Mathematics

Research

31 pages, 11738 KiB

Open AccessArticle

Computational Evaluation of Heat and Mass Transfer in Cylindrical Flow of Unsteady Fractional Maxwell Fluid Using Backpropagation Neural Networks and LMS

by Waqar Ul Hassan, Khurram Shabbir, Muhammad Imran Khan and Liliana Guran

Mathematics 2024, 12(23), 3654; https://doi.org/10.3390/math12233654 - 21 Nov 2024

Viewed by 297

Abstract

Fractional calculus plays a pivotal role in modern scientific and engineering disciplines, providing more accurate solutions for complex fluid dynamics phenomena due to its non-locality and inherent memory characteristics. In this study, Caputo’s time fractional derivative operator approach is employed for heat and [...] Read more.

Fractional calculus plays a pivotal role in modern scientific and engineering disciplines, providing more accurate solutions for complex fluid dynamics phenomena due to its non-locality and inherent memory characteristics. In this study, Caputo’s time fractional derivative operator approach is employed for heat and mass transfer modeling in unsteady Maxwell fluid within a cylinder. Governing equations within a cylinder involve a system of coupled, nonlinear fractional partial differential equations (PDEs). A machine learning technique based on the Levenberg–Marquardt scheme with a backpropagation neural network (LMS-BPNN) is employed to evaluate the predicted solution of governing flow equations up to the required level of accuracy. The numerical data sheet is obtained using series solution approach Homotopy perturbation methods. The data sheet is divided into three portions i.e.,

80 %

is used for training,

10 %

for validation, and

10 %

for testing. The mean-squared error (MSE), error histograms, correlation coefficient (R), and function fitting are computed to examine the effectiveness and consistency of the proposed machine learning technique i.e., LMS-BPNN. Moreover, additional error metrics, such as R-squared, residual plots, and confidence intervals, are incorporated to provide a more comprehensive evaluation of model accuracy. The comparison of predicted solutions with LMS-BPNN and an approximate series solution are compared and the goodness of fit is found. The momentum boundary layer became higher and higher as there was an enhancement in the value of Caputo, fractional order α = 0.5 to α = 0.9. Higher thermal boundary layer (TBL) profiles were observed with the rising value of the heat source. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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24 pages, 7309 KiB

Open AccessArticle

Advanced Computational Framework to Analyze the Stability of Non-Newtonian Fluid Flow through a Wedge with Non-Linear Thermal Radiation and Chemical Reactions

by Muhammad Imran Khan, Ahmad Zeeshan, Rahmat Ellahi and Muhammad Mubashir Bhatti

Mathematics 2024, 12(10), 1420; https://doi.org/10.3390/math12101420 - 7 May 2024

Cited by 1 | Viewed by 1079

Abstract

The main idea of this investigation is to introduce an integrated intelligence approach that investigates the chemically reacting flow of non-Newtonian fluid with a backpropagation neural network (LMS-BPNN). The AI-based LMS-BPNN approach is utilized to obtain the optimal solution of an MHD flow [...] Read more.

The main idea of this investigation is to introduce an integrated intelligence approach that investigates the chemically reacting flow of non-Newtonian fluid with a backpropagation neural network (LMS-BPNN). The AI-based LMS-BPNN approach is utilized to obtain the optimal solution of an MHD flow of Eyring–Powell over a porous shrinking wedge with a heat source and nonlinear thermal radiation (

R d

). The partial differential equations (PDEs) that define flow problems are transformed into a system of ordinary differential equations (ODEs) through efficient similarity variables. The reference solution is obtained with the bvp4c function by changing parameters as displayed in Scenarios 1–7. The label data are divided into three portions, i.e., 80% for training, 10% for testing, and 10% for validation. The label data are used to obtain the approximate solution using the activation function in LMS-BPNN within the MATLAB built-in command ‘nftool’. The consistency and uniformity of LMS-BPNN are supported by fitness curves based on the MSE, correlation index (

R

), regression analysis, and function fit. The best validation performance of LMS-BPNN is obtained at 462, 369, 642, 542, 215, 209, and 286 epochs with MSE values of 8.67 × 10⁻¹⁰, 1.64 × 10⁻⁹, 1.03 × 10⁻⁹, 302 9.35 × 10⁻¹⁰, 8.56 × 10⁻¹⁰, 1.08 × 10⁻⁹, and 6.97 × 10⁻¹⁰, respectively. It is noted that

f^{'} (η)

,

θ (η),

and

ϕ (η)

satisfy the boundary conditions asymptotically for Scenarios 1–7 with LMS-BPNN. The dual solutions for flow performance outcomes (

{C f}_{x}

,

{N u}_{x}

, and

S h_{x}

) are investigated with LMS-BPNN. It is concluded that when the magnetohydrodynamics increase (

M = 0.01, 0.05, 0.1

), then the solution bifurcates at different critical values, i.e.,

λ_{c} = - 1.06329, - 1.097, - 1.17694

. The stability analysis is conducted using an LMS-BPNN approximation, involving the computation of eigenvalues for the flow problem. The deduction drawn is that the upper (first) branch solution remains stable, while the lower branch solution causes a disturbance in the flow and leads to instability. It is observed that the boundary layer thickness for the lower branch (second) solution is greater than the first solution. A comparison of numerical results and predicted solutions with LMS-BPNN is provided and they are found to be in good agreement. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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25 pages, 19783 KiB

Open AccessArticle

Ternary Hybrid Nanofluid Flow Containing Gyrotactic Microorganisms over Three Different Geometries with Cattaneo–Christov Model

by Moh Yaseen, Sawan Kumar Rawat, Nehad Ali Shah, Manoj Kumar and Sayed M. Eldin

Mathematics 2023, 11(5), 1237; https://doi.org/10.3390/math11051237 - 3 Mar 2023

Cited by 67 | Viewed by 3167

Abstract

The movement of microorganism cells in fluid influences various biotic processes, including septicity and marine life ecology. Many organic and medicinal applications need to look into the insight of mechanism in nanofluids containing a microbial suspension. The current paper concerns the bioconvection of [...] Read more.

