Nonlinear mixed convection flow of a tangent hyperbolic fluid with activation energy

In this communication, the dynamics of a non‐Newtonian tangent hyperbolic fluid with nanoparticles past a nonuniformly thickened stretching surface is discussed. We examine the impact of nonlinear mixed convection flow of a hyperbolic tangent fluid with the Cattaneo‐Christov heat and mass diffusion model past a bidirectional stretching surface. The effects of activation energy and magnetic field are incorporated in the analysis. The variables of transformations are used to change the nonlinear partial differential equations into ordinary differential equations (ODEs). Then, these ODEs are numerically solved using the Matlab routine of the bvp4c algorithm. The derailed analysis of the influences of the governing parameters on velocities along the x‐ and y‐axes, temperature and concentration profiles are presented using tables and figures. The outcomes of these parameters reveal that the velocities along the x‐ and y‐axes are decreased for the values of We increasing but the opposite behavior is observed as the value of $A$ increases. The results also show that the values of $e$ and $Nb$ rise as the temperature profiles increase. Similar influences are observed on the profile of concentration as the values of $F$ and $f$ rise. As the values of $N1$ go from 0.27 to 0.25, the skin‐friction coefficient increases, and similarly, as $Nb$ goes from 0.3 to 0.1, $?\theta \prime (0)$ is enhanced.     

Mixed convection flow of a Maxwell nanofluid with Hall and ion‐slip impacts employing the spectral relaxation method

This article investigates the Hall and ion‐slip impacts on the mixed convection flow of a Maxwell nanofluid over an expanding surface in a permeable medium. The impacts of Brownian movement and thermophoresis parameters, Soret, Dufour, viscous dissipation, chemical reaction, and suction parameters, are, moreover, considered. Using the similitude changes, the partial differential equations with regard to the momentum, energy, and concentration equations are transformed to an arrangement of nonlinear ordinary differential equations, which are handled numerically utilizing a spectral relaxation method (SRM). The impacts of noteworthy physical parameters on the velocities, thermal, and concentration distributions are investigated graphically. Moreover, the numerical values of skin‐friction coefficients, local Nusselt number, and Sherwood number for different values of the mixed convection parameter $\left(\gamma \right),$ Deborah number $\left(\lambda \right),$ Hall parameter $({\beta }_{\mathrm{H}}),$ ion‐slip parameter $({\beta }_{\mathrm{i}}),$ Dufour number (Du), and Soret number $\left(\mathit{Sr}\right)$ are computed and tabulated. It is discovered that ascent in Deborah number reduces both the stream and transverse velocity profiles, while the inverse pattern is seen with augmentation in the mixed convection parameter. In addition, inverse patterns of the stream and transverse velocity profiles are seen with expansion in magnetic, Hall, and ion‐slip parameters. Besides this, the temperature and concentration disseminations decline with augmentation in Dufour number and chemical reaction parameters, respectively. It is likewise seen that both the skin‐friction coefficients lessen with expansion in Deborah number, and they ascend with upgrade in blended convection and ion‐slip parameters, while the opposite condition is noticed with augmentation in Hall parameter. Furthermore, the reverse trends of local Nusselt and Sherwood numbers are discovered with expansion in the Dufour and Soret numbers.     

Three-dimensional numerical simulation of free convection and entropy generation in a cubic cavity containing a heat source

The present article provides a three-dimensional numerical investigation of thermal convection and entropy generation. The lattice Boltzmann method, coupled with the finite difference approach, is applied to perform numerical simulations. The validation of these numerical approaches for thermal convection simulation and entropy calculation is performed by comparing our numerical results with those in the published literature for the case of benchmark problems. The physical geometry studied in this paper concerns a hot obstacle having the shape of a plus sign (+) placed in the center of a cubic enclosure. This cube is filled with air of a Prandtl number of 0.71 and characterized by two cold vertical walls. The heat exchange between the fluid and the hot body is studied as a function of the Rayleigh number (${10}^3\le \mathit{Ra}\le {10}^7$). The performed simulations show that the heat transfer rate can be increased by about 429% by switching from $\mathit{Ra}={10}^3$ to ${10}^7$. The entropy generation due to fluid friction, heat transfer, and total entropy are also calculated and discussed. For an irreversibility coefficient $\mathit{\phi }={10}^{-4}$, the analysis of the results showed that for low values of the Rayleigh number ($\mathit{Ra}={10}^3$), the entropy production due to temperature gradients predominates over that produced by viscous effects. In the cases of $\mathit{Ra}={10}^4$ and ${10}^5$, entropy generation is due to both fluid friction and heat transfer. However, when the Rayleigh number becomes large ($\mathit{Ra}{\ge 10}^6$), entropy generation due to viscosity predominates over entropy production related to heat exchange. These results have important implications for the optimization and design of heat transfer systems in various industrial applications.     

