Search Results (6)

This paper concerns control BVPs, driven by ODEs $x^{\prime }\left(t\right)=u\left(t\right)$, using controls $u^0\left(·\right)$ $\&u^1\left(·\right)$ in $L^1\left(\left(a,b\right),{\mathbb{R}}^2\right)$. We ask these two controls to satisfy a very simple restriction: at points where their first coordinates coincide, also their second coordinates must coincide; which allows one to write $(u^1-u^0)\left(·\right)=v\left(·\right)\left(1,f\left(·\right)\right)$ for some $f\left(·\right)$. Given a relaxed non bang-bang solution $\overline{x}\left(·\right)\in W^{1,1}\left(\left[a,b\right],{\mathbb{R}}^2\right)$, a question relevant to applications was first posed three decades ago by A. Cellina: does there exist a bang-bang solution $\widehat{x}\left(·\right)$ having lower first-coordinate ${\widehat{x}}_1\left(·\right)\le {\overline{x}}_1\left(·\right)$? Being the answer always yes in dimension $d=1$, hence without $f\left(·\right)$, as proved by Amar and Cellina, for $d=2$ the problem is to find out which functions $f\left(·\right)$ “are good”, namely “allow such 1-lower bang-bang solution $\widehat{x}\left(·\right)$ to exist”. The aim of this paper is to characterize “goodness of $f\left(·\right)$” geometrically, under “good data”. We do it so well that a simple computational app in a smartphone allows one to easily determine whether an explicitly given $f\left(·\right)$ is good. For example: non-monotonic functions tend to be good; while, on the contrary, strictly monotonic functions are never good. Full article

For efficient heating and cooling applications, minimum wall shear stress and maximum heat transfer rate are desired. The current study optimized the local skin friction coefficient and Nusselt number in Al₂O₃-Cu/water hybrid nanofluid flow over a permeable shrinking rotating disk. First, the governing equations and boundary conditions are solved numerically using the bvp4c solver in MATLAB. Von Kármán’s transformations are used to reduce the partial differential equations into solvable non-linear ordinary differential equations. The augmentation of the mass transfer parameter is found to reduce the local skin friction coefficient and Nusselt number. Higher values of these physical quantities of interest are observed in the injection case than in the suction case. Meanwhile, the increase in the magnitude of the shrinking parameter improved and reduced the local skin friction coefficient and Nusselt number, respectively. Then, response surface methodology (RSM) is conducted to understand the interactive impacts of the controlling parameters in optimizing the physical quantities of interest. With a desirability of 66%, the local skin friction coefficient and Nusselt number are optimized at 1.528780016 and 0.888353037 when the shrinking parameter ($\lambda$) and mass transfer parameter ($S$) are −0.8 and −0.6, respectively. Full article

The mixed convection flow with thermal characteristics of a water-based Cu-Al₂O₃ hybrid nanofluid towards a vertical and permeable wedge was numerically and statistically analyzed in this study. The governing model was constructed using physical and theoretical assumptions, which were then reduced to a set of ordinary differential equations (ODEs) using similarity transformation. The steady flow solutions were computed using the Matlab software bvp4c. All possible solutions were presented in the graphs of skin friction coefficient and thermal rate. The numerical results show that the flow and thermal progresses are developed by enhancing the controlling parameters (wedge parameter, volumetric concentration of nanoparticles, and suction parameter). Moreover, the response surface methodology (RSM) with analysis of variance (ANOVA) was employed for the statistical evaluation and conducted using the fit general linear model in the Minitab software. From the standpoint of statistical analysis, the wedge parameter and volumetric nanoparticle concentration have a considerable impact on all responses; however, the suction parameter effect is only substantial for a single response. Full article

In the last two decades, academicians have concentrated on the nanofluid squeezing flow between parallel plates. The increasing energy demands and their applications have seen the focus shifted to the hybrid nanofluid flows, but so much is still left to be investigated. This analysis is executed to explore the symmetry of the MHD squeezing nanofluid (MoS₂/H₂O) flow and the hybrid nanofluid (MoS₂–SiO₂/H₂O–C₂H₆O₂) flow between the parallel plates and their heat transport property. The heat transport phenomenon is analyzed with the magnetic field, thermal radiation, heat source/sink, suction/injection effect, and porous medium. In the present model, the plate situated above is in the movement towards the lower plate, and the latter is stretching with a linear velocity. The prevailing PDEs depicting the modeled problem with the aforementioned effects are transformed via similarity transformations and solved via the “bvp4c” function, which is an inbuilt function in MATLAB software. The control of the factors on the fields of velocity and temperature, heat transfer rate, velocity boundary layer patterns, and streamlines is investigated. The solution profiles are visually shown and explained. Furthermore, the Nusselt number at the bottom plate is larger for the (MoS₂–SiO₂/H₂O–C₂H₆O₂) hybrid nanofluid than for the (MoS₂/H₂O) nanofluid flow. In the presence of suction/injection, the streamlines appear to be denser. In addition, the magnetic field has a thinning consequence on the velocity boundary layer region. The results of this study apply to several thermal systems, engineering, and industrial processes, which utilize nanofluid and hybrid nanofluid for cooling and heating processes. Full article

The present paper studies the flow and heat transfer of the hybrid nanofluids flows induced by a permeable power-law stretching/shrinking surface modulated orthogonal surface shear. The governing partial differential equations were converted into non-linear ordinary differential equations by using proper similarity transformations. These equations were then solved applying a numerical technique, namely bvp4c solver in MATLAB. Results of the flow field, temperature distribution, reduced skin friction coefficient and reduced Nusselt number were deduced. It was found that increasing mass flux parameter slows down the velocity and, hence, decreases the temperature. Furthermore, on enlarging the stretching parameter, the velocity and temperature increases and decreases, respectively. In addition, that the radiation parameter can effectively control the thermal boundary layer. Finally, the temperature decreases when the values of the temperature parameter increases. We apply similarity transformation in order to transform the governing model into a system of ODEs (ordinary differential equations). Numerical solutions for particular values of involved parameters are in very good agreement with previous calculations. The most important and interesting result of this paper is that for both the cases of shrinking and stretching sheet flows exhibit dual solutions in some intervals of the shrinking and stretching parameter. In spite of numerous published papers on the flow and heat transfer over a permeable stretching/shrinking surface in nanofluids and hybrid nanofluids, none of the researchers studied the present problem. Therefore, we believe that the results of the present paper are new, and have many industrial applications. Full article

This study comprehensively explores the generalized form of two-dimensional peristaltic motions of incompressible fluid through temperature-dependent physical properties in a non-symmetric channel. Generation of entropy in the system, carrying Joule heat and Lorentz force is also examined. Viscous dissipation is not ignored, for viewing in-depth, effects of heat transmission and entropy production. The modeling of equations is tracked first in fixed and then in wave frame. The resultant set of coupled non-linear equations are solved numerically by utilizing NDSolve in Mathematica. Comparison between NDSolve and the numerical results obtained through bvp4c MATLAB is made for the validation of our numerical codes. The attained results are found to be in excellent agreement. The impact of control parameters on the velocity profiles, pressure gradient, heat transfer, streamlines and entropy production are studied and discussed graphically. It is witnessed that entropy production and heat transfer are increased significantly subject to the enhancement of Hartman number, Brinkman number and electrical conductivity parameter. Hence, choosing appropriate values of physical parameters, performance and efficiency of flow structure and system can be improved. The results reported provide a virtuous insight into bio energy systems providing a useful standard for experimental and extra progressive computational multiphysics simulations. Full article