The movement of microorganism cells in fluid influences various biotic processes, including septicity and marine life ecology. Many organic and medicinal applications need to look into the insight of mechanism in nanofluids containing a microbial suspension. The current paper concerns the bioconvection of a ternary hybrid nanofluid (Al₂O₃-Cu-CNT/water) flow containing motile gyrotactic microorganisms toward three different geometries (a flat plate, a wedge, and a cone) in the occurrence of natural convection, radiation, and heat source/sink. The Cattaneo–Christov theory is employed to develop the model. The equations are solved by using the “bvp4c function in MATLAB”. The influence of the crucial significant factors on the motile microorganisms’ density, velocity, temperature, nanoparticles’ concentration, microbe density gradient, and transmission rates of heat and mass is discussed. The results depict that the heat transmission rate is highest for the flow toward the cone, whereas the mass transmission rate and microbe density gradient are highest for the flow toward the wedge. In addition, the higher estimates of the thermal relaxation parameter corresponding to the Cattaneo–Christov theory act to enhance the rate of heat transmission. The results of the current study will be useful to many microbial-enhanced oil recovery systems, carriage processes, architectural design systems, medicinal fields that utilize nanofluids, and so on. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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13 pages, 901 KiB

Open AccessArticle

Significance of Darcy–Forchheimer Law, Activation Energy, and Brownian Motion of Tiny Particles on the Dynamics of Rotating MHD Micropolar Nanofluid

by Meznah M. Alanazi, Awatif A. Hendi, Bagh Ali, Sonia Majeed, Ahmed Kadhim Hussein and Nehad Ali Shah

Mathematics 2023, 11(4), 866; https://doi.org/10.3390/math11040866 - 8 Feb 2023

Cited by 4 | Viewed by 1770

Abstract

The time-independent performance of a micropolar nanofluid under the influence of magneto hydrodynamics and the existence of a porous medium on a stretching sheet has been investigated. Nano-sized particles were incorporated in the base fluid because of their properties such as their extraordinary [...] Read more.

The time-independent performance of a micropolar nanofluid under the influence of magneto hydrodynamics and the existence of a porous medium on a stretching sheet has been investigated. Nano-sized particles were incorporated in the base fluid because of their properties such as their extraordinary heat-enhancing ability, which plays a very important role in modern nanotechnology, cooling electronic devices, various types of heat exchangers, etc. The effects of Brownian motion and thermophoresis are accounted for in this comprehensive study. Using similarity conversion, the leading equations based on conservation principles are non-dimensionalized with various parameters yielding a set of ODEs. The numerical approach boundary value problem fourth-order method (bvp4c) was implemented as listed in the MATLAB computational tool. The purpose of this examination was to study and analyze the influence of different parameters on velocity, micro-rotation, concentration, and temperature profiles. The primary and secondary velocities reduced against the higher inputs of boundary concentration, rotation, porosity, and magnetic parameters, however, the base fluid temperature distribution grows with the increasing values of these parameters. The micro-rotation distribution increased against the rising strength of the Lorentz force and a decline is reported against the growing values of the micropolar material and rotational parameters. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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27 pages, 4662 KiB

Open AccessArticle

Levenberg–Marquardt Training Technique Analysis of Thermally Radiative and Chemically Reactive Stagnation Point Flow of Non-Newtonian Fluid with Temperature Dependent Thermal Conductivity

by Khalil Ur Rehman, Wasfi Shatanawi and Andaç Batur Çolak

Mathematics 2023, 11(3), 753; https://doi.org/10.3390/math11030753 - 2 Feb 2023

Cited by 6 | Viewed by 1332

Abstract

We have examined the magnetized stagnation point flow of non-Newtonian fluid towards an inclined cylindrical surface. The mixed convection, thermal radiation, viscous dissipation, heat generation, first-order chemical reaction, and temperature-dependent thermal conductivity are the physical effects being carried for better novelty. Mathematical equations [...] Read more.

We have examined the magnetized stagnation point flow of non-Newtonian fluid towards an inclined cylindrical surface. The mixed convection, thermal radiation, viscous dissipation, heat generation, first-order chemical reaction, and temperature-dependent thermal conductivity are the physical effects being carried for better novelty. Mathematical equations are constructed for four different flow regimes. The shooting method is used to evaluate the heat transfer coefficient at the cylindrical surface with and without heat generation/thermal radiation effects. For better examination, we have constructed artificial neural networking models with the aid of the Levenberg–Marquardt training technique and Purelin and Tan-Sig transfer functions. The Nusselt number strength is greater for fluctuations in the Casson fluid parameter, Prandtl number, heat generation, curvature, and Eckert number when thermal radiations are present. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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14 pages, 1956 KiB

Open AccessFeature PaperArticle

by Muhammad Idrees Afridi, Zhi-Min Chen, Theodoros E. Karakasidis and Muhammad Qasim

Mathematics 2022, 10(21), 4159; https://doi.org/10.3390/math10214159 - 7 Nov 2022

Cited by 7 | Viewed by 1882

Abstract

The present study aims to examine the effects of uniform lateral mass flux on the boundary layer flow induced by a non-linearly stretching surface. For uniform mass flux, the boundary layer flow does not conform to a similarity solution. The problem may be [...] Read more.

The present study aims to examine the effects of uniform lateral mass flux on the boundary layer flow induced by a non-linearly stretching surface. For uniform mass flux, the boundary layer flow does not conform to a similarity solution. The problem may be resolved by the similarity solution only when the transverse velocity at the boundary of the porous stretching surface is of the form

v_{w} \sim x^{\frac{p - 1}{2}} .