Heat and mass transfer effects on fluid past an exponentially accelerated vertical plate due to time-fractional MHD-free convection

This article focuses on the study of heat and mass transfer (HMT) fluid flow over an exponentially accelerated vertical plate, which is subjected to an applied magnetic field and viscous dissipation. The research has applications in various manufacturing processes such as wire/fiber drawing, hot rolling, continuous casting, and hot extrusion, where heat transfer to the ambient medium and the hot moving material are of utmost importance. The findings could also be relevant to aerospace engineering applications. The study investigates the time-fractional natural convection phenomenon and utilizes conservation laws to derive the flow guiding equations, which are then made nondimensional. Finite difference discretization is utilized to solve the dimensionless equations implicitly. Then the flow simulation results such as concentration, temperature, and velocity profiles are discussed based on the variation in parameters such as Prandtl number ($Pr$), thermal/mass Grashof number ($\mathit{Gr}/\mathit{Gc}$), Eckert number ($\mathit{Ec}$), magnetic parameter ($M$), time-fractional order ($\lambda$), and Schmidt number $\left(\mathit{Sc}\right)$. Also, the HMT rate is depicted using the Nusselt number and skin friction plots. It is noted that HMT increases when $\mathit{Sc}$ increases and $\lambda$ decreases. The change in time-fractional order affects the velocity profiles adjacent to the wall and is more significant in the case of lower values of the Prandtl number.     

Convection of 3D MHD non-Newtonian couple stress nanofluid flow via stretching surface

This study involvesthe numerical modeling of steady thermal radiation and chemical reaction on non-Newtonian fluid motion via a bidirectional stretching surface. We have taken convective boundary conditions, and heat sources on the stretching surface. The working fluid of the present study is Casson fluid (“non-Newtonian”) with couple stress. The self-similarity forms of the nonlinear thermal radiative flow model are obtained by using similarity variables. Furthermore, the numerical results are computed with the help of fourth-order Runge–Kutta–Fehlberg method with a shooting algorithm after reducing nonlinear partial differential equations have been translated into strong ordinary differential equations (ODEs). Impacts of the various flow physical parameters especially Biot number, nonlinear thermal radiation, and heat source parameters containing nonlinear ODEs are discussed in detail for distinct numerical values. A comparison of calculated results with the known numerical results made with the previously published literature is mentioned and obtained a good agreement. Finally, we found that the $R{e}_x^{1/2}{C}_{fx}$ (“coefficient of skin friction”) declines along $x*,y*$ directions, respectively, with $\beta$ via $\lambda$ while the opposite direction follows $M$ with respect to $\lambda$ and the $R{e}_x^{-1/2}N{u}_x$ (“heat transfer rate”), $R{e}_x^{-1/2}Sh$ (“mass transfer rate”) increase with $\Gamma$ via ${\gamma }_1$ while opposite direction follows ${\gamma }_1$ with respect to ${\gamma }_2$.     

