Mathematics

Research

20 pages, 5122 KiB

Open AccessArticle

Implicit Finite Difference Simulation of Prandtl-Eyring Nanofluid over a Flat Plate with Variable Thermal Conductivity: A Tiwari and Das Model

by Nidal H. Abu-Hamdeh, Abdulmalik A. Aljinaidi, Mohamed A. Eltaher, Khalid H. Almitani, Khaled A. Alnefaie, Abdullah M. Abusorrah and Mohammad Reza Safaei

Mathematics 2021, 9(24), 3153; https://doi.org/10.3390/math9243153 - 7 Dec 2021

Cited by 24 | Viewed by 3244

Abstract

The current article presents the entropy formation and heat transfer of the steady Prandtl-Eyring nanofluids (P-ENF). Heat transfer and flow of P-ENF are analyzed when nanofluid is passed to the hot and slippery surface. The study also investigates the effects of radiative heat [...] Read more.

The current article presents the entropy formation and heat transfer of the steady Prandtl-Eyring nanofluids (P-ENF). Heat transfer and flow of P-ENF are analyzed when nanofluid is passed to the hot and slippery surface. The study also investigates the effects of radiative heat flux, variable thermal conductivity, the material’s porosity, and the morphologies of nano-solid particles. Flow equations are defined utilizing partial differential equations (PDEs). Necessary transformations are employed to convert the formulae into ordinary differential equations. The implicit finite difference method (I-FDM) is used to find approximate solutions to ordinary differential equations. Two types of nano-solid particles, aluminium oxide (Al₂O₃) and copper (Cu), are examined using engine oil (EO) as working fluid. Graphical plots are used to depict the crucial outcomes regarding drag force, entropy measurement, temperature, Nusselt number, and flow. According to the study, there is a solid and aggressive increase in the heat transfer rate of P-ENF Cu-EO than Al₂O₃-EO. An increment in the size of nanoparticles resulted in enhancing the entropy of the model. The Prandtl-Eyring parameter and modified radiative flow show the same impact on the radiative field. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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24 pages, 4161 KiB

Open AccessArticle

Bioconvection Reiner-Rivlin Nanofluid Flow between Rotating Circular Plates with Induced Magnetic Effects, Activation Energy and Squeezing Phenomena

by Muhammad Bilal Arain, Muhammad Mubashir Bhatti, Ahmad Zeeshan and Faris Saeed Alzahrani

Mathematics 2021, 9(17), 2139; https://doi.org/10.3390/math9172139 - 2 Sep 2021

Cited by 41 | Viewed by 4050

Abstract

This article deals with the unsteady flow in rotating circular plates located at a finite distance filled with Reiner-Rivlin nanofluid. The Reiner-Rivlin nanofluid is electrically conducting and incompressible. Furthermore, the nanofluid also accommodates motile gyrotactic microorganisms under the effect of activation energy and [...] Read more.

This article deals with the unsteady flow in rotating circular plates located at a finite distance filled with Reiner-Rivlin nanofluid. The Reiner-Rivlin nanofluid is electrically conducting and incompressible. Furthermore, the nanofluid also accommodates motile gyrotactic microorganisms under the effect of activation energy and thermal radiation. The mathematical formulation is performed by employing the transformation variables. The finalized formulated equations are solved using a semi-numerical technique entitled Differential Transformation Method (DTM). Padé approximation is also used with DTM to present the solution of nonlinear coupled ordinary differential equations. Padé approximation helps to improve the accuracy and convergence of the obtained results. The impact of several physical parameters is discussed and gives analysis on velocity (axial and tangential), magnetic, temperature, concentration field, and motile gyrotactic microorganism functions. The impact of torque on the lower and upper plates are deliberated and presented through the tabular method. Furthermore, numerical values of Nusselt number, motile density number, and Sherwood number are given through tabular forms. It is worth mentioning here that the DTM-Padé is found to be a stable and accurate method. From a practical point of view, these flows can model cases arising in geophysics, oceanography, and in many industrial applications like turbomachinery. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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17 pages, 4720 KiB

Open AccessArticle

Magnetic Field Effect on Sisko Fluid Flow Containing Gold Nanoparticles through a Porous Curved Surface in the Presence of Radiation and Partial Slip

by Umair Khan, Aurang Zaib and Anuar Ishak

Mathematics 2021, 9(9), 921; https://doi.org/10.3390/math9090921 - 21 Apr 2021

Cited by 48 | Viewed by 2808

Abstract

The radiation and magnetic field effects of nanofluids play a significant role in biomedical engineering and medical treatment. This study investigated the performance of gold particles in blood flow (Sisko fluid flow) over a porous, slippery, curved surface. The partial slip effect was [...] Read more.