In other words, the flow becomes non-similar; to date, this has not been reported in the literature. That is why, in the current study, the local-similarity approximation up to the third level of truncation is utilized to solve the problem. The pseudo-similarity variable, stream function and transformed streamwise coordinate are defined such that the continuity equation is identically satisfied, and the momentum equation reduces to a non-similar dimensionless boundary layer equation. We derived the non-similar equations of the first, second and third levels of truncations and compared the numerical results obtained from different levels of truncations. In order to find numerical solutions to these equations, the built-in MATLAB routine, known as bvp4c, is used. Further, all non-similar terms that appear in the momentum equations are retained without any approximations. The approximations are introduced only in the subsidiary equations and relative boundary conditions. For the case of suction, the rate of increase in the numerical values of skin friction coefficient obtained from the first level of truncation with increasing velocity index parameter is found to be underestimated, while overestimation is found in the case of injection. The numerical results that were obtained from the third level of truncations are plotted against the embedding physical parameters and are then discussed. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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10 pages, 966 KiB

Open AccessArticle

A Correct Benchmark Problem of a Two-Dimensional Droplet Deformation in Simple Shear Flow

by Junxiang Yang, Yibao Li and Junseok Kim

Mathematics 2022, 10(21), 4092; https://doi.org/10.3390/math10214092 - 2 Nov 2022

Cited by 5 | Viewed by 2103

Abstract

In this article, we numerically investigate a two-dimensional (2D) droplet deformation and breakup in simple shear flow using a phase-field model for two-phase fluid flows. The dominant driving force for a droplet breakup in simple shear flow is the three-dimensional (3D) phenomenon via [...] Read more.

In this article, we numerically investigate a two-dimensional (2D) droplet deformation and breakup in simple shear flow using a phase-field model for two-phase fluid flows. The dominant driving force for a droplet breakup in simple shear flow is the three-dimensional (3D) phenomenon via surface tension force and Rayleigh instability, where a liquid cylinder of certain wavelengths is unstable against surface perturbation and breaks up into individual droplets to reduce the total surface energy. A 2D droplet breakup does not occur except in special cases because there is only one curvature direction of the droplet interface, which resists breakup. However, there have been many numerical simulation research works on the 2D droplet breakups in simple shear flow. This study demonstrates that the 2D droplet breakup phenomenon in simple shear flow is due to the lack of space resolution of the numerical grid. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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19 pages, 1431 KiB

Open AccessArticle

Thermal and Phase Change Process in a Locally Curved Open Channel Equipped with PCM-PB and Heater during Nanofluid Convection under Magnetic Field

by Walid Aich, Fatih Selimefendigil, Talal Alqahtani, Salem Algarni, Sultan Alshehery and Lioua Kolsi

Mathematics 2022, 10(21), 4070; https://doi.org/10.3390/math10214070 - 1 Nov 2022

Cited by 7 | Viewed by 1679

Abstract

Thermal performance and phase-change dynamics in a channel having a cavity equipped with a heater and phase-change material (PCM)-packed bed (PB) region are analyzed during nanoliquid convection under an inclined magnetic field. Curvature of the upper wall above the PCM zone is also [...] Read more.

Thermal performance and phase-change dynamics in a channel having a cavity equipped with a heater and phase-change material (PCM)-packed bed (PB) region are analyzed during nanoliquid convection under an inclined magnetic field. Curvature of the upper wall above the PCM zone is also considered by using the finite element method. Impacts of curvature of the upper wall (between 0.01H and 0.6H, H-channel height), strength of magnetic field (MGF) (Hartmann number between 0 and 40), height (between 0.1H and 0.4H) and number (between 5 and 17) of heaters on the thermal performance and phase-change dynamics are studied. In the interior and wall near regions of the PCM-PB, the curvature effects become opposite, while phase completion time (tF) rises by about 42% at the highest radius of the curvature. Imposing MGF and increasing its strength has positive impacts on the phase change and thermal performance. There is a reduction in tF by about

45.2 %

and

41.8 %

when MGF is imposed at Ha = 40 for pure fluids and nanofluids. When thermal performance for all different cases is compared, using MGF+nanofluid+PCM provides the most favorable case. When the reference case (only pure fluid without MGF and PCM) is used, including nanoparticles results in an improvement of 33.7%m while it is further increased to 71.1% when PCM-PB is also installed. The most favorable case by using MGF, nanofluid and PCM-PB results in thermal performance improvement of about 373.9% as compared to the reference configuration. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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22 pages, 13650 KiB

Open AccessArticle

Investigation of Efficient Optimization Approach to the Modernization of Francis Turbine Draft Tube Geometry

by Ivana Lučin, Ante Sikirica, Marija Šiško Kuliš and Zoran Čarija

Mathematics 2022, 10(21), 4050; https://doi.org/10.3390/math10214050 - 1 Nov 2022

Cited by 8 | Viewed by 1915

Abstract

The installation of new and revitalization of existing hydropower requires the optimal design of hydraulic turbine parts, which were previously mostly based on simplified analytical methods and engineering practices. With the increase in computational power, optimization methods now present an effective approach to [...] Read more.

The installation of new and revitalization of existing hydropower requires the optimal design of hydraulic turbine parts, which were previously mostly based on simplified analytical methods and engineering practices. With the increase in computational power, optimization methods now present an effective approach to the improvement of the geometry of turbine parts, thus enabling fine geometry-tuning adjustments of site specifics. However, despite continually increasing computational power, such numerical optimizations still require substantial resources and time. Therefore, in the proposed paper, extensive analysis of different numerical simplifications was conducted to determine the best trade-off between accuracy and the required computational resources for draft tube optimization. Additionally, the influence of geometry parametrization in terms of greater geometry flexibility, different goal functions, and optimization parameters was investigated. Within the considered constraints, a novel draft tube geometry was obtained, with the elbow displacements toward the draft tube exit, a shape that could not be obtained with the usual draft tube parametrization. It was also observed that geometry improvements for a single operating point did not provide improvements for other operating points. The obtained findings regarding the numerical and optimization setup can be used as guidelines for future optimization research, where multiobjective optimization for multiple operating points should be further investigated. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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17 pages, 5485 KiB

Open AccessArticle

Analysis of Mixed Convection on Two-Phase Nanofluid Flow Past a Vertical Plate in Brinkman-Extended Darcy Porous Medium with Nield Conditions

by Hatem Gasmi, Umair Khan, Aurang Zaib, Anuar Ishak, Sayed M. Eldin and Zehba Raizah

Mathematics 2022, 10(20), 3918; https://doi.org/10.3390/math10203918 - 21 Oct 2022

Cited by 7 | Viewed by 1780

Abstract

The rapid advancement in technology in recent years has shown that nanofluids are very vital to further development in science and technology. Moreover, many industrial specifications cannot be met by allowing natural convection only, hence the need to incorporate forced convection and natural [...] Read more.