Effect of arc-shaped vertical control plate on heat and mass transfer in uniform flow past an isothermally heated circular cylinder

The current work investigates the effect of an arc-shaped vertical control plate on the heat and mass transfer in uniform flow past an isothermally heated circular cylinder. The control plate is positioned downstream at various distances from the circular cylinder's surface. The governing equations are discretized by the higher order compact (HOC) finite difference scheme, and then this system of discretized equations is solved by using a bi-conjugate gradient stabilized iterative method for Prandtl number $Pr=0.7$, Reynolds number $Re=150$. To investigate the effect of an arc-shaped control plate on heat and mass transfer, we consider a range of nondimensional distances between the circular cylinder and the control plate, $0.5\le d∕R_0\le 8$, where $d$ is the control plate's distance and $R_0$ is the cylinder's radius. The exact timing and location of the bifurcation points are calculated by using topological aspect-based structural bifurcation analysis. Significant effects of various locations of the control plate on periodic wake and heat transfer are observed. It is found that the increasing distance of the control plate from the cylinder delays the occurrence of the structural bifurcation and shifts the bifurcation points upwards in the upper half and downwards in the lower half of the cylinder. Our study shows that the specific location of the control plate can fully suppress the vortex shedding. Time-averaged total Nusselt number can be drastically reduced by increasing the distance between the control plate and the cylinder. With proper positioning, the vertical control plate can lessen the time-averaged drag force by up to $22.5\%$ when compared to a cylinder without a control plate. Overall, this work presents many new phenomena that have not been reported before.     

Parametric studies of mixed convective fluid flow around cylinders of different cross-sections

A numerical study of mixed convective heat transfer in a lid-driven square enclosure containing a hot elliptic cylinder is conducted. The impacts of the Grashof number $({10}^3\le \mathit{Gr}\le 10^6)$, Reynolds number $(1.0\le Re\le 100)$, cylinder tilt angle $(0^{°}\le \varphi \le {90}^{°})$, and aspect ratio $(1.0\le AR\le 3.0)$ have been examined for a fluid of $Pr$ of 0.71. The horizontal enclosure walls are insulated, while its vertical walls are restricted to a nonvarying temperature T_c, whereas a sinusoidal temperature of $T_h+∆T\sin (\pi x/L)$ is imposed on the wall of the elliptical cylinder. The governing equations are solved using COMSOL Multiphysics 5.6 software. The fluid dynamic and the heat transport profiles between the enclosure and the elliptical cylinder walls are represented by the stream function, isothermal contours, and average Nusselt number. Results established that for all the considered aspect ratios, the thermal heating range of ${10}^3\le \mathit{Gr}\le 10^4$ is predominantly a conduction mechanism. The critical position of the ellipse where the inclination effect becomes insignificant is determined by the Grashof number and aspect ratio when the Re = 100. The strength of vortices and cell numbers are significantly influenced by the aspect ratio, particularly when the $\mathit{Gr}=10^4$. When $AR=1.0$, the average heat transfer from the cylinder remains the same regardless of the cylinder's orientation. The impact of cylinder orientation on heat transfer from the cylinder wall is minimal for $1.5\le AR\le 2.0$. For AR values of $2.5\le AR\le 3.0$, increasing the inclination angle does not result in improved heat transfer. The influence of the increasing inclination angle on the right wall diminishes as the angle increases, except when the Grashof number is greater than 10⁵, where the rate of heat transfer is enhanced for inclination angles beyond 45°.     

Nanoscale heat transfer investigation of an array of impinging jet systems with different working fluids under crossflow with and without pin fins

In the current numerical study, the thermal and flow field performance of an array of confined multiple jets with air, water, and water‐Al₂O₃ nanofluid in the maximum crossflow configuration over the target plate with and without pin fins is investigated. The numerical results are validated with the experimental data; it is found that a reasonable prediction related to heat transfer can be made. For this study, steady‐state Reynolds‐averaged Navier‐Stokes simulations with the shear‐stress transport $k‐\omega$ turbulence model in ANSYS Fluent were performed. The simulations are performed with volumetric concentration $?=0.2\%$ to 3% and the jet's Reynolds number Re = 15 000 to 35 000. In all cases, the jet outlet‐to‐target plate distance $Z/D$ is 3. It is found that the increase in values of the volumetric concentration of nanoparticles results in a decrease of the Nusselt number and an increase of the convective heat transfer coefficient. This is because there is an increase in thermal conductivity of the working fluid with the increase in the volumetric concentration of nanoparticles for the same Reynolds number. About 81.5% and 89.1% enhancement in the average heat transfer flux values is observed for flat and pin fin‐roughened target plates, respectively, for $?=3\%$.     

