- The free convection phenomenon from a heated square cylinder submerged in Bingham Plastic fluids is numerically investigated. The governing equations are solved for a wide range of physical dimensionless parameters, such as Rayleigh number (102 ≤ Ra ≤ 105), Prandtl number (10 ≤ Pr ≤ 100) and Bingham number (0 ≤ Bn ≤ 107). The heat transfer characteristics are investigated in terms of local Nusselt number distribution ov er the surface of the cylinder surface average Nusselt number. Streamlines, isothermal contours, yielded and unyielded regions are visualized in detail. Index ed Terms - Bingham plastic fluids, square cylinder, Free -convection, Nusselt number, Bingham number, Ray leigh number, Prandtl number I. INTRODUCTION Bingham plastic fluids are one type of viscoplastic fluid which are characterized by the existence of yield stress or threshold stress. If the applied stress is lower than the yield stress, the fluid behaves like an elastic solid, and if the applied pressure ex- cedes the yield stress, then the fluid behaves as fluid; hence the flow domain in this class of fluids is co spanned by both fluid -like (unyielding region) and solid like (yield region). Heat transfer in the unyielded region occurs mainly by conduction, whereas convection is limited to the yielded part of the system [1 –5]. To overcome the fluid’s yield stress, a minimum strength of buoyancy induced flow is needed for a given value of fluid yield stress [6]. Most substances of multi -phase nature like foam, suspensions, yoghurt, molten chocolate, cosmetics are examples of Bingham plastic fluids. Heat transfer from a square cylinder has many industrial applications, including the desi gn of novel heat exchanger, cooling of electronic components, thermal treatment of various foodstuffs, fluidized bed drying of fibrous substances. Besides, free convection or natural convection, which al - ways contributes in various proportions to the over all heat transfer, and indeed there are numerous applications in food processing where the only mode of heat transfer is by natural convection. Even in mixed convection applications, natural convection is found to be dominating [7, 8] . Over the years, researchers have done significant research, and they investigated the heat transfer characteristics from cylinders of various cross - sections, including circular, elliptical, triangular and square in Newtonian and Power -law fluids for forced convection, free convection and mixed connection. K.S.Chang [1] investigated laminar natural convection over a sharp -edged horizontal bar situated in an infinite fluid medium has been investigated numerically and experimentally. Finite -difference solution s to the two -dimensional Naiver -Stokes and energy equations were obtained with a fixed Prandtl number of 0.7 for the two configurations: a hat mate of finite thickness and a square bar. The difficulty associated with the complex physical flow domain was overcome by using body -fitted coordinates. In the numerical study, they found no indication of flow separation for the flat plate, in the range of Rayleigh number 103 ≤ Ra ≤ 105. For the square bar, however, the boundary layer separated easily at the upper s harp edges for Ra ≥ 5 x 105 and well -defined twin vortices were identified above the upper horizontal surface. Comparison of the two results, the numerical and the experimental, offered good agreement. O. Zeitoun and M. Ali. [2] examined over a broad variety of Rayleigh numbers and aspect ratios in ducts with square and rectangular cross sections based on natural convec tion. Streamlines demonstrate that at a constant Rayleigh number, as the aspect ratio rises, separation and circulation occur above the duct’s top © JUL 2021 | IRE Journals | Volume 5 Issue 1 | ISSN: 2456 -8880 IRE 170 2874 ICONIC RESEARCH AND ENGINEERING JOURNALS 305 surface, and the isotherms confirm this behavior. Additionally, they discovered that increasing the Rayleigh number reduces the thermal boundary layer along the duct’s vertical and bottom surfaces, but not on the top surface, at the centerline, where the plume was rising. Increasing Ra has a negligible effect on the thermal boundary layer on the top surface, at the centerline, where the plume was rising, and its thickness depends on the solution domain chosen. Additionally, velocity distributions were shown, and velocity profiles showed the circulation and reverse flow detected using streamlines. The total heat transfer from the perimeter of the duct was determined, and the average heat transfer coefficients for different aspect ratios were converted to dimensionless Nusselt values. Zeitoun et al [3] conducted experimental investigations into the average natural convection heat transfer fr om the isothermal vertical surfaces of a vertical short and slender square cylinder to air and concluded that, below certain Rayleigh numbers, due to curvature effects, the average natural convection heat transfer from the vertical square cylinder’s side walls was greater than the data for a vertical flat plate. Kumar Arnab De, Amaresh Dalal [9] examined natural convection around a