Effects Of Biot Number Research Articles

Thermal response of TBC systems subjected to non-uniform thermal shocks and periodic thermal load: Effect of Biot number and interface imperfection

This study analytically examines the temperature variations within thermal barrier coating (TBC) systems under diverse kinds of thermal loadings, such as concentrated/distributed convective thermal shocks and oscillating thermal loads. In this study, the TBC system comprises ceramic top coat (TC) and NiCrAlY alloy bond coat (BC). The transient behavior of the TBC system facing various kinds of non-uniform convective thermal shocks is elaborated via the dual-phase-lag (DPL) heat conduction model. The effect of imperfect bonding of the TC/BC interface on transient responses of the regions in the neighborhood of the TC/BC interface is studied by considering impermeable, partially permeable and completely permeable heat flux flows. Laplace and Fourier integral transforms are employed to derive the exact solutions of time-dependent temperature fields governing equations. The prominent effects of sudden thermal impulse and steady alternating thermal loading on the variations of temperature within the TC, BC and substrate are inspected. The analysis reveals that the kind of thermal load, loading parameters, thermal properties and thickness of ceramic TC directly affect the transient/periodic temperature fields inside the TBC constituents. Meanwhile, the interface imperfection and the Biot number of convective thermal load play an impressive role in the severity of thermal response of the TBC constituents.

Linear stability analysis of micropolar nanofluid flow across the accelerated surface with inclined magnetic field

PurposeThe purpose of this paper is to study the two-dimensional micropolar fluid flow with conjugate heat transfer and mass transpiration. The considered nanofluid has graphene nanoparticles.Design/methodology/approachGoverning nonlinear partial differential equations are converted to nonlinear ordinary differential equations by similarity transformation. Then, to analyze the flow, the authors derive the dual solutions to the flow problem. Biot number and radiation effect are included in the energy equation. The momentum equation was solved by using boundary conditions, and the temperature equation solved by using hypergeometric series solutions. Nusselt numbers and skin friction coefficients are calculated as functions of the Reynolds number. Further, the problem is governed by other parameters, namely, the magnetic parameter, radiation parameter, Prandtl number and mass transpiration. Graphene nanofluids have shown promising thermal conductivity enhancements due to the high thermal conductivity of graphene and have a wide range of applications affecting the thermal boundary layer and serve as coolants and thermal management systems in electronics or as heat transfer fluids in various industrial processes.FindingsResults show that increasing the magnetic field decreases the momentum and increases thermal radiation. The heat source/sink parameter increases the thermal boundary layer. Increasing the volume fraction decreases the velocity profile and increases the temperature. Increasing the Eringen parameter increases the momentum of the fluid flow. Applications are found in the extrusion of polymer sheets, films and sheets, the manufacturing of plastic wires, the fabrication of fibers and the growth of crystals, among others. Heat sources/sinks are commonly used in electronic devices to transfer the heat generated by high-power semiconductor devices such as power transistors and optoelectronics such as lasers and light-emitting diodes to a fluid medium, thermal radiation on the fluid flow used in spectroscopy to study the properties of materials and also used in thermal imaging to capture and display the infrared radiation emitted by objects.Originality/valueMicropolar fluid flow across stretching/shrinking surfaces is examined. Biot number and radiation effects are included in the energy equation. An increase in the volume fraction decreases the momentum boundary layer thickness. Nusselt numbers and skin friction coefficients are presented versus Reynolds numbers. A dual solution is obtained for a shrinking surface.

Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink

Abstract Recognizing the flow behaviours across a Riga plate can reveal information about the aerodynamic efficiency of aircraft, heat propagation, vehicles, and other structures. These data are critical for optimizing design and lowering drag. Therefore, the purpose of the current analysis is to examine the energy and mass transfer across the mixed convective nanofluid flows over an extending Riga plate. The fluid flow is deliberated under the influences of viscous dissipation, exponential heat source/sink, activation energy, and thermal radiation. The Buongiorno’s concept is utilized for the thermophoretic effect and Brownian motion along with the convective conditions. The modelled are simplified into the lowest order by using similarity transformation. The obtained set of non-dimensional ordinary differential equations is then numerically solved through the parametric continuation method. For accuracy and validation of the outcomes, the results are compared to the existing studies. From the graphical analysis, it can be observed that the fluid velocity boosts with the rising values of the divider thickness parameter. The fluid temperature also improves with the effect of Biot number, Eckert number, and heat source factor. Furthermore, the effect of heat source sink factor drops the fluid temperature.