Natural convection coupled to surface radiation in an air-filled square cavity containing two heat-generating bodies

The present numerical study focuses on the cooling by natural convection and surface radiation of two electronic components generating two different and uniform volumetric powers. These components are modeled by two square bodies placed inside a closed square cavity with a cold straight wall. Two configurations are analyzed based on the position of the two heat-generating bodies. In the first one (horizontal position configuration), the two bodies are located at the same height of the cavity, while they are placed at different heights in the second case (vertical position configuration). The effects of two Rayleigh numbers ($0\le ({\mathit{Ra}}_1,{\mathit{Ra}}_2)\le {10}^6$), the conductivity ratio ($0.01\le K\le 100$), and the emissivity ($0\le \epsilon \le 1$) on the heat transfer characteristics and the flow structure are analyzed. The data is displayed as streamlines, isotherms, velocity, and maximum temperature profiles, and local heat transfer on the active wall. The obtained results indicate that the choice of the appropriate configuration depends mainly on the deviation between the two Rayleigh numbers. Furthermore, the maximum temperature of a specific block decreases as the quantity of heat generated by the other block rises. We can also see that the maximum temperature of the two blocks decreases by about $50\%$ with the increase in the emissivity (from $0$ to $1$) or the conductivity ratio (from $0.1$ to $1$).     

Cavity inclination effect on convection flow in a triangular geometry

In this study, the water convection flow within a right-angled, inclined, and isosceles triangle enclosure for various inclination angles was numerically analyzed using the lattice Boltzmann method with the multirelaxation time model. On the hypotenuse side, the enclosure is thermally insulated, while the left and horizontal walls are kept, respectively, at cold and hot temperatures. This study was conducted to show the effects of two key parameters, the tilt angle $\varphi$ and the Rayleigh number $Ra$, whose changes span from $0^{\circ }$ to $315^{\circ }$ and $5\times 10^3$ to $10^6$, respectively. The effect of these variables is presented in terms of streamlines, isotherms, velocity profiles, temperature plots, and the average Nusselt number. Furthermore, the impact of the size of a hot square obstruction inside the cavity on the isotherms and streamlines has been investigated. The findings demonstrate that the rate of heat transport is enhanced as the Rayleigh number increases. This result is in good agreement with earlier research without tilting the cavity. Depending on the Rayleigh number, the tilt angle has a significant effect on the rate of heat transmission.     

Time-dependent pressure-driven flow in a horizontal cylinder filled with a bidisperse porous material

Some properties of time-dependent that modify Brinkman equations for fluid flow in a cylindrical tube filled with Bidisperse Porous Material are discussed in this article. The fluid velocities through the fracture and porous phases of the Bidisperse Porous Medium (BDPM) resulting from the application of pressure gradient are described by two coupled second-order partial differential equations. Laplace transform technique, D'Alembert and Riemann-Sum Approximation Methods are used to obtain a semianalytical solution for the model. The choice of the D'Alembert is made to systematically decouple the coupled governing equations without altering their initial orders. The role of the coupling parameter: The coefficient of momentum transfer $(\eta )$ in the flow formation is considered. Accordingly, three cases are analyzed: (a) weak coupling $(\eta =0)$ which described the fluid flow in the absence of the coupling parameter, (b) the strong coupling resulting from a large value of the coupling parameter $(\eta \to \infty )$, and (c) fluid momentum for any arbitrary value of $\eta$. It is observed that fluid stability is attained when ${\mathit{Da}}_f$ and ${\mathit{Da}}_p$ are decreased; a finding that agrees with the findings of Nield and Kuznetsov and Magyari. Also, the maximum velocity in the fracture phase of the BDPM is attained when the coefficient of momentum transfer is neglected $(\eta =0)$ while an opposing flow formation is demonstrated in the fracture and porous phases of BDPM as $\eta$ is increased.     

