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This study explains the steady flow of a hybrid nanofluid due to a permeable shrinking sheet close to a stagnation region with suction/injection. By proper transformations, the leading differential equations, partial in nature, are reduced to nonlinear differential equations with a single independent variable. A combination of the Runge–Kutta method and a shooting technique is applied to unravel those nonlinear equations. The effects of various parameters on the flow and thermal fields are presented graphically and analyzed. Also, for various physical parameters, wall shear stress, and the heat transport coefficient are computed numerically and discussed in detail. For the boundary value problem, multiple (two) solutions are obtained for a definite choice of governing parameters. Hence, a stability analysis is carried out, indicating that the first branch of the solution is stable and physically realistic. The unsteady equations are considered for implementing the stability analysis. Furthermore, in a hybrid nanofluid flow, a higher heat transfer rate than a nanofluid is observed. The separation of the boundary layer occurs. Also, with a rise in the value of the nanoparticle volume fraction of Al2O3, the critical value of the suction parameter increases. [ABSTRACT FROM AUTHOR]

Enhancement of "heat transfer" using "nanofluid" has diverse potential applications in heat exchangers, thermal management of electric devices, cooling of tractors, solar thermal systems, manufacturing of paper, and many others. Hence, the aim of the current investigation is to explore the impacts of "mixed convection" on "nanofluid flow" over a permeable rotating disk, which is stretched radially in a porous medium. Variable wall "temperature" and "convective boundary conditions" are also considered here. This makes the present investigation different from others. The suitable "similarity transformations" are imposed to alter the governing partial differential equations into a set of coupled ordinary differential equations (ODEs). Then, these ODEs are solved numerically by the "4th order Runge‐Kutta method" using the "shooting technique" with the help of the bvp4c package in MATLAB software. The effects of fluid controlling "parameters" on "flow and thermal fields" as well as "skin friction coefficient" and "Nusselt number" are presented graphically and explained physically. Due to enhanced rotation of the disk, the radial and azimuthal velocity of the fluid increase and the temperature of the fluid decreases. Most importantly, it is observed that when the disk rotates faster than the stretching rate, the temperature of the nanofluid decreases rapidly, which has wider applications for cooling purposes. It is also noted that when the suction parameter increases its value from −1 to 1, for Ag–water nanofluid, the "skin friction coefficient" decreases by 73.56%, and the Nusselt number also decreases by 24.11%, and for Fe3O4–water nanofluids, the "skin friction coefficient" decreases by 71.25% and the Nusselt number decreases by 24.47%. [ABSTRACT FROM AUTHOR]

AbstractEyring–Prandtl nanofluid is used in several areas for the purpose of flow control, improvement of stability, enhancement of heat transfer, and many others. Motivated by these uses, this work aims to investigate the transmission of heat in two-dimensional Eyring–Prandtl nanofluid flow across a moving plate under the influence of Stefan blowing effects in a Darcy-porous medium. The new aspect of the flow problem arises due to the moving plate having nonlinear velocity as well as inclusion of Stefan blowing and consideration of nanofluid. This creates a boundary layer flow over the plate. Similarity transformations are employed for obtaining self-similar structure of the leading equations. Self-similar solutions are found for a specific power law velocity of the moving plate. Numerical technique is adopted for finding solutions of the self-similar nonlinear equations. A thorough discussion and demonstration of the effects of fluid material parameters, medium permeability, Prandtl number, Lewis number, and other parameters on temperature, concentration, and velocity are made. The possible physical explanations are provided systematically. The permeability of the porous medium determines how the fluid is flowing in the porous medium. It restricts the velocity but temperature and concentration are enhanced. However, the Stefan blowing parameter has shown reverse effects on velocity and concentration. This study indicates that the two material fluid parameters involved in the momentum equation have opposite effects on temperature, concentration, and velocity. The basic essence of this study is to find the boundary layer flow structure developed over a moving plate in porous medium and heat, mass transfer influenced by the nanofluid. [ABSTRACT FROM AUTHOR]