Electrically conducting mixed convective nanofluid flow past a nonlinearly slender Riga plate subjected to viscous dissipation and activation energy

This paper reports the mass and energy transmission characteristics of an electrically conducting mixed convective nanofluid flow past a stretching Riga plate. An additional effect of viscous dissipation, Arrhenius activation energy and heat source is also studied. The energy and mass transmissions are evaluated by a zero-mass flux of nanoparticle and convective boundary conditions. Buongiorno’s relations are proposed for the Brownian motion and thermophoretic diffusion. The similarity substitutions are employed to derive the non-dimensional set of modeled equations. The obtained set of equations is numerically processed via parametric continuation method (PCM). Several flow factors affecting the velocity, energy, and mass distributions are graphically discussed. It has been perceived that the fluid velocity field declines with the influence of velocity power index (m), while improves with the upshot of modified Hartmann number (Q). The effect of Schmidt number and chemical reaction diminishes the concentration profile [Formula: see text]. Furthermore, the energy curve enhances with the effect of thermophoresis factor, Biot and Eckert number.

Modeling the stability of thin liquid film flows on a uniformly heated slippery inclined substrate: A realistic approach

We investigate the stability of gravity-driven, Newtonian, thin liquid film falling down a uniformly heated slippery rigid inclined wall. The authors of previous research works considered specified temperature (ST) boundary condition to study the effects of slip length. However, the ST boundary condition does not include the effects of heat fluxes at wall–air and wall–liquid interfaces and so fails to incorporate the real situation. Consequently, we consider heat flux/mixed-type boundary condition as the thermal boundary condition on the rigid plate. This boundary condition involves the heat flux from the rigid plate to the surrounding liquid and the heat losses from the wall to the ambient air. Using long-wave expansion method, we construct a highly nonlinear evolution equation in terms of the film thickness at any instant. Using normal mode approach, the linear study reveals the stabilizing (destabilizing) behavior of the wall film Biot number (dimensionless slip length). It is found that the destabilizing tendency of the slip length is more in the absence of thermocapillary stress. The linear study reveals that the destabilizing role of MB may be controlled to some extent by increasing the wall film Biot number Bw. Using asymptotic expansions of the flow variables in terms of the small wave number k, the Orr–Sommerfeld boundary value problem gives an onset of instability in terms of critical Reynolds number. It slightly differs from that of the same as obtained by Benney's long-wave expansion method, due to the consideration of small free surface Biot number [B=O(ϵ)]. For arbitrary wave numbers, using Chebyshev spectral collocation method, the effect of Marangoni number (Ma), slip length (δ), and wall film Biot number (Bw) on the H, S, P, and shear modes of instability are discussed in detail. Near the threshold, both Ma and δ show the destabilizing effect on H mode of instability, whereas Bw gives the stabilizing effect. Interestingly, their roles on H mode of instability becomes diametrically opposite far from the onset of instability. For S mode, both Ma and Bw display the destabilizing effect, whereas δ plays the dual role. For P mode, both Ma and δ show the destabilizing effect, whereas Bw plays the stabilizing role. The slip length (δ) plays the stabilizing role, in the case of shear mode. In the absence of thermocapillary effect, the vorticity balance at the liquid–air interface explains that the amplitude of the vorticity perturbation amplifies the surface deformation due to the presence of inertia and the slip length. In the absence of the slip length, a weakly nonlinear study transforms the evolution equation to the famous Kuramoto–Sivashinsky (KS) equation.