Transient free convective flow through a vertical cylinder filled with a porous material

This article aims to explore the impressive impact of emerging parameters on transient fully evolved free convective flow inside a vertical cylinder containing a porous material. The mathematical formulation of the model related to the considered physical circumstance is presented under compatible boundary conditions. Closed‐form solutions are received for the velocity field, the temperature distribution, mass flux, skin friction, and the Nusselt number in terms of Bessel functions and modified Bessel functions of the first kind. Impressive effects of parameters such as the Darcy number $Da$, Prandtl number $\text{Pr}$, viscosity ratio $M$, and also time $t$ on both the velocity and temperature distribution have been explored employing graphs and tables. It is irradiated by analysis that flow erection, heat transfer rate, skin friction, and mass flux are admirably impacted by the Prandtl number, the Darcy number, viscosity ratio parameter, and time. It is found that both the velocity and temperature field profiles rise with the rising value of time and ultimately attain their steady state. Moreover, the Prandtl number and the viscosity ratio parameter reduce the velocity profiles, while the reverse phenomenon occurs with the Darcy number.     

Conjugate local thermal nonequilibrium and non-Darcy flow inside porous enclosure: Analysis of localized heating and cooling arrangements

This study covers a simulation on conjugate free convective in a porous enclosure containing a side wall thickness and partially heated and cooled from sides under the considerations of local thermal nonequilibrium (LTNE) and non-Darcy flow. Interest has been focused on how the side wall thickness and the locations of cooled and heated parts affect the effectiveness of the Nusselt number (Nu). Three different cases of localized heating and cooling locations have been implemented for the following ranges: scaled heat transfer coefficient ($0.1\le H\le 100$), wall to fluid thermal conductivity ratio ($0.1{\le R}_k\le 100$), modified Rayleigh number ($200\le \mathit{Ra}*\le 1000$), wall width ($0.1\le \hat{Z}\le 0.5$), inertial parameter (${10}^{-4}{\le F}_s/P_r^{*}\le {10}^{-2}$), and thermal conductivity ratio ($0.1\le K_r\le 100$). Outcomes show that $\hat{Z}$ and the locations of cooled and heated parts have remarkable impacts on all the Nusselt numbers. The intensity of LTNE region considerably relies on $\mathit{Ra}*$, $K_r$ and $H$. The total average Nu_T is highly dependent on $R_k$, $\mathit{Ra}*$, $\hat{Z}$, $F_s/P_r^{*}$, and $K_r$ as compared to H. The increase in $\hat{Z}$ leads to change of the convective mechanism to conductive mode. The rise in $R_k$ guides to increase Nu, where $R_k$ can control the flow strength. The actions of $F_s/P_r^{*}$ on Nu_f is more evident than Nu_s. For low H and K_r, the size of LTNE zone is considerably affected by H as compared to K_r although K_r has a high influence on Nu. For high K_r and H, the LTNE zone has closely vanished. Findings display that the Case 2 provided the highest Nu for all tested parameters except the case of $K_r=0.1$. Finally, it is evident that for the problems that employed solid conduction wall with localized heating and cooling sections, Case 2 is recommended for future use in the applications that implement a porous medium and depend on free convection.     

Multiple slip impacts on unsteady MHD micropolar fluid past a stretching sheet with non-Darcy porous medium and uneven heat source/sink

The major scope of this research is to scrutinize the effects of multiple slips on unstable magnetohydrodynamic micropolar fluid past a stretched sheet with a non-Darcy porous medium. In the momentum equation, the non-Darcy porous medium effect is also taken into consideration. The effects of uneven heat source/sink and thermal radiation in the energy equation are also analyzed. By implementing the similarity transmission, the mathematical modeling of the set of managing partial differential equations is reframed into nonlinear ordinary differential equations. These equations are numerically solved by applying Matlab built-in solver bvp5c. The implications of foremost parameters such as micropolar parameter, magnetic parameter, permeability parameter, Prandtl, Eckert, and Schmidt numbers, Chemical reaction, slip parameters on velocity, microrotation, temperature as well as concentration profiles are displayed pictorially and explained. It is worthwhile to mention that the improving values of micropolar parameter $K$ escalate the velocity as well as microrotation profiles. However, the upsurge in non-Darcy porous medium $F_s$ will cause a declining nature in the velocity profile. Also, an enhancement in the unsteadiness parameter $A$ brings about a lessening in all the profiles. Increment in all the three usual slip parameters will bring a declining nature in the respective profiles. An increase in Schmidt number will give a deduction nature in velocity as well as concentration profile. Moreover, the physical quantities are defined and Nusselt numbers are formulated in the table, and it enlarges while boosting up $Pr$ and $R$, whilst a reverse nature is noticed for others. This present study compared with the earlier studies in special cases holds a better agreement.     