tilted heated square cylinder kept in an enclosure has been studied in the range of 103 ≤ Ra ≤ 106. The Navier –Stokes’s equation was numerically solved using the finite - difference technique in a non -orthogonal body -fitted coordinate system. The authors provide de- tailed flow and heat transfer characteristics for two distinct thermal boundary situations. The effects of the enclosure shape were evaluated using three different aspect ratios and varying the height of the square cylinder from the bottom. The notion of heat function was used to determine the direction of heat transfer. They discovered a quanti fiable differ ence between uniform wall temperature heating and uniform wall heat flux heating. If the aspect ratio was changed, the flow pattern and thermal stratification are altered. Overall heat transmission was also affected by the aspect ratio. Therefore, they determined that constant wall temperature heating was more efficient than constant wall heat flux heating for the current physical condition in terms of total heat transfer. The location of the heated cylinder was irrelevant. R.P.Chhabra, C.Sasmal [10] conducted numerical simulations of laminar natural convection heat transfer from a heated long cylinder with a square cross - section immersed in stationary power -law fluids . The governing differential equations (continuity, momentum, and thermal ener gy) have been solved over wide ranges of the relevant dimension - less parameters, namely, Grashof number (10 ≤ Gr ≤ 105), Prandtl number (0.72 ≤ Pr ≤ 100), and power -law index (0.3 ≤ n ≤ 1.8), covering both shear -thinning and shear - thickening type fluid. They investigated the flow’s intricate structure in terms of stream - line and isotherm patterns, while analyzing the heat transfer characteristics in terms of the local Nusselt number distribution throughout the cylinder’s surface as well as its surface av eraged values. In general, the flow stays connected to the surface until the Grashof number is greater than one in shear -thinning fluids (n < 1) than in Newtonian media (n =1). Similarly, shear - thinning behavior increases heat transmission when all other f ac- tors are equal. Indeed, given the right circumstances, i.e., values of the Grashof number, Prandtl number, and power -law index, it is feasible to increase the rate of heat transfer by up to 100%. Shear - thickening fluid behavior has a detrimental effect on heat transfer rate. In the limiting situation of Newtonian fluid behavior (n = 1), it was established that for constant values of Rayleigh number, Prandtl number, and Reynolds number. Nirmalkar, N., Bose, A., and Chabra, R.P. [8] Rayleigh number 102 ≤ Ra ≤ 106; Prandtl number 30 ≤ Pr ≤ 100; and Bingham number 0 ≤ Bn ≤ 104 for laminar free convection heat transfer from a horizontal heated cylinder to Bingham plastic fluids across a variety of circumstances was studied. They found that the Nusselt number exhibits a positive relationship with the Rayleigh number, but an inverse relationship with the Bingham and Prandtl number. Heat transfers are greater in shear -thinning fluids than in shear - thickening fluids. Nirmalkar, R.P.Chhabra, and R.J.Poole [11] investigated the momentum and heat transfer properties of a heated square cylinder submerged in a streaming Bingham plastic media . The governing differential equations (continuity, momentum, and thermal energy) have been numerically solved for a variety of ci rcumstances, including the plastic Reynolds number of 0.1 ≤ Re ≤ 40, the Prandtl number © JUL 2021 | IRE Journals | Volume 5 Issue 1 | ISSN: 2456 -8880 IRE 170 2874 ICONIC RESEARCH AND ENGINEERING JOURNALS 306 of 1 ≤ Pr ≤ 100, and the Bing - ham number of 0 ≤ Bn ≤ 100. The flow is anticipated to be symmetrical and constant throughout this range of circumstances. The flow and t emperature fields near the cylinder surface are studied in depth using streamline and isotherm profiles. Due to the existence of the yield stress, the flow domain is demarcated into yielded (fluid -like) and unyielded (solid -like) areas as functions of the Reynolds and Bingham numbers. They achieve this goal by doing a thorough analysis of the velocity and shear rate profiles along the centre planes. Additional insights are gained into the pressure and local Nusselt number distributions throughout the cylind er surface, as well as their average values in terms of drag coefficients and mean Nusselt number. They reached an agreement. Additional reliance on Prandtl number exists in the temperature field and heat transfer properties. By and large, all unyielded ar eas grow as the Bingham number increases. By using an effective viscosity, it is possible to reconcile Nusselt number findings over a broad range of Bingham numbers. P.Mishra, R.P.Chhabra [12] The heat transfer characteristics of steady flow of Bing - ham p lastic fluids over a square cylinder confined in a plane slit have been numerically explored in terms of isotherm contours, local Nusselt number, and average Nusselt number for the following ranges of conditions: Bingham number 0.01 ≤ Bn ≤ 100, Reynolds nu mber 0.1 ≤ Re ≤ 40, Prandtl