULTS AND DISCUSSION
The nature of the stream function, temperature zone, local and average Nusselt number for certain values of involved parameters are interpreted vividly.From the literature on bi-viscosity flows, the simulation is seized for certain values of parameter Ha, Gr, Re and Ri [16].The radiation parameter N R is restricted from 0 ≤ N R ≤ 10 for reference followed by [23].A similar approach has not been evaluated for the cavities.In the current study, β is taken pretty much in the same manner as taken in reference [16].The yield stress and shear thinning aspect of

MESH INDEPENDENCE TEST
Figure 3 depicts a direct relationship between bi-viscosity parameter and stream function.For different values of β different patterns of stream, lines are plotted.For β = 0.01, anticlockwise primary circulation is observed, and maximum value obtained for stream function is 0.094.Secondary circulation is appeared at β = 0.5 and β = 1.For β = 0.5 the value of stream function is 0.14.While in case of β = 1 the ψ = 0.17.
Figure 4 reveals the interesting aspect of current study.For β = 0.01, the maximum value of stream function which is observed is 0.093.A secondary circulation at β = 0.5 is detected with same value of stream unction as in the case of β = 0.01.When the value of bi-viscosity parameter gained fluid can be elaborated by the bi-viscosity model.Blood slurries are the prevailing examples of bi-viscosity fluid.The Prandtl number Pr for the blood ranges from 10 to above.While there is no presence of Prandtl number ranges for the slurries.That's why the simulations in the current study are carried out for a fluid whose Prandtl ranges from 0 ≤ Pr ≤ 10.
Figure 2 is the structure of the coarse mesh of current geometry.It consists of 191 triangles.However, we use the fine mesh consisting of 1651 triangles.Here in Figure 2, the red zone on the wall represents the heater.It clearly shows how we make the distribution of hot and cold walls.The motion of the bottom heater exposed the inertial impact on the bottom wall.While N R = 0 gives a slightly high value of ψ as compared to other values of thermal radiation parameter.
From Figure 8 it can be noticed that for Pr = 10 the relationship between thermal radiation parameter and stream function is inverse.The anticlockwise secondary circulation is recognized.The maximum motion of the fluid is observed for N R = 0.The velocity of motion decreased as the value of the thermal radiation parameter increased.
Figure 9 illustrates the behaviour of the local Nusselt number along the bottom wall of the cavity against the bi-viscosity parameter.For β = 1 the maximum value of Nusselt number is attained.For x = 0.7 all three values of the bi-viscosity parameter give maximum value of Nusselt number.For 0.75 ≤ x ≤ 1.05, the Nusselt number shows the decreasing behaviour.After that graph depicts the increasing behaviour for the interval 1.1 ≤ x ≤ 1.3.For Pr = 10, it's analysed that for three different values of the bi-viscosity parameter the Nusselt number showed different demeanour than for Pr = 0.7 at the bottom wall.The difference between the maximum values of Nusselt number for β = 1 and β = 0.5 is comparatively smaller than for β = 0.01.
At x = 0.7 Nusselt number gained its maximum value after that it starts to decrease.For β = 1 and β = 0.5 Nusselt number exhibited decreasing behaviour for 0.7 Figure 10 elaborate on the nature of the local Nusselt number at Pr = 0.7 for the bottom wall of the cavity against the thermal radiation parameter.Graph exhibit the decreasing attitude for the interval 0.7 ≤ x ≤ 1.While the interval 1.05 ≤ x ≤ 1.3 expounds the increasing behaviour    Figure 12 exhibits the trend of Nusselt number against the heat generation parameter ranges from -10 ≤ ∈ ≤ 10, for the bottom wall.There exists an inverse relationship between the heat generation parameter and Nusselt number.In the case of Pr = 0.7, the value of ∈ decreases as the magnitude of Nusselt number increases.The maximum value of Nusselt number is achieved in the interval 1 ≤ x ≤ 1.3, against the ∈ = -10.While in case of Pr = 10, the behaviour of Nusselt number is decreasing in the interval 0.7 ≤ y ≤ 1, and increasing in 1 ≤ x ≤ 1.3.The profile of     • For convection dominate case a slight difference can be seen in the isotherms for Hartmann number, thermal radiation, and bi-viscosity parameter, while in the case of Pr = 0.7 the effect of Hartmann number, bi-viscosity and thermal radiation parameter on the isotherm is negligible.The results conferred that the settings of the properties of the fluid are the key role to achieve the desired thermal effects.Maximum intensity of circulation was observed for large value of bi-viscosity parameter i.e. β = 1, for Pr = 0.7.A significant change in isotherm is observed for Pr = 10, by increasing the value of the bi-viscosity parameter β, and Hartmann's number Ha.Hence, the bi-viscosity parameter should be large to improve the thermal operations of fluid.the linearly decreasing demeanour.For side wall heater, the maximum value of the average Nusselt number is observed for β = 0.5.While β = 0.01 retained its behaviour and gives almost a constant value of average Nusselt number.Figure 14 illustrate the trend of average Nusselt number against the broad range of thermal radiation parameter.For Pr = 0.7 observations are carried out for the bottom and sidewall of the cavity.In the case of the bottom wall for Pr = 0.7 and 10, the average Nusselt number showed a linearly increasing trend.While in the case of side wall heater for Pr = 0.7 average Nusselt number depicts the decreasing behaviour.As we increase the value of Prandtl number Pr = 10 a little bit different demeanour of Nusselt number is recognized.