The rapid advancement in technology in recent years has shown that nanofluids are very vital to further development in science and technology. Moreover, many industrial specifications cannot be met by allowing natural convection only, hence the need to incorporate forced convection and natural convection into a single flow regime. The research aims to quantify the mixed convective two-phase flow past a vertical permeable surface in a Brinkman-Extended Darcy porous medium (BEDPM) induced by nanofluid, with heat and mass transfer. In addition, the Nield condition is also incorporated. The model of the problem was initially constructed in the vital form of leading governing equations (LGEs). These LGEs are specifically called partial differential equations (PDEs) (because of two or more independent variables) which were later converted into a set of the single independent variable of ordinary differential equations (ODEs) by implementing the similarity transformations. The set of single independent ODEs was numerically solved via the boundary value problem of fourth-order (bvp4c) technique. The bvp4c is one of the most frequently recommended built-in MATLAB subroutines based on the three-stage Labatto formula. The impact of several physically embedded influential parameters on the fluid flow, along with mass and thermal properties of the nanofluid in a Brinkman-Extended Darcy porous medium for the cases of buoyancy assisting flow (BAF) and buoyancy opposing flow (BOF), were investigated and argued. The numerical outcomes clarify that the porosity parameter reduces the velocity, whereas the concentration and the temperature enhance in the case of the buoyancy assisting and buoyancy opposing flows. In addition, the wall drag force elevates for the larger value of the dimensionless permeability parameter

K_{1}

and the buoyancy ratio parameter

N

, while it declines for the modified porosity parameter

ε_{1}

. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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10 pages, 1077 KiB

Open AccessArticle

Steady-State Solutions for Two Mixed Initial-Boundary Value Problems Which Describe Isothermal Motions of Burgers’ Fluids: Application

by Constantin Fetecau, N. Ameer Ahammad, Dumitru Vieru and Nehad Ali Shah

Mathematics 2022, 10(19), 3681; https://doi.org/10.3390/math10193681 - 8 Oct 2022

Cited by 3 | Viewed by 1241

Abstract

Steady-state solutions of two mixed initial-boundary value problems are presented in equivalent forms. They describe isothermal permanent motions of incompressible Burgers’ fluids over an infinite flat plate that applies time-dependent shear stresses to the fluid. More exactly, they are the first exact solutions [...] Read more.

Steady-state solutions of two mixed initial-boundary value problems are presented in equivalent forms. They describe isothermal permanent motions of incompressible Burgers’ fluids over an infinite flat plate that applies time-dependent shear stresses to the fluid. More exactly, they are the first exact solutions for motions of Burgers’ fluids with differential expressions of the shear stress or velocity on the boundary. The obtained results are designed to make equivalent solutions for motions caused by an infinite plate moving in its plane at velocities that seem to be similar to previous shear stresses. It is simple to limit all results for the purpose of providing efficient results for incompressible Oldroyd-B, Maxwell, second grade and Newtonian fluids undergoing comparable motions. They may also be used to estimate how long it will take to get to a steady or permanent state. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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18 pages, 5471 KiB

Open AccessArticle

Comparative Analysis of a Cone, Wedge, and Plate Packed with Microbes in Non-Fourier Heat Flux

by Halavudara Basavarajappa Santhosh, Mamatha Sadananda Upadhya, N. Ameer Ahammad, Chakravarthula Siva Krishnam Raju, Nehad Ali Shah and Wajaree Weera

Mathematics 2022, 10(19), 3508; https://doi.org/10.3390/math10193508 - 26 Sep 2022

Cited by 7 | Viewed by 1603

Abstract

In this study, we investigated a radiative chemically reactive Casson fluid above a cone, plate, and wedge with gyrotactic microorganisms subjected to the Cattaneo–Christov heat flux model. Newton’s method and the Runge–Kutta methods were employed to solve the physical problem, and a graphical [...] Read more.

In this study, we investigated a radiative chemically reactive Casson fluid above a cone, plate, and wedge with gyrotactic microorganisms subjected to the Cattaneo–Christov heat flux model. Newton’s method and the Runge–Kutta methods were employed to solve the physical problem, and a graphical representation of the numerous impacts of non-dimensional parameters on temperature, velocity, and concentration was created. In addition, we also compared recently published solutions with our current solution, which showed good agreement. From this investigation, we concluded that the motile organisms’ momentum, temperature, and concentration density were non-uniform in nature. Here, for engineering importance, we also present the mass transfer and thermal transfer rate over the cone, wedge, and plate cases in a tabular form. We concluded that the mass and heat transfer rate was larger over the cone when compared to the same case over a plate or wedge. The results also highlighted that the local Nusselt and Sherwood numbers and the mass density of the microorganisms depreciated as the Casson fluid parameter decreased. In summary, we concluded that the gyrotactic microorganisms played a role in enhancing the local Sherwood number. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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19 pages, 10943 KiB

Open AccessArticle

Combined Convective Energy Transmission Performance of Williamson Hybrid Nanofluid over a Cylindrical Shape with Magnetic and Radiation Impressions

by Firas A. Alwawi, Feras M. Al Faqih, Mohammed Z. Swalmeh and Mohd Asrul Hery Ibrahim

Mathematics 2022, 10(17), 3191; https://doi.org/10.3390/math10173191 - 4 Sep 2022

Cited by 10 | Viewed by 1666

Abstract

This analysis focuses on extending and developing some previous studies of energy transport through nanofluids to include the states of combined convection flow of a Williamson hybrid nanofluid that flows around a cylinder. Mathematical models that simulate the behavior of these upgraded nanofluids [...] Read more.