Impact of size and location of outflow opening vent on mixed convective heat transfer induced by two aligned heated cylinders immersed in a partially open channel

The current work deals with the numerical investigation of mixed convection occurring from two hot cylinders placed inside vented ducts under different sizes of inflow and locations of outflow opening vents. The mathematical formulation is solved based on the finite volume approach. The location of the outflow section is changed to left, center, and right of enclosure. Impacts of inflow section sizes ($\mathit{O}$₁ = 0.125 and $\mathit{O}$₁ = 0.25), locations of outflow section ($O_2$ = $0.125$), space between cylinders ($0.3\le S\le 0.45$), Reynolds number ($200\le Re\le 400$), and Richardson number ($0.1\le \mathit{Ri}\le 20$) are implemented. Findings display that the average Nusselt number (Nu) is exceeded by 20% for low $\mathit{Ri}$ due to the effect of nonsymmetrical inflow and outflow sections as compared with the case of equal opening vents. For large Ri, a higher Nu is obtained for nonequal opening vents in comparison with the case of equal opening vents by 10%. The difference in the average Nu between left and right cylinders is about 11%. Findings display that the enhancement in Nu due to the large spacing size between cylinders as compared with those of low spacing size is about 35%. The variations of Nu display the opening vents ratio, Re, $S$, and $\mathit{Ri}$ have an extreme action on the characteristics of flow and temperature fields.     

Heat transfer characteristics in non-Newtonian fluid flow due to a naturally permeable curved surface and chemical reaction

In this research endeavor, Casson fluid flow and melting heat transfer due to a curved nonlinearly stretching sheet are investigated. The sheet is naturally permeable and the flow is considered in a porous medium. For flow in a porous medium, a modified Darcy's resistance term for Casson fluid is considered in the momentum equation. In the energy equation, heat transport characteristics, including viscous dissipation, are taken into account. Mass transport is also studied together with the impact of chemical reaction of higher order. The governing nonlinear partial differential equations of flow, heat, and mass transport are reduced to nondimensional ordinary differential equations using adequate similarity transformations and then solved numerically employing the bvp4c technique and Runge–Kutta fourth-order method on MATLAB. The impacts of numerous occurring parameters on relevant fields (velocity field, temperature field, and concentration field) are depicted and discussed by plotting graphs. We concluded the curvature parameter, $K$ reduces the pace of the flow. The impacts of the stretching index, $m$ and melting parameter, $Me$ are also found to reduce flow and temperature field. Furthermore, we noted that the reaction parameter, $K_n$ and its order, $n$ exhibit opposite impacts on the concentration field. Moreover, the numerical values of skin-friction coefficient and Nusselt number calculated employing bvp4c and Runge–Kutta fourth-order technique are expressed in tabular mode, and these are found in an excellent match. For validation of the results, skin-friction coefficient values were computed using the Runge–Kutta fourth-order technique and bvp4c solver, compared with the existing results, and a good agreement was found.     

Transient natural convective heat transfer and fluid flow in an undulated cavity: Effects of localized heat sources