NOMENCLATURE
The values of Nusselt number linearly increase for thermal radiation parameter ranges from 0 ≤ N R ≤ 1.But in the case of N R = 5 the average Nusselt number illustrates a decreasing behaviour.
Figure 15 demonstrate the profile of the average Nusselt number over a wide range of heat generation parameter.In both cases of the bottom and side wall for Pr = 0.7 and Pr = 10, an increasing trend is noticed.

CONCLUSION
The analysis is carried out for the study of a bi-viscosity fluid through mixed convection in a triangular cavity under the influence of heating and magnetic effects.The uniformly discrete heating effect is applied at the center of each wall.The heater at the bottom wall is assumed to be moving along positive x-direction.While the heaters at the inclined walls are kept fixed.The numerical simulations of governing equations are carried out by using the standard Galerkin Finite Element Method.Based on the obtained results in the present study, the important conclusions are drawn as follow: • By increasing the magnitude of a bi-viscosity parameter, secondary circulation is observed for convection dominate case.The rate of heat transfer has a direct relation with the bi-viscosity parameter.• For thermal radiation parameter N R , strong secondary circulations are spotted for Pr = 0.7,10.• For the different value of Hartmann's number, only primary circulations are observed in both cases (Pr = 0.7,10).• The maximum value of local and average Nusselt number is observed for convection dominate case at the bottom wall of the cavity against the heat generation parameter ∈ = -10.• Maximum value of stream function is observed for Pr = 0.7 at β = 1, while the minimum value is noticed against the Hartmann number Ha = 60 for convection dominate case.

Figure 1 .
Figure 1.Geometry of the flow problem.
function Ψ exhibit the motion of fluid in the geometry.The relation between stream function Ψ and velocity components (U X ,U Y ) are given as below:

Figure 6
Figure 6 expounds that for Pr = 10, β = 0.3, N R = 1, Re = 10 2 the relationship between Hartmann number and stream function retained as in the previous figure.It can be concluded that an increment in the Hartmann number did not affect the pattern of streamlines in both cases for Pr = 0.7 and Pr = 10.The anticlockwise behaviour also retained in Figure6.Figure7displays the stream lines pattern and temperature contour over a wide range of thermal radiation parameter and Pr = 0.7 for Re = 10 2 and β = 0.05.The anticlockwise secondary circulation is detected for 0 ≤ N R ≤ 10.For N R = 5 and N R = 10 same motion of fluid is observed.

Figure 7
Figure 6 expounds that for Pr = 10, β = 0.3, N R = 1, Re = 10 2 the relationship between Hartmann number and stream function retained as in the previous figure.It can be concluded that an increment in the Hartmann number did not affect the pattern of streamlines in both cases for Pr = 0.7 and Pr = 10.The anticlockwise behaviour also retained in Figure 6.Figure 7 displays the stream lines pattern and temperature contour over a wide range of thermal radiation parameter and Pr = 0.7 for Re = 10 2 and β = 0.05.The anticlockwise secondary circulation is detected for 0 ≤ N R ≤ 10.For N R = 5 and N R = 10 same motion of fluid is observed.value 1, a strong anticlockwise secondary circulation is being observed.Stream function showed a low value for β = 1 as compared to β = 0.5 and β = 0.01.These observations are carried out for Pr = 10.From Figure 5 its quite evident that for Pr = 0.7 there exists an inverse relationship between Hartmann number and motion of the fluid.For Ha = 0 the stream function gives a value of 0.095.An anticlockwise circulation is observed.As we increase the Hartmann number the motion of fluid decreases.For Ha = 30 the value of stream function is ψ = 0.088, and for Ha = 60, ψ = 0.076.These observations are made for Pr = 0.7.

Table 1 .
Mesh independence test