This analysis focuses on extending and developing some previous studies of energy transport through nanofluids to include the states of combined convection flow of a Williamson hybrid nanofluid that flows around a cylinder. Mathematical models that simulate the behavior of these upgraded nanofluids are constructed by expanding the Tiwari and Das model, which are then solved numerically via Keller box approaches. The accuracy of the results is emphasized by comparing them with the previous published outcomes. Nanosolid volume fraction

0 \leq χ \leq 0.1,

combined convection

- 1 \leq λ \leq 5,

radiation factor

0.1 \leq R \leq 6,

Weissenberg number

0.2 \leq W e \leq 0.9,

and magnetic factor

0.1 \leq M \leq 1

are the factors that have been taken into consideration to examine the energy transfer performance of Williamson hybrid nanofluid. Numerical and graphical outcomes are obtained using MATLAB, analyzed, and discussed in depth. According to the outcomes, the Weissenberg number reduces energy transfer and friction forces. Both the combined convective coefficient and the radiation factor improved the rate of energy transfer and increased the velocity of the host fluid. The fluid velocity and rate of energy transfer can be reduced by increasing the magnetic factor. The nanoparticle combination of silver and aluminum oxide (Ag-Al₂O₃) has demonstrated superiority in enhancing the energy transfer rate and velocity of the host fluid. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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15 pages, 2668 KiB

Open AccessArticle

Agglomeration Regimes of Particles under a Linear Laminar Flow: A Numerical Study

by Yunzhou Qian, Shane P. Usher, Peter J. Scales, Anthony D. Stickland and Alessio Alexiadis

Mathematics 2022, 10(11), 1931; https://doi.org/10.3390/math10111931 - 4 Jun 2022

Cited by 2 | Viewed by 2583

Abstract

In this work, a combined smoothed particle hydrodynamics and discrete element method (SPH-DEM) model was proposed to model particle agglomeration in a shear flow. The fluid was modeled with the SPH method and the solid particles with DEM. The system was governed by [...] Read more.

In this work, a combined smoothed particle hydrodynamics and discrete element method (SPH-DEM) model was proposed to model particle agglomeration in a shear flow. The fluid was modeled with the SPH method and the solid particles with DEM. The system was governed by three fundamental dimensionless groups: the Reynolds number Re (1.5~150), which measured the effect of the hydrodynamics; the adhesion number Ad (6 × 10⁻⁵~6 × 10⁻³), which measured the inter-particle attraction; and the solid fraction α, which measured the concentration of particles. Based on these three dimensionless groups, several agglomeration regimes were found. Within these regimes, the aggregates could have different sizes and shapes that went from long thread-like structures to compact spheroids. The effect of the particle–particle interaction model was also investigated. The results were combined into ‘agglomeration maps’ that allowed for a quick determination of the agglomerate type once α, Re, Ad were known. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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13 pages, 687 KiB

Open AccessArticle

Thermal Conductivity and Thermophoretic Impacts of Micropolar Fluid Flow by a Horizontal Absorbent Isothermal Porous Wall with Heat Source/Sink

by Hossam A. Nabwey, Ahmed M. Rashad, Abd El Nasser Mahdy and Shaaban M. Shaaban

Mathematics 2022, 10(9), 1514; https://doi.org/10.3390/math10091514 - 2 May 2022

Cited by 6 | Viewed by 1560

Abstract

Boundary layer analysis is invoked to clarify the aspects of variable thermal conductivity and thermophoretic forces on a steady state of MHD micropolar fluid flow in the existence of a uniform transverse magnetic field along an isothermal horizontal plate. The micropolar pattern permits [...] Read more.

Boundary layer analysis is invoked to clarify the aspects of variable thermal conductivity and thermophoretic forces on a steady state of MHD micropolar fluid flow in the existence of a uniform transverse magnetic field along an isothermal horizontal plate. The micropolar pattern permits the rotational freedom degrees that lead to couple stresses and a non symmetric stress tensor. The initiated PDEs governing the case pattern are mutated into a non-dimensional system due to proper transformations. The transformed mathematical governing equations are solved by implementing a very potent computer algebra software MATLAB code. The plotted graphs analyzed the attitude of multiple physical aspects involving factors on the flow attitude of micropolar velocity and angular velocity and temperature. Through the involved factors, the couple stress, skin friction and Nusselt number are manifested and interpreted amply. A new outcome for drag force and heat gradient experienced by the key factors is portrayed. Augmentation in Ω results in the thermophoretic forces that encapsulate the mass transmission. The local Nusselt number strengthened as the thermal conductivity, heat absorption factors or wall suction velocity were improved, and weakened due to the existence of viscous dissipation or heat generation impacts. As a particular case, the governing field equations of a classical Newtonian liquid are given by dropping the micropolar parameter impacts. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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20 pages, 2179 KiB

Open AccessArticle

Performance Optimization of a Thermoelectric Device by Using a Shear Thinning Nanofluid and Rotating Cylinder in a Cavity with Ventilation Ports

by Nidhal Ben Khedher, Fatih Selimefendigil, Lioua Kolsi, Walid Aich, Lotfi Ben Said and Ismail Boukholda

Mathematics 2022, 10(7), 1075; https://doi.org/10.3390/math10071075 - 27 Mar 2022

Cited by 9 | Viewed by 1822

Abstract

The combined effects of using a rotating cylinder and shear thinning nanofluid on the performance improvements of a thermoelectric generator (TEG)-installed cavity with multiple ventilation ports are numerically assessed. An optimization algorithm is used to find the best location, rotational speed and size [...] Read more.

The combined effects of using a rotating cylinder and shear thinning nanofluid on the performance improvements of a thermoelectric generator (TEG)-installed cavity with multiple ventilation ports are numerically assessed. An optimization algorithm is used to find the best location, rotational speed and size of the cylinder to deliver the highest power generation of the TEG. The power generation features with varying Rew are different for the first nanofluid (NF1) when compared to the second one (NF2). The power rises with higher Rew when NF1 is used, and up to 49% enhancement is obtained. The output power variation between nanofluids NF1 and NF2 is the highest at Rew = 0, which is obtained as 68.5%. When the cylinder location is varied, the change in the output power becomes 61% when NF2 is used. The optimum case has 11.5%- and 161%-higher generated power when compared with the no-object case with NF1 and NF2. The computational effort of using the high-fidelity coupled system is reduced when optimization is considered. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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15 pages, 596 KiB

Open AccessArticle

Electroviscous Effect of Water-Base Nanofluid Flow between Two Parallel Disks with Suction/Injection Effect

by Muhammad Sohail Khan, Sun Mei, Shabnam, Unai Fernandez-Gamiz, Samad Noeiaghdam, Aamir Khan and Said Anwar Shah

Mathematics 2022, 10(6), 956; https://doi.org/10.3390/math10060956 - 17 Mar 2022

Cited by 20 | Viewed by 2026

Abstract

This article, investigates the behaviour of an ionized nanoliquid motion regarding heat transmission between two parallel discs. In the proposed model, the squeezing flow of Cu-water nanofluid with electrical potential force is analysed for studying the flow properties and an uniform magnetic field [...] Read more.