Combined impacts of prescribed surface heat and mass fluxes on nanoliquid flow over a vertical plate with shear flow have been inspected. A two‐phase model for nanofluid has been considered. By means of similarity alterations, partial differential equations which govern the problem under consideration are changed to ordinary ones and nonlinear equations are solved numerically by shooting technique and Runge–Kutta method. The related effects of appropriate parameters on streamwise velocity, velocity, velocity gradient, temperature, and concentrations are analyzed elaborately. The importance of unlike factors on heat and mass transport as well as fluid flow phenomena are detected and analyzed physically as far as practicable. Fluid velocity rises but temperature and concentration are noted to diminish due to mixed convection. Velocity gradient at the wall rises with the rise in mixed convection parameter but wall temperature and wall concentration are found to diminish with the growing values of mixed convection parameter. As Brownian motion parameter and thermophoresis parameter increase, the wall velocity gradient declines until a certain point and from that point it starts to behave in the opposite way. Additionally, the temperature at the wall increases as these parameters grow, but temperature at the wall decreases with the rising values of the mixed convection parameter. On the other hand, the wall concentration increases with the increasing values of the thermophoresis parameter, but wall concentration decreases with the swelling values of the Brownian motion parameter and mixed convection parameter. The results of this study will be of immense help to the engineers and scientists to obtain superior flow phenomenon and heat, mass transport features. [ABSTRACT FROM AUTHOR]

The aim of this study is to present forced convective nanofluid flow over a moving plate embedded in an absorbent medium. Following Darcy law's for porous medium, the flow analysis is explored in attendance of warmth basis/drop. The main objective of this study is to explore the effects of Brownian motion and thermophoresis. The plate is considered to move in both directions: in the way of movement of fluid and in the opposite route of fluid movement. Similarity alterations have been applied to alter the leading partial differential equations (PDEs) to ordinary differential equations (ODEs). Numerical solutions have been obtained with the help of MATHEMATICA software. Dual solutions have been obtained when the plate and liquid go in reverse ways. Wall shear stress, Nusselt and Sherwood numbers all are found to rise with the rising permeability parameter of absorbent medium. For Nusselt and Sherwood numbers, ranges of dual solutions diminish by the mounting values of permeability parameter K. The critical values for porosity parameter K = 0. 0 1 , 0.02, 0.03 are R c 1 = 1. 8 7 2 9 0 9 , R c 2 = 1. 9 2 7 2 1 1 , R c 3 = 1. 9 8 2 4 2 8 4 , respectively. For decreasing values of s, range of dual solutions decreases. For s = − 0. 4 5 , dual solutions exist in the range (1. 1 9 , 1. 2 0). [ABSTRACT FROM AUTHOR]

The combined impacts of Thompson‐Troian slip and Stefan blowing on flow of nanoliquid over a vertical plate in attendance of shear flow have been studied. Nanofluid is generally used to enhance the thermal properties as well as heat transport of liquid. Moreover, Stefan blowing can enhance the heat transfer coefficient by lessening the thermal resistance which is widely used in different types of cooling system. For this, such type of problem has its significant importance in various technological sectors and it attracts the attention of the researchers. The forced convection flow was considered by the authors whereas the mixed convection flow in this aspect is new and fills the gap in the literature as we have addressed this issue in this article. Here lies the novelty of our work. Using proper “similarity transformations”, the principal (partial differential equations) PDEs are altered to (ordinary differential equations) ODEs, and abridged ODEs are numerically solved with the support of Runge‐Kutta method and shooting technique. The outcomes of this investigation have significant impression on the physical appearance for flow, heat and mass transports. Fluid velocity reduces due to rise of Stefan blowing parameter but temperature is experienced to diminish. For growing slip speed and the critical shear speed, the liquid speed augments, while the hotness reduces for growing velocity slip. Compared to no‐slip case, rate of heat transfer increases 73.73% and 17.95% rise in concentration gradient at the wall is noted whereas 5.35% reduction in wall shear stress is observed for δ=0.2$\delta = 0.2$. The outcomes of this learning will be of immense help to the engineers and scientists to acquire superior flow and warmth convey features. [ABSTRACT FROM AUTHOR]