Heat Transfer in Hybrid Nanofluid Flow Past an Infinite Orthogonal Plate with Biot Number and Velocity Slip Effects

Nanofluids are made up of nanoscale particles such as copper, carbides, graphite, and alumina, which help base fluids transmit heat more efficiently. These nanofluids have a broad range of applications in today’s framework of cooling and heating, solar-powered cells, hybrid-powered engines, new fuel generation, cancer therapy, and pharmaceuticals. This present investigation emphasizes the importance of a specific type of fluid called a hybrid nanofluid, which consists of (Cu and TiO2) nanoparticles suspended in H2O (water). This fluid is subjected to a combination of several complex phenomena of heat transfer in hybrid nanofluid flow past an infinite orthogonal plate, including velocity slip and Biot number using fractional calculus. The system of governing partial differential equations (PDEs) are transformed into a set of first-order ordinary differential equations (ODEs) using appropriate mathematical transformations. These equations were then solved numerically using fractional power series method (FPSM). FPSM is a very powerful method in solving fractional differential equations arising from different types of scientific problems. The study investigated the behaviour of velocity profiles, temperature, skin friction and heat transfer for various values of the parameters involved.The rate of heat transfer decreases with increasing the hybrid nanofluid parameter but it increases with increasing the fractional order, velocity slip and Biot number Additionally, the skin friction decreases with increasing both the hybrid nanofluid parameter and velocity slip, but it increases with increasing of the fractional order. However, there is no change in skin friction when the Biot number increases. It is also clear that the velocity increases for increasing both fractional order, velocity slip, it decreases for increasing the nanofluid parameter. The temperature profile rises when both the nanofluid parameter and Biot number increase. Also temperature profile decreases when the values of the fractional order and slip parameters increase.

Analytical and numerical (FEA) solution for steady-state heat transfer in generic FGM cylinder coated with two layers of isotropic material under convective-radiative boundary conditions

An investigation was carried out in order to develop an accurate analytical solution and a numerical (FEA) solution for steady-state heat transfer in a circular sandwich structure incorporated with convective-radiative boundary conditions. The dimensional governing equations and boundary conditions were developed in the form of a 4th order algebraic equation, and then the solution was obtained using Ferrari's method. By solving for the roots of the quartic equation, we were able to determine the dimensionless temperature fields of the FG sandwich composite. The findings obtained utilizing the exact analytical solution for the FG sandwich composite under thermal loads were satisfactorily validated against those data obtained using the Galerkin finite element approximation. The impact of geometric and thermo-physical characteristics, such as Biot number (Bii=1,2), Inner and outer surface thickness ratio (ri=1,2Ro), ambient temperature ratio (θd), radiation-conduction parameter (Nr), and thermal conductivity ratio (λ3λ1) on the efficiency of heat transfer, has also been studied. This study reveals the distinct effect of Biot number on the inner and outer layers of the composite cylinder. It shows that Bi1 has a negligent effect on temperature distribution; on the other hand, the outer surface (Bi2≤1) minimizes temperature variation. However, for design consideration, a thicker inner face sheet is not recommended in high thermal load, as Nr>4 has an insignificant impact on inner surface thickness on top surface temperature. Moreover, the outer surface temperature appears to be more sensitive to θd than the radiation-convection side. Furthermore, the given analytical solution is adequately verified against the proposed FEA method, having an error of less than 1.5%.

The Effect of Biot Number on a Generalized Heat Conduction Solution

Abstract Analytical solutions for thermal conduction problems are extremely important, particularly for verification of numerical codes. Temperatures and heat fluxes can be calculated very precisely, normally to eight or ten significant figures, even in situations involving large temperature gradients. It can be convenient to have a general analytical solution for a transient conduction problem in rectangular coordinates. The general solution is based on the principle that the three primary types of boundary conditions (prescribed temperature, prescribed heat flux, and convective) can all be handled using convective boundary conditions. A large convection coefficient closely approximates a prescribed temperature boundary condition and a very low convection coefficient closely approximates an insulated boundary condition. Since a dimensionless solution is used in this research, the effect of various values of dimensionless convection coefficients, or Biot number values, are explored. An understandable concern with a general analytical solution is the effect of the choice of convection coefficients on the precision of the solution, since the primary motivation for using analytical solutions is the precision offered. An investigation is made in this study to determine the effects of the choices of large and small convection coefficients on the precision of the analytical solutions generated by the general convective formulation. Results are provided, in tabular and graphical form, to illustrate the effects of the choices of convection coefficients on the precision of the general analytical solution.