This article, investigates the behaviour of an ionized nanoliquid motion regarding heat transmission between two parallel discs. In the proposed model, the squeezing flow of Cu-water nanofluid with electrical potential force is analysed for studying the flow properties and an uniform magnetic field is applied to that fluid, by taking the surface of the bottom disc porous. We have also studied the effects of different nanomaterials on the transmission of heat through nanofluids. Furthermore, the influence of various physical parameters in the proposed model of nanofluids flow like volume fraction of nanomaterials, squeezing number, Hartmann number, Eckert number, and Prandtl number are analysed and discussed quantitatively through various tables and graphs. The system of nonlinear partial differential equations (PDE’s) has been used to formulate the proposed flow model and later converted to a set of nonlinear ODE’s by mean similarity transformation. Further, the reduced form of ODEs has been solved by Parametric Continuation Method (PCM), which is a stable numerical scheme. The outcomes obtained from the proposed model could also be used to analyse nanofluid flow in several fields, such as polymer processing, power transfer and hydraulic lifts. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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20 pages, 1880 KiB

Open AccessArticle

Multiple Impinging Jet Cooling of a Wavy Surface by Using Double Porous Fins under Non-Uniform Magnetic Field

by Lioua Kolsi, Fatih Selimefendigil, Kaouther Ghachem, Talal Alqahtani and Salem Algarni

Mathematics 2022, 10(4), 638; https://doi.org/10.3390/math10040638 - 18 Feb 2022

Cited by 7 | Viewed by 1714

Abstract

Coupled effects of double porous fins and inhomogeneous magnetic field on the cooling performance of multiple nanojet impingement for a corrugated surface were numerically analyzed. Different values of magnetic field parameters (strength, inclination, and amplitude of spatially varying part) and double porous fin [...] Read more.

Coupled effects of double porous fins and inhomogeneous magnetic field on the cooling performance of multiple nanojet impingement for a corrugated surface were numerically analyzed. Different values of magnetic field parameters (strength, inclination, and amplitude of spatially varying part) and double porous fin parameters (inclination and permeability) were used, while finite element method is used as the solution method. When parametric computational fluid dynamics (CFD) simulations were performed, there were 162.5% and 34% Nusselt number (Nu) enhancement with magnetic field for flat and wavy surfaces, respectively. The variations of average Nu became

36 %

and

24 %

when varying the inclination and amplitude of inhomogeneous magnetic for a flat surface, while the amounts were

43.7 %

and

32 %

for a corrugated one. The vortex distribution in between the jets and cooling performance was affected by the variation of double porous fin permeability and inclination. An optimization method was used for the highest cooling performance, while the optimum set of parameters was obtained at (Ha, Amp, Da,

Ω

) = (0.224, 0.5835,

7.59 \times 10^{- 4}

, 0.1617). By using the double porous fins and inhomogeneous magnetic field, excellent control of the cooling performance of multiple impinging jet was obtained. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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23 pages, 10763 KiB

Open AccessArticle

Computational Efficiency Assessment of Adaptive Mesh Refinement for Turbulent Jets in Crossflow

by Ante Sikirica, Luka Grbčić, Marta Alvir and Lado Kranjčević

Mathematics 2022, 10(4), 620; https://doi.org/10.3390/math10040620 - 17 Feb 2022

Cited by 3 | Viewed by 2219

Abstract

Numerical analyses of environmental discharges are commonly conducted on pre-generated numerical grids with refinements implemented in regions of interest or influence on the flow field. This approach to problem formulation relies on insights into the flow specifics so that appropriate attention is given [...] Read more.

Numerical analyses of environmental discharges are commonly conducted on pre-generated numerical grids with refinements implemented in regions of interest or influence on the flow field. This approach to problem formulation relies on insights into the flow specifics so that appropriate attention is given to relevant segments of the domain. In this paper we investigated the applicability of adaptive mesh refinement (AMR) on a commonly considered environmental problem—a jet in crossflow. The assessment was made using the OpenFOAM toolbox. Several RANS turbulence models and grid generation approaches were compared in terms of accuracy to previous studies and experimental results. Main emphasis is given to the computational efficiency of the methodology with a focus on load distribution. Our findings indicate that the results are acceptable in terms of accuracy with load balancing providing significant computational savings thus enabling AMR methodology to outperform the conventional approach. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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14 pages, 3178 KiB

Open AccessArticle

Influence of Bioconvection and Chemical Reaction on Magneto—Carreau Nanofluid Flow through an Inclined Cylinder

by Hossam A. Nabwey, Sumayyah I. Alshber, Ahmed M. Rashad and Abd El Nasser Mahdy

Mathematics 2022, 10(3), 504; https://doi.org/10.3390/math10030504 - 4 Feb 2022

Cited by 31 | Viewed by 2330

Abstract

The present contribution focuses on heat transmission in the conjugate mixed bioconvection flow of Carreau nanofluid with swimming gyrotactic microorganisms through an inclined stretchable cylinder with variable magnetic field impact and binary chemical reaction. Additionally, the investigation involves the aspects of variable decrease [...] Read more.

The present contribution focuses on heat transmission in the conjugate mixed bioconvection flow of Carreau nanofluid with swimming gyrotactic microorganisms through an inclined stretchable cylinder with variable magnetic field impact and binary chemical reaction. Additionally, the investigation involves the aspects of variable decrease or increase in heat source and non-uniform thermal conductivity. A passively controlled nanofluid pattern is used to estimate this nano-bioconvection flow case, which is believed to be more physically accurate than the earlier actively controlled nanofluid typically employed. One of essential features of this investigation is the imposition of a zero-mass flux condition at the surface of the cylinder. Through the implementation of an appropriate transformation, the nonlinear PDE system is mutated into similar equations. The flow equations thus obtained are solved numerically to explore the influence of the physical constraints involved through implementation with the aid of the MATLAB bvp4c code. The solutions were captured for both zero and non-zero bioconvection Rayleigh number, i.e., for flow with and without microorganisms. The present numerical results are compared with the available data and are determined to be in excellent agreement. The significant result of the present article is that the degree of nanoparticle concentration in the nanofluid exhibits an increasing trend with higher values of activation energy constraint. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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18 pages, 8909 KiB

Open AccessArticle

Numerical Study of Natural Convection of Power Law Fluid in a Square Cavity Fitted with a Uniformly Heated T-Fin

by Sardar Bilal, Noor Zeb Khan, Imtiaz Ali Shah, Jan Awrejcewicz, Ali Akgül and Muhammad Bilal Riaz

Mathematics 2022, 10(3), 342; https://doi.org/10.3390/math10030342 - 24 Jan 2022

Cited by 17 | Viewed by 3304

Abstract

Flow of a liquid in an enclosure with heat transfer has drawn special focus of researchers due to the abundant thermal engineering applications. So, the aim of present communication is to explore thermal characteristics of natural convective power-law liquid flow in a square [...] Read more.