The purpose of this research work is to investigate two-dimensional transient natural convective heat transfer and fluid flows in an undulated cavity by placing solid objects with isolated heated surfaces on the bottom wall. We discretize the coupled nonlinear transport equations using a higher-order compact finite-difference scheme. First, we test our scheme using existing experimental and numerical data. Then, we analyze the transient and steady-state natural convective flow phenomena for distributed heat sources on corrugations on the lower wall for a range of the Rayleigh number $\left(Ra=10^3-10^6\right)$ and Prandtl number $\left(Pr=0.71\right)$. These simulated outcomes are presented in the form of central-line velocity $\left(u,v\right)$, local $(N{u}_h^{*},N{u}_v^{*})$ and averaged $(\overline{N{u}_h^{*}},\overline{N{u}_v^{*}})$ Nusselt numbers, streamlines $\left(\psi \right)$, dispersion of isotherms $\left(T\right)$, and so forth. It is found that the transient fluid flow behavior is more magnificent than the steady-state solutions and shows the dominant behavior of the prominent primary cells over secondary cells, where it influences the heat transfer rates inside the entire enclosure. In steady states, at high Rayleigh numbers; convection dominates, formation of thermal boundary layers, compression of isotherms, and stratification of isotherms are significantly observed. Our results show many interesting flow phenomena that have not been analyzed previously.     

Dual solution of heat transfer in hydromagnetic micropolar fluid flow over a stretching sheet in a porous media: A numerical approach

In this study, the researcher looks at the heat transmission of an incompressible magnetohydrodynamics micropolar fluid across a moving stretched surface in a Darcian permeable medium. The proper boundary conditions are used to facilitate the numerical solution (bvp4c) of the transformed governing equations. Graphical discussions have been made of the influence of the physical parameters on the velocity, angular velocity (microrotation), and temperature, and the distributions are accentuated on the plots via MATLAB. The study is validated by the previous work and it is found appropriate for investigation, where the absolute difference between the previous work and the present investigation by adopting the finite difference scheme is smaller than which implies that the scheme is stable and convergent. The microrotation has a great impact on the micropolar fluid with the influences of buoyancy forces, source, and suction over the stretching surface in a Darcian regime. With a rise in the heat source parameter, both velocity and microrotational profiles lessen, but the opposite is true for temperature. Eringen number () rises with the flow velocity, whereas temperature and microrotational profiles show the reverse relationship. The current study focused on particular applications in non-Newtonian fluid mechanics, polymer flows in filtration systems and metallurgical procedures that included cooling unbroken strips or filaments via a static fluid.     

Triple solutions of micropolar nanofluid in the presence of radiation over an exponentially preamble shrinking surface: Convective boundary condition

In this study, we attempt to obtain all probable multiple solutions of the magnetohydrodynamic (MHD) steady flow of micropolar nanofluid on an exponentially shrinking surface by the consideration of concentration slip, thermal radiation, and convective boundary condition with help of the revised model of Buongiorno. The significance of the mass suction on the existence of multiple solutions is integrated. The suitable pseudo‐exponential similarity variables have been adopted to transfer the system of nonlinear partial differential equations into a system of nonlinear quasi‐ordinary ordinary differential equations. The resultant system has been solved by employing the Runge–Kutta fourth‐order method along with the shooting method. Three different ranges of solutions are noticed, namely triple solutions and single solution. When ranges of the suction parameter are $S\ge S_{c1}$ and $S\ge S_{c2}$, then there exist triple solutions otherwise there exists only single solution. The effect of the numerous emerging parameters on the velocity profile, angular velocity profile, temperature profile, concentration profile, coefficient of skin friction, and local Nusselt and Sherwood numbers are demonstrated graphically. Results reveal that the velocity of the rotating fluid particles near the rigid surface declines regularly by the rise of the micropolar parameter $K$ in the second and first solutions.     

Investigation of Oldroyd-B fluid flow and heat transfer over a stretching sheet with nonlinear radiation and heat source

The given investigation concerns the study of non-Newtonian Oldroyd-B fluid flow across a permeable surface along with nonlinear thermal radiation, chemical reactions, and heat sources. Equations modified are thus numerically evaluated by employing bvp4c-technique. Obtained outcomes are exhibited graphically. Pictorial notations are used to investigate the consequences of necessary parameters of velocity, energy, and mass. Acquired outcomes provide promising agreement with already established consequences provided in the open literature. The obtained results guided that magnetic field parameter ($M$), porosity parameter ($Kp$), Deborah number ${\beta }_1$ reduce momentum boundary layer thickness, furthermore, growth in the relevant Deborah number ${\beta }_2$ improves the corresponding momentum boundary layer.