Investigation of Nanofluid Natural Convection Inside a Square Cavity for Two Orientations Using Lattice Boltzmann Method

In this study, natural convection heat transfer of a water based nanofluid inside a square cavity is numerically investigated for two different orientations of a wall-heated cavity. The enclosure is heated by applying a constant heat flux while cooled at ambient conditions. Lattice Boltzmann method (LBM) is used to simulate nanofluid natural convection. The Brownian motion of nanoparticles is considered. LBM simulation is validated by comparison with experimental and numerical results of the literature. The effect of Rayleigh number (Ra = 103, 104, 105, 106), Biot number (Bi = 0.1, 1, 10, 100) and volume fraction of nanoparticles (Φ = 0, 1, 3 and 5%) on the isotherms, streamlines, velocity components, local and Nusselt number is analyzed for two oriented cavities. The bottom-heated cavity shows higher heat transfer rate than that of the cavity heated from the sidewall. The average Nusselt number increases by up to 6.81%. Furthermore, Biot number, Rayleigh number, and volume fraction of nanoparticles show significant effects on the heat transfer rate.

Impact of Arrhenius activation energy on MHD nanofluid flow past a stretching sheet with exponential heat source: A modified Buongiorno’s model approach

Nanofluids have a wide range of applications in biological research. They are employed in targeted medication administration, hyperthermia (for cancer treatment) and differential diagnostics like magnetic resonance image (MRI). In light of these medical applications, the impact of an external magnetic field and an exponential heat source on the dynamics of [Formula: see text]–[Formula: see text] over a nonlinearly stretched surface has been investigated. A realistic modified Buongiorno model has been used which includes the effects of reaction rate, Biot number and activation energy. The boundary value problem governing the model is solved on MATLAB R2022a using the solver, BVP5C. Further, the consequences of different parameters on rate of heat transfer coefficient (Nusselt number), rate of mass transfer coefficient (Sherwood number), drag coefficient, velocity, temperature and volume fraction profile are observed graphically. It is noted that volume fraction and uniform heat source intensity have a positive effect on the Nusselt number and negative effect on Sherwood number. The effects of thermal radiation and magnetic field on volume fraction profile are, respectively, positive and negative. The current physics of flow across a vertical stretching surface is expected to serve as the foundation for various medical science, engineering and technology applications.

Temperature Dependent Property Considerations for Biot Number Matching With Composite Materials

Abstract High-temperature materials such as ceramic matrix composites (CMCs) are promising for advancements in gas turbine engines; however, experimental techniques are necessary to characterize the thermal behavior of high-temperature materials in scaled representative environments. In turbine cooling applications, the results of low-temperature experiments in laboratories must be scaled to the intended operating conditions to predict the thermal state at high-temperature engine conditions where such measurements are generally not feasible. Scaling of turbine cooling experiments with metallic components has been aided by the fortuitous fact that the thermal conductivity of turbine-relevant metal alloys varies with temperature in such a way that the Biot number is relatively insensitive to temperature, at least to the extent that Biot number affects the overall effectiveness. However, unlike traditional metallic alloys, materials such as CMCs have unique anisotropic thermal conductivities which vary with temperature in quite different ways than traditional metals. The through-thickness and in-plane thermal conductivities of CMCs may vary with temperature in different ways, thereby adding further complexity. Since most existing low-temperature experimental methods were developed with isotropic metallic alloys in mind, it is necessary to consider the potential for errors due to the unique characteristics of composite materials. Analyses to determine the expected Biot number errors for various composite materials were conducted at a range of experimental conditions. These are compared to a traditional metallic alloy for context and computational simulations were performed on the various materials at engine and laboratory conditions. Additionally, a novel technique was implemented to isolate and quantify the effects of both Biot number and fluid property mismatches at low temperatures. The results show overall effectiveness experiments conducted at low temperatures using composite materials will likely have a large error and could significantly skew the expected thermal characteristics of a turbine component. Careful experimental design should be practiced to mitigate the effect that a mismatched Biot number can have on experimental results.