number, 10 ≤ Pr ≤ 100, and blockage ratio, (l/H) = 0.2, 0.3, 0.4. Additionally, the effect of two distinct thermal boundary conditions, namely constant heat flow condition (CHF) and constant wall temperature (CWT), on the square cylinder’s surface has been defined. The focus is on clarifying the impact of blockage ratio on total heat transmission since it affects temperature gradients and adds dissipation at confining walls. Heat transmission occurs in Bingham plastic fluids thro ugh two competing mechanisms: convection in fluid -like areas and conduction in solid unyielded regions. As a result, the total heat transmission is complexly controlled by these two competing processes. It is thus clear that most of the literature is avail able on natural convection from horizontal square cylinder in Newtonian and power law fluids. Besides, literature on natural convection heat transfer in Bingham plastic fluids is available for circular and elliptical cylinders. To the best of authors knowl edge, there is no prior study on natural convection heat transfer in quiescent Bingham plastic fluids from a heated horizontal square cylinder. Limited results are available on heat transfer in forced convection regimes. This work aims to fill the gap in t he current literature. Hence, laminar heat transfers from a heated square cylinder have been numerical studied for over a wide range of parameters such as Rayleigh number (102 ≤ Ra ≤ 105), Prandtl number (10 ≤ Pr ≤ 100) and Bingham number (0 ≤ Bn ≤ 107) II. PROBLEM FORMULATION AND GOVERNING EQUATIONS The heat transfer from a horizontal square cylinder with side L (infinitely long in z - direction) whose surface is kept at a constant temperature Tw, submerged in a quiescent Bingham plastic fluid at the constant temperature T o, as illustrated in Fig 1. The density of the fluid will be lowest near the cylinder due to the existing temperature differential between the fluid and the square bar, and will progressively rise away from the cylinder, eventually reaching the value corresponding to the distant fluid temperature T0. As a result, buoyancy -induced flow occurs, resulting in heat transfer through free (or natural) convection. The viscous dissipation effect is believed to be minimal at this point since the buoyancy generated flow is known to be very modest. Furthermore, the resulting flow is considered to be laminar, steady, two-dimensional, and incompressible (except for the body force term in the y- momentum © JUL 2021 | IRE Journals | Volume 5 Issue 1 | ISSN: 2456 -8880 IRE 170 2874 ICONIC RESEARCH AND ENGINEERING JOURNALS 307 equation) over the range of conditions spanned here, despite the fact that no conclusive information is available in the literature for the transition from one regime to another. Except for the density term in the y-momentum equation, the flow is assumed to be symmetric about the vertical centerline of the cylinder and the other thermo -physical properties of the fluid, such as Bingham viscosity, threshold stress, thermal conductivity, and heat capacity, are assumed to be temperature independent. The temperature differential T= Tw -T0 utilized here is never more than 5 K. For a lower difference between th e surface and fluid temperatures, the Boussinesq approxima tion is employed in the change of fluid density. The Boussinesq approximation is given as: 0 0 0 ()TT    − = − Where is the coefficient of thermal expansion at a constant temperature, and it is defined 1 TT=− temperature fields in dimensionless forms. Continuity equation: 0y xU U xy+= X-momentum equation: () () Pr X y yx xx XXUU UU p x y x Ra x y        + =− + +            Y-momentumequation : () () Pr y y xy yy XUU UyU pTx y y Ra x y        + =− + + +            Thermal energy equation : 22 22() ()1/ .Pry xUT UT TTRax y x y   + = +      To make the problem more accessible, in the energy equation, the viscous dissipation is neglected here. for Bingham plastic fluid, the extra stress tensor τ in di mensionless form is returned as follows: 1Bn =+ if Bn 0= if Bn Since the Bingham constitutive relation is non - differentiable and disconti nuous, this form of relation poses difficulties. To avoid these difficulties and provide convergence characteristics and smooth transitions between the yielded and unyielded region, the Herschel -Bulkley model predicts the Bingham plastic model. In the equ ations as mentioned earlier, length L, velocity lg T , viscosity B , pressure 0lg T , the rate of strain and stress component are scaled as / g T l , /Bg T l . The temperature is non-dimensioalized as 00 ( ) / ( )W T T T T T= − − . The dimensional consideration of these parameters are functions of the three dimensionless numbers, namely, Bingham number (Bn). Rayleigh number (Ra), Prandtl number (Pr). For Bingham plastic fluid, these parameters are defined as follows: Bingham number: 0 BlBngT = Bn→0 denotes the limiting case of Newtonian fluid behavior. Besides, Bn→∞ denotes the plastic sold like behavior. Rayleigh number: 23 0.P Bc g TlRa Gr Prk == Grashof number: 23 0 2 Bg TlGr = Prandtl number: PrpBc k= These definitions are commonly based on these particular p