Flow of a liquid in an enclosure with heat transfer has drawn special focus of researchers due to the abundant thermal engineering applications. So, the aim of present communication is to explore thermal characteristics of natural convective power-law liquid flow in a square enclosure rooted with a T-shaped fin. The formulation of the problem is executed in the form of partial differential expressions by incorporating the rheological relation of the power-law fluid. The lower wall of the enclosure along with the fin is uniformly heated and vertical walls are prescribed with cold temperature. For effective heat transfer within the cavity the upper boundary is considered thermally insulated. A finite element based commercial software known as COMSOL is used for simulations and discretization of differential equations and is executed incorporating a weak formulation. Domain discretization is performed by dividing it into triangular and rectangular elements at different refinement levels. A grid independence test is accomplished for quantities of engineering interest like local and average Nusselt numbers to attain accuracy and validity in results. Variation in the momentum and thermal distributions against pertinent parameters is analyzed through stream lines and isothermal contour plots. Measurement of the heat flux coefficient along with the calculation of kinetic energy against involved parameters is displayed through graphs and tables. After the comprehensive overview of attained results it is deduced that kinetic energy elevates against the upsurging magnitude of the Rayleigh number, whereas contrary behavior is encapsulated versus power-law index

(n)

. Elevation in the Nusselt number for the shear thinning case

(i . e ., n = 0.5)

adheres as compared to Newtonian

(i . e ., n = 1)

and shear thickening cases

(i . e ., n = 1.5)

. It is perceived that by the upsurging power-law index viscosity augmentations and circulation zones increases. Heat is transferred quickly against Rayleigh number (Ra) due to production of temperature difference in flow domain. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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23 pages, 12488 KiB

Open AccessArticle

Effect of Heat Source Position in Fluid Flow, Heat Transfer and Entropy Generation in a Naturally Ventilated Room

by Mohammed A. Alghaseb, Walid Hassen, Abdelhakim Mesloub and Lioua Kolsi

Mathematics 2022, 10(2), 178; https://doi.org/10.3390/math10020178 - 7 Jan 2022

Cited by 3 | Viewed by 1465

Abstract

In this study, a 3D numerical study of free ventilated room equipped with a discrete heat source was performed using the Finite Volume Method (FVM). To ensure good ventilation, two parallel openings were created in the room. A suction opening was located at [...] Read more.

In this study, a 3D numerical study of free ventilated room equipped with a discrete heat source was performed using the Finite Volume Method (FVM). To ensure good ventilation, two parallel openings were created in the room. A suction opening was located at the bottom of the left wall and another opening was located at the top of the opposite wall; the heat source was placed at various positions in order to compare the heating efficiency. The effects of Rayleigh number (10³ ≤ Ra ≤ 10⁶) for six heater positions was studied. The results focus on the impact of these parameters on the particle trajectories, temperature fields and on the heat transfer inside the room. It was found that the position of the heater has a dramatic effect on the behavior and topography of the flow in the room. When the heat source was placed on the wall with the suction opening, two antagonistic behaviors were recorded: an improvement in heat transfer of about 31.6%, compared to the other positions, and a low Rayleigh number against 22% attenuation for high Ra values was noted. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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12 pages, 2189 KiB

Open AccessArticle

Slip Microrotation Flow of Silver-Sodium Alginate Nanofluid via Mixed Convection in a Porous Medium

by Hossam A. Nabwey, Ahmed M. Rashad and Waqar A. Khan

Mathematics 2021, 9(24), 3232; https://doi.org/10.3390/math9243232 - 14 Dec 2021

Cited by 5 | Viewed by 2233

Abstract

In the previous decennium, considerable applications ofnanoparticles have been developed in the area of science. Nanoparticles with micropolar fluid suspended in conventional fluids can increase the heat transfer. Micropolar fluids have attracted much research attention because of their use in industrial processes. Exotic [...] Read more.

In the previous decennium, considerable applications ofnanoparticles have been developed in the area of science. Nanoparticles with micropolar fluid suspended in conventional fluids can increase the heat transfer. Micropolar fluids have attracted much research attention because of their use in industrial processes. Exotic lubricants, liquid crystal solidification, cooling of a metallic plate in a bath, extrusion of metals and polymers, drawing of plastic films, manufacturing of glass and paper sheets, and colloidal suspension solutions are just a few examples. The primary goal of this studywas to see how radiation and velocity slip affect the mixed convection of sodium alginate nanofluid flow over a non-isothermal wedge in a saturated porous media.In this communication, theTiwari and Das model was employed to investigate the micropolarnanofluid flow via mixed convection over aradiated wedge in a saturated porous medium with the velocity slip condition. Nanoparticles of silver (Ag) wreused in asodium alginate base fluid. The intended system of governing equations is converted to a set of ordinary differential equations and then solved applying the finite difference method. Variousfluid flows, temperatures, and physical quantities of interest were examined. The effects of radiation on the skin friction are negligible in the case of forced and mixed convection, whereas radiation increases the skin friction in free convection. It is demonstrated that the pressure gradient, solid volume fraction, radiation, and slip parameters enhance the Nusselt number, whereas the micropolar parameter reduces the Nusselt number. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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18 pages, 1192 KiB

Open AccessArticle

Combined Effects of Sequential Velocity and Variable Magnetic Field on the Phase Change Process in a 3D Cylinder Having a Conic-Shaped PCM-Packed Bed System

by Lioua Kolsi, Fatih Selimefendigil, Mohamed Omri and Lotfi Ladhar

Mathematics 2021, 9(23), 3019; https://doi.org/10.3390/math9233019 - 25 Nov 2021

Cited by 3 | Viewed by 1601

Abstract

Effects of sequential velocity and variable magnetic field on the phase change during hybrid nanofluid convection through a 3D cylinder containing a phase-change material packed bed (PCM-PB) system is analyzed with the finite element method. As the heat transfer fluid, 40% ethylene glycol [...] Read more.