Accelerating heat transfer modeling in material extrusion additive manufacturing: From desktop to big area

Thermal modeling informs the understanding of cooling phenomena and structural evolution during material extrusion (MatEx) additive manufacturing to help predict and tune final part properties. For fused filament fabrication (FFF) and big area additive manufacturing (BAAM), in particular, thermal models are well-suited to augment parameter optimization and closed-loop control for high throughput manufacturing of custom parts and products. However, limitations in computational time, speed, and cost pose a challenge to modeling large structures with sufficient multi-dimensional granularity over extended timescales, with many models confined either to high-resolution studies on short length and timescales or low-resolution, lumped approximations for larger timescale simulations. Thus, the goal of this work is to develop a high-fidelity thermal model applicable from desktop to BAAM systems with minimal computational constraint. Here we present a scalable, 2D finite volume “proof-of-concept” model and algorithm to rapidly simulate thermal histories across both FFF and BAAM length and time scales. Our approach implements a two-dimensional implicit numerical strategy that confers reduced computational complexity while offering fine scale granularity in capturing critical inter- and intralayer cooling dynamics and radiative heat transfer. Further, analysis of the effective Biot number (Bi) reveals that, on the BAAM scale, layer width and radiative heat transfer both drive the formation of intralayer temperature gradients, exceeding 51 °C between the layer center and edge. We further simulate the fabrication of single road-width acrylonitrile-butadiene-styrene (ABS) walls at the BAAM scale (50 layers) to demonstrate both the relative speed of the computational approach and its ability to capture the complex dynamics of a multilayer build. The model agrees with published experimental measurements, and results gauge the influence of layer thickness and proximity to print bed on cooling behavior, with larger layers and separation from the print bed lending to longer periods for interlayer welds to form above the ABS glass transition temperature (Tg). Finally, a BAAM case study highlights the model’s utility in design for additive manufacturing (DfAM) and print optimization, demonstrating how changes in bed temperature and layer time can impact interlayer bonding.

Exploration of energy‐based volumetric heating on ferrothermal porous convection: Effects of MFD viscosity and boundary conditions

AbstractThe impact of energy‐based volumetric internal heating and magnetic field‐dependent (MFD) viscosity on the onset of ferrothermal porous convection for different types of boundary conditions is investigated. The lower and upper boundaries are considered to be either rigid or stress‐free. The lower boundary is insulating, while at the upper boundary a Robin type of thermal condition is applied. Besides this, a general type of boundary conditions is invoked on the magnetic potential. The stability eigenvalue problem is solved numerically using the Galerkin‐type of weighted residuals technique with the Rayleigh number, as the eigenvalue. The results obtained for different boundary combinations are found to be qualitatively similar though not quantitatively. The effect of Biot number, MFD viscosity, and porous parameters is to hinder, while the influence of internal heat source strength, magnetic number, and the nonlinearity of fluid magnetization is to advance the onset of convection. The rigid and conducting boundaries offer a more stabilizing influence on the onset when compared to stress‐free and insulating boundaries. The convection cells get contracted the most when the boundaries are rigid and also when the upper boundary is conducting. The results delineated under the limiting cases are found to be in good agreement with those available in the literature.