The unsteady three-dimensional flow of nanofluid over a stretching surface in the presence of a magnetic field and thermal radiation are investigated. Zero nanoparticle flux at the wall has been considered. Buongiorno's model for nanofluid has been used in this study. Employing similarity transformations, the governing partial differential equations are transformed to ordinary ones and the transformed equations are then solved numerically by using the shooting technique with the help of Runge–Kutta method. The computed results are compared with previously reported work and found in excellent agreement. Fluid velocity initially decreases with the increase of the unsteadiness parameter. Temperature and nanoparticle volume fraction both increase with the increase of unsteadiness parameter. Due to the increase of thermal radiation temperature increases, nanoparticle volume fraction decreases initially but increases away from the surface. The effect of the other governing parameters on velocity, temperature and concentration fields are discussed through their graphical representations. Besides, the local skin friction coefficient, local Nusselt number and local Sherwood number are presented through graphs. [ABSTRACT FROM AUTHOR]

An investigation is carried out to analyze the flow, heat and mass transport phenomena of nanofluids over a disk stretched radially in a porous medium. Besides nonlinear radiative heat flux and Arrhenius activation energy, multiple slips are considered in this model. This ensures the novelty of the present research. The main goal of the current investigation is to explore the mutual impacts of nonlinear radiative heat flux, various slip factors and Arrhenius activation energy on flow, heat and mass transfers. The highly nonlinear central equations which are partial differential equations (PDEs) are transformed into nonlinear ordinary ones (ODEs) by employing suitable similarity alterations and then for finding the approximate numerical solutions, shooting technique with 4th order "Runge–Kutta method" are applied. The impact of a variety of parameters on "skin friction coefficient," "Nusselt number," "Sherwood number," velocity of the fluid, heat and mass transfers are computed and presented graphically. It is interesting to observe that fluid velocity diminishes with the growing porosity, suction and velocity slip parameters. Nanofluid's temperature reduces for enhanced values of thermal slip parameter but for rising thermal radiation parameter, temperature is boosted up. Also, nanoparticle concentration diminishes for increasing values of mass slip parameter. [ABSTRACT FROM AUTHOR]

An expanding sheet problem is more relevant when the thickness of the sheet is variable and it bears frequent applications in polymer press, paper production, metallic plate cooling, etc. On the other hand, activation energy is an important phenomenon of chemical reaction in flow dynamics of Newtonian and non-Newtonian fluids. The activation energy and chemical reaction have vital applications in food preparing, the mechanism of water and oil emulsions, chemical engineering, and more. So in this project, the impacts of activation energy and binary chemical reaction on MHD two-dimensional boundary layer flow of Williamson nanofluid on an expanding surface of variable thickness embedded in Darcy–Forchheimer porous medium are investigated. Using suitable transformations, the governing equations are transformed into a set of non-linear ordinary differential equations (ODEs). Later, numerical solutions have been achieved by well-known MATLAB inbuilt function 'bvp4c'. Several vital results are explored for variations of involved physical parameters and those are presented in graphical and tabular modes. The achieved results suggest that when wall thickness parameter increases, there is a contrast in behaviors of velocity, temperature and nanoparticle concentration if there is a condition that the shape parameter is greater than or less than unity. For the former case, the above flow properties reduce with wall thickness parameter, whereas, for the latter case, those are showing significant growth. The Brownian motion of nanoparticles causes an increase in temperature and a reduction in nanoparticle concentration, whereas due to thermophoretic force, both temperature and nanoparticle concentration rise. Due to the presence of activation energy in chemical reaction, the nanoparticle concentration enhances, while, temperature decreases(increases) near(away from) the sheet. With increasing reaction rate parameters, nanoparticle concentration diminishes, but temperature increases near the sheet. The surface drag force decreases with Williamson fluid parameter, while it increases with the magnetic parameter, inverse Darcy number, and Forchheimer parameter. On the other hand, the surface heat flux and surface mass flux are decreasing functions of Williamson fluid parameter, magnetic parameter, inverse Darcy number, and Forchheimer parameter. It also reveals that surface heat flux reduces with increasing reaction rate parameters, whereas surface mass flux increases. Finally, for the growth of activation energy parameter, initially surface heat flux rises and surface mass flux declines, but for its larger values, the quantities turn out to be constants. Also, the surface heat and mass fluxes are decreasing functions of thermophoresis parameter. [ABSTRACT FROM AUTHOR]