Effects of sequential velocity and variable magnetic field on the phase change during hybrid nanofluid convection through a 3D cylinder containing a phase-change material packed bed (PCM-PB) system is analyzed with the finite element method. As the heat transfer fluid, 40% ethylene glycol with hybrid TiO

_{2}

-Al

_{2}

O

_{3}

nanoparticles is considered. Impacts of the sequential velocity parameter (K, between 0.5 and 1.5), geometric factor of the conic-shaped PCM-PB (M, between 0.2 and 0.9), magnetic field strength (Ha number between 0 and 50) and solid volume fraction of hybrid nanoparticles (vol.% between

0.02 %

and

0.1 %

) on the phase change dynamics are explored. Effects of both constant and varying magnetic fields on the phase change process were considered. Due to the increased fluid velocity at the walls, the phase change becomes higher with higher values of the sequential velocity parameter (K). There is a 21.6% reduction in phase transition time (tF) between the smallest and highest values of K both in the absence and presence of a constant magnetic field. The value of tF is reduced with higher magnetic field strength and the amount of reduction depends upon the sequential velocity parameter. At K = 1.5, the reduction amount with the highest Ha number is 14.7%, while it is 26% at K = 0.5. When nanoparticle is loaded in the base fluid, the value of tF is further reduced. In the absence of a magnetic field, the amount of phase-transition time reduction is 6.9%, while at Ha = 50, it is 11.7%. The phase change process can be controlled with varying magnetic field parameters as well. As the wave number and amplitude of the varying magnetic field are considered, significant changes in the tF are observed. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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17 pages, 3804 KiB

Open AccessArticle

Duality Solutions in Hydromagnetic Flow of SWCNT-MWCNT/Water Hybrid Nanofluid over Vertical Moving Slender Needle

by Nur Adilah Liyana Aladdin and Norfifah Bachok

Mathematics 2021, 9(22), 2927; https://doi.org/10.3390/math9222927 - 17 Nov 2021

Cited by 10 | Viewed by 2057

Abstract

Recently, the topic of convection of heat transfer has created an interest among researchers because of its numerous applications in the daily life. The objective of this paper was to study theoretically the problem of mixed convection boundary layer flow and heat transfer [...] Read more.

Recently, the topic of convection of heat transfer has created an interest among researchers because of its numerous applications in the daily life. The objective of this paper was to study theoretically the problem of mixed convection boundary layer flow and heat transfer of single-wall carbon nanotube (SWCNT) and multi-wall carbon nanotube (MWCNT) in presence of hydromagnetic effects. The problem was initiated by formulating a mathematical model in partial differential equation (PDE) for the hybrid nanofluid flow with appropriate boundary conditions. The similarity equation was used to transform the PDE into an ordinary differential equation (ODE) and solved using bvp4c in MATLAB. The graphical results on variation of skin friction coefficient,

C_{f}

, local Nusselt number,

N u_{x}

, shear stress,

f^{″} (c)

and local heat flux,

- θ^{'} (c)

with the effects of magnetic,

M,

size of needle, c, mixed convection parameter,

λ

and volume fraction of nanoparticles,

φ

were presented and discussed in detail. The study revealed that duality of solutions appears when the buoyance force is in opposing flow of the fluid motion,

λ < 0

. The presence of M in hybrid nanofluid reduced the skin friction coefficient and heat transfer. On the other hand, the

C_{f}

and

N u_{x}

increased as different concentrations of

φ_{1}

and c were added. It gives an insight into the medical field, especially in treating cancer cells. By means, it reveals that CNTs hybrid nanofluid shows high potential in reaching the site of tumors faster compared with nanofluid. A stability analysis has to be carried out. It is noticed that the first solution was stable and physically realizable. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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20 pages, 7283 KiB

Open AccessArticle

Darcy–Brinkman–Forchheimer Model for Nano-Bioconvection Stratified MHD Flow through an Elastic Surface: A Successive Relaxation Approach

by Anwar Shahid, Mohamed S. Mohamed, Muhammad Mubashir Bhatti and Mohammad Hossein Doranehgard

Mathematics 2021, 9(19), 2514; https://doi.org/10.3390/math9192514 - 7 Oct 2021

Cited by 7 | Viewed by 2357

Abstract

The present study deals with the Darcy–Brinkman–Forchheimer model for bioconvection-stratified nanofluid flow through a porous elastic surface. The mathematical modeling for MHD nanofluid flow with motile gyrotactic microorganisms is formulated under the influence of an inclined magnetic field, Brownian motion, thermophoresis, viscous dissipation, [...] Read more.

The present study deals with the Darcy–Brinkman–Forchheimer model for bioconvection-stratified nanofluid flow through a porous elastic surface. The mathematical modeling for MHD nanofluid flow with motile gyrotactic microorganisms is formulated under the influence of an inclined magnetic field, Brownian motion, thermophoresis, viscous dissipation, Joule heating, and stratifi-cation. In addition, the momentum equation is formulated using the Darcy–Brinkman–Forchheimer model. Using similarity transforms, governing partial differential equations are reconstructed into ordinary differential equations. The spectral relaxation method (SRM) is used to solve the nonlinear coupled differential equations. The SRM is a straightforward technique to develop, because it is based on decoupling the system of equations and then integrating the coupled system using the Chebyshev pseudo-spectral method to obtain the required results. The numerical interpretation of SRM is admirable because it establishes a system of equations that sequentially solve by providing the results of the first equation into the next equation. The numerical results of temperature, velocity, concentration, and motile microorganism density profiles are presented with graphical curves and tables for all the governing parametric quantities. A numerical comparison of the SRM with the previously investigated work is also shown in tables, which demonstrate excellent agreement. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics II)

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