Theoretical Analysis of MHD Mixed Convection From A Vertical Plate Embedded in a Porous Medium with a Convective Boundary Condition

An analytical approach has been used to study the heat and mass transfer from a vertical plate embedded in a porous medium experiencing a first-order chemical reaction and exposed to a transverse magnetic field. Instead of the commonly used conditions of constant surface temperature or constant heat flux, a convective boundary condition is employed which makes this study unique and the results more realistic and practically useful. The momentum, energy, and concentration equations derived as coupled second-order, ordinary differential equations are solved analytically a highly accurate and thoroughly tested using Homotopy Perturbation Method. The effects of Biot number, thermal Grashof number, permeability parameter, Hartmann number, Eckert number, Sherwood number and Schmidt number on the velocity, temperature, and concentration profiles are illustrated graphically. Proportional to the plate surface temperature, the local skin-friction coefficient, the local Nusselt number and the local Sherwood number were also presented analytically. The discussion focuses on the physical interpretation of the results as well their comparison with the results of previous studies.

Effect of Richardson number on double-diffusive mixed convective slip flow, Heat and Mass transfer of MHD Casson fluid

An analysis of the heat and mass transfer in a mixed-convective double diffusive flow with convective boundary conditions is carried out in this paper. The governing equations are solved numerically by using Runge-Kutta method with shooting procedure with Matlab software. An accuracy of the numerical procedure has been validated through a results of the current work when compared with prior available results in the literature. The values of shear surface stress, Nusselt and Sherwood number are increasing with increasing values of Prandtl number. The effect of Biot number [Formula: see text] on flow and heat transfer are also investigated and further it is observed that friction coefficient, Nusselt number and Sherwood number are increasing due to enhancing values of Biot number. In the present paper it is noticed that the increasing in estimations of Casson parameter slowdown the heat transfer rate and accelerates mass transfer rate. The results are in good agreement with existing findings The impact of pertinent constraints on distinct flow parameters are determined and analysed through tables and graphs.

Analysis of bio-convective MHD Blasius and Sakiadis flow with Cattaneo-Christov heat flux model and chemical reaction

The combined impacts of bio-convection and magnetic field on boundary layer magnetohydrodynamic unsteady Sakiadis and Blasius flow of nanofluid with accretion/ablation of the leading-edge are investigated. Moreover, the effects of Biot number, thermal radiation, chemical reaction, and the convective boundary condition have been observed. Perpendicular to the sheet, a uniform magnetic field remains operative with a source of radiative heat and convective conditions at the boundary. The basic formulation is constituted in the form of coupled non-linear PDEs and transformed into higher-order ODEs by using suitable similarity functions. A Galerkin numerical technique is harnessed to yield a solution to the system of differential equations. A graphical behavior of the effects of governing physical parameters on the velocity components, temperature, and concentration is expressed. The most pertinent consequence of this study is that the growing strength of the magnetic field reduced the velocity of the boundary layers in both the Sakiadis and Blasius flows. It is observed that the concentration profile directly decreases in response to the Lewis number, bioconvective Peclet number, and the chemical reaction. The temperature of the fluid rises proportionally with the increasing values of each of the radiative parameter (Rd), Biot number (Bi), and thermophoretic parameter (Nt). The convergence of numerical technique is checked by meshing variation and the accuracy of results is verified by related studies.

Effects of Richardson and Biot number on double‐diffusive Casson fluid flow with viscous dissipation

AbstractAn analysis is done of the effect of Richardson and Biot number on double‐diffusive mixed convective Casson fluid stream with viscous dissipation on warmth and mass stream with convective limit conditions and radiating vertical plate. The R‐K method with shooting procedure is used to solve the transformed equations mathematically. The accuracy of the numerical procedure has been validated through a comparison of the current work compared with prior available results. The sheer surface stress, Nusselt, and Sherwood number are increased with enhancement in Prandtl number. The Biot number Βi > 0.1 is investigated and increasing Biot number is observed to enhance the friction coefficient, Nusselt, and Sherwood number are increased. The influence of pertinent constraints on distinct flow parameters is determined and analyzed through tables and graphs.