A numerical analysis has been presented for stream of a mixture nanoliquid over a slendering stretched sheet in company of an external magnetic meadow under boundary layer approximations. This investigation is performed to look at the impacts of variable thickness on balanced two-dimensional (2D) flow of a hybrid nanofluid under border-level approximations as it has wider technological applications in various industries. A variable magnetic field has been considered here. An effort has been taken in this piece to report the joint impacts of variable thickness and MHD on hybrid nanoliquid flow over a slendering stretched sheet as no one has reported such effects. Here lies the novelty of the work. By applying appropriate similarity transformations, the leading partial differential equations (PDEs) are converted to ordinary differential equations (ODEs) and the solutions of the nonlinear equations are subsequently obtained numerically using shooting technique. To hold up the soundness of the statistical outcomes, an assessment is completed with the accessible known outcomes for a few meticulous cases of this study. The consequences of a variety of parameters on motion of fluid and heat transport are analyzed and talked about exhaustively with the help of tabular and graphical representations. It is found that Lorentz force slows down the border-level flow and thickens the momentum border level width. Skin-friction coefficient reduces with the decreasing values of m for m > 0 which is obvious as superior values of m help the liquid to move faster over the exterior and the thickness of the boundary layer augments with the mounting values of m (m > 0). Thus, it can be concluded that the fluid motion is controlled by the power-law index m (m < 1) since the motion slows down for m < 1 and goes faster for m > 1. [ABSTRACT FROM AUTHOR]

The aim of this paper is to examine the combined effects of unsteadiness and thermal radiation on the flow of a mixture nanoliquid due to a shrinking disk. Investigation of flow near a stagnation-point naturally arises in several industrial sectors and is useful owing to its significance in thermal enhancement. The motion is persuaded by a radially stretched/shrinked exterior of the disk in adding to the flow near a stagnation point. As a result, this research examines the thermophysical belongings of the unsteady flow near a stagnation-point past a stretching/shrinking disk by using twin-kind nanoelements, viz. a mixture nanoliquid. Cu and Al2O3 nanoparticles have been considered in water to produce hybrid nanofluid. The numerous uses like electronic cooling system, heat exchangers and radiators are available. Using similarity transformations, self-similar equations have been obtained which are then solved numerically using MATHEMATICA software. Comparisons have been made with the available data in the literature and found a complete agreement for some special case of the problem. Double solutions subsist for equally stretched and shrinked disks, whereas a unique solution exists for the static disk. For flow past a stretching or a shrinking disk, a 27.79% of increase in warmth transport speed is noted for mixture nanofluid, contrasted to the nanoliquid. Hence, this study will be of interest to several scientists and engineers, owing its capability to augment the speed of heat transport in contrast to the regular nanofluids. [ABSTRACT FROM AUTHOR]

An exploration is carried out to examine mass diffusion of unsteady 'boundary layer' (bl) motion of viscous liquid passed a stretched leaky piece with variable mass flux. For several engineering applications, moving free stream is considered here. This makes this research unique. The leading 'partial differential equations' (PDEs) accompanied by the 'boundary conditions' are converted to 'ordinary differential equations' (ODEs) with the help of 'similarity transformations' and 'numerical solutions' are attained by MATLAB software. The effect of pertinent 'parameters' on fluid 'flow, concentration, skin friction coefficient' and wall concentration are discussed 'graphically' and numerically. When suction/blowing parameter increases from -0.2 to 0.2, skin friction coefficient decreases 18.235%. Fluid concentration reduces with growing values of velocity ratio parameter X for all cases considered. Compared to the case for static free stream, fluid velocity is higher when the free stream moves. Also higher concentration is noted in presence of moving free stream. The presence of moving free stream causes to diminish the effect of suction/blowing on flow and concentration fields. The increasing strength of suction causes to decrease the fluid velocity more significantly than that for blowing. [ABSTRACT FROM AUTHOR]