Influence of Biot number and geometric parameters on the overall cooling effectiveness of double wall structure with pins

In this paper, the overall cooling effectiveness of double wall cooling structures with pins is experimentally investigated on the low-speed wind tunnel. The analysis of the single cooling technology contributions to improve the double wall cooling performance is provided. The effect of Biot number (Bi = 0.03, 0.32, 2.25) and the film hole size (de= 2 mm, 3 mm, 5 mm) on the overall cooling effectiveness are also studied. In order to further improve the overall cooling performance, the present study proposes two new shapes of film hole patterns with compound angles (compound configuration, double configuration). By numerically analyzing the cooling effectiveness of the outer film cooling and the heat transfer performance of the inner target wall, the mechanism by which film hole size and new film hole patterns affect the overall cooling performance of double wall structures is clarified. The results show that the area-averaged overall cooling effectiveness of the double cooling technology is more than 30% higher than that of the two single cooling technologies and the contribution of impingement cooling is higher than that of effusion cooling at high blowing ratios. The cooling performance of the double wall structure is greatly influenced by the Biot number of effusion plates. Low Biot number means higher thermal conductivity, resulting in higher overall cooling effectiveness under the same thermal load conditions. Increasing the film hole size could enhance the film cooling performance outside of the effusion plate, suppressing the unfavorable results of lower heat transfer performance at the inner side. The fitting correlation equation obtained using the binary linear regression method agrees well with the experimental results, with a maximum deviation of less than 4%. Both of the two compound structures can effectively improve the overall cooling performance of the double wall structure. The double configuration has the best cooling performance and the area-averaged cooling effectiveness improves by 15.1% compared to the reference configuration under the condition of M = 1.5. The experimental results provide a vital database for the future double wall cooling design.

MHD Flow and Heat Transfer of a Jeffrey Fluid over a Porous Stretching/Shrinking Sheet with a Convective Boundary Condition

This work explores the heat transfer flow characteristics of an incompressible non-Newtonian Jeffrey fluid over a stretching/shrinking surface with thermal radiation and heat source. The sheet is linearly stretched in the presence of a transverse magnetic field with convective boundary conditions. Appropriate similarity variables are used to transform the basic governing equations (PDEs) into ODEs. The resulting equations are solved by utilizing MATLAB bvp4c. The impact of distinctive physical parameters and dimensionless numbers on the flow field and heat transfer is analysed graphically. It is noticed that the measure of heat raised with increasing the Biot number and opposite effect with the rise of the suction parameter.

Numerical Scrutinization of Darcy-Forchheimer Relation in Convective Magnetohydrodynamic Nanofluid Flow Bounded by Nonlinear Stretching Surface in the Perspective of Heat and Mass Transfer.

The aim of this research is mainly concerned with the numerical examination of Darcy-Forchheimer relation in convective magnetohydrodynamic nanofluid flow bounded by non-linear stretching sheet. A visco-elastic and strictly incompressible liquid saturates the designated porous medium under the direct influence of the Darcy-Forchheimer model and convective boundary. The magnetic effect is taken uniformly normal to the flow direction. However, the model is bounded to a tiny magnetic Reynolds number for practical applications. Boundary layer formulations are taken into consideration. The so-formulated leading problems are converted into highly nonlinear ordinary problems using effectively modified transformations. The numerical scheme is applied to solve the governing problems. The outcomes stipulate that thermal layer receives significant modification in the incremental direction for augmented values of thermal radiation parameter Rd. Elevation in thermal Biot number γ1 apparently results a significant rise in thermal layer and associated boundary layer thickness. The solute Biot number is found to be an enhancing factor the concentration profile. Besides the three main profiles, the contour and density graphs are sketched for both the linear and non-linear cases. Furthermore, skin friction jumps for larger porosity and larger Forchheimer number. Both the heat and mass flux numbers receive a reduction for augmented values of the Forchheimer number. Heat flux enhances, while mass flux reduces, the strong effect of thermal Biot number. The considered problem could be helpful in any several industrial and engineering procedures, such as rolling, polymeric extrusion, continuously stretching done in plastic thin films, crystal growth, fiber production, and metallic extrusion, etc.