The radiation and magnetic field effects of nanofluids play a significant role in biomedical engineering and medical treatment. This study investigated the performance of gold particles in blood flow (Sisko fluid flow) over a porous, slippery, curved surface. The partial slip effect was considered to examine the characteristics of nanofluid flow in depth. The foremost partial differential equations of the Sisko model were reduced to ordinary differential equations by using suitable variables, and the boundary value problem of the fourth-order (bvp4c) procedure was applied to plot the results. In addition, the effects of the parameters involved on temperature and velocity were presented in light of the parametric investigation. A comparison with published results showed excellent agreement. The velocity distribution was enhanced due to the magnetic field, while the temperature increased due to the effects of a magnetic field and radiation, which are effective in therapeutic hyperthermia. In addition, the nanoparticle suspension showed increased temperature and decelerated velocity. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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18 pages, 3831 KiB

Open AccessArticle

Double Solutions and Stability Analysis of Micropolar Hybrid Nanofluid with Thermal Radiation Impact on Unsteady Stagnation Point Flow

by Nur Syazana Anuar and Norfifah Bachok

Mathematics 2021, 9(3), 276; https://doi.org/10.3390/math9030276 - 30 Jan 2021

Cited by 36 | Viewed by 3253

Abstract

The mathematical modeling of unsteady flow of micropolar Cu–Al₂O₃/water nanofluid driven by a deformable sheet in stagnation region with thermal radiation effect has been explored numerically. To achieve the system of nonlinear ordinary differential equations (ODEs), we have employed [...] Read more.

The mathematical modeling of unsteady flow of micropolar Cu–Al₂O₃/water nanofluid driven by a deformable sheet in stagnation region with thermal radiation effect has been explored numerically. To achieve the system of nonlinear ordinary differential equations (ODEs), we have employed some appropriate transformations and solved it numerically using MATLAB software (built-in solver called bvp4c). Influences of relevant parameters on fluid flow and heat transfer characteristic are discussed and presented in graphs. The findings expose that double solutions appear in shrinking sheet case in which eventually contributes to the analysis of stability. The stability analysis therefore confirms that merely the first solution is a stable solution. Addition of nanometer-sized particle (Cu) has been found to significantly strengthen the heat transfer rate of micropolar nanofluid. When the copper nanoparticle volume fraction increased from 0 to 0.01 (1%) in micropolar nanofluid, the heat transfer rate increased roughly to an average of 17.725%. The result also revealed that an upsurge in the unsteady and radiation parameters have been noticed to enhance the local Nusselt number of micropolar hybrid nanofluid. Meanwhile, the occurrence of material parameter conclusively decreases it. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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21 pages, 8222 KiB

Open AccessArticle

Effects of Leading-Edge Modification in Damaged Rotor Blades on Aerodynamic Characteristics of High-Pressure Gas Turbine

by Thanh Dam Mai and Jaiyoung Ryu

Mathematics 2020, 8(12), 2191; https://doi.org/10.3390/math8122191 - 9 Dec 2020

Cited by 15 | Viewed by 4159

Abstract

The flow and heat-transfer attributes of gas turbines significantly affect the output power and overall efficiency of combined-cycle power plants. However, the high-temperature and high-pressure environment can damage the turbine blade surface, potentially resulting in failure of the power plant. Because of the [...] Read more.

The flow and heat-transfer attributes of gas turbines significantly affect the output power and overall efficiency of combined-cycle power plants. However, the high-temperature and high-pressure environment can damage the turbine blade surface, potentially resulting in failure of the power plant. Because of the elevated cost of replacing turbine blades, damaged blades are usually repaired through modification of their profile around the damage location. This study compared the effects of modifying various damage locations along the leading edge of a rotor blade on the performance of the gas turbine. We simulated five rotor blades—an undamaged blade (reference) and blades damaged on the pressure and suction sides at the top and middle. The Reynolds-averaged Navier–Stokes equation was used to investigate the compressible flow in a GE-E³ gas turbine. The results showed that the temperatures of the blade and vane surfaces with damages at the middle increased by about 0.8% and 1.2%, respectively. This causes a sudden increase in the heat transfer and thermal stress on the blade and vane surfaces, especially around the damage location. Compared with the reference case, modifications to the top-damaged blades produced a slight increase in efficiency about 2.6%, while those to the middle-damaged blades reduced the efficiency by approximately 2.2%. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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17 pages, 5120 KiB

Open AccessArticle

Impact of Heat Generation on Magneto-Nanofluid Free Convection Flow about Sphere in the Plume Region

by Anwar Khan, Muhammad Ashraf, Ahmed M. Rashad and Hossam A. Nabwey

Mathematics 2020, 8(11), 2010; https://doi.org/10.3390/math8112010 - 11 Nov 2020

Cited by 21 | Viewed by 1899

Abstract

The main aim of the current study is to analyze the physical phenomenon of free convection nanofluids heat transfer along a sphere and fluid eruption through boundary layer into a plume region above the surface of the sphere. In the current study, the [...] Read more.

The main aim of the current study is to analyze the physical phenomenon of free convection nanofluids heat transfer along a sphere and fluid eruption through boundary layer into a plume region above the surface of the sphere. In the current study, the effect of heat generation with the inclusion of an applied magnetic field by considering nanofluids is incorporated. The dimensioned form of formulated equations of the said phenomenon is transformed into the non-dimensional form, and then solved numerically. The developed finite difference method along with the Thomas algorithm has been utilized to approximate the given equations. The numerical simulation is carried out for the different physical parameters involved, such as magnetic field parameter, Prandtl number, thermophoresis parameter, heat generation parameter, Schmidt number, and Brownian motion parameter. Later, the quantities, such as velocity, temperature, and mass distribution, are plotted under the impacts of different values of different controlling parameters to ascertain how these quantities are affected by these pertinent parameters. Moreover, the obtained results are displayed graphically as well in tabular form. The novelty of present work is that we first secure results around different points of a sphere and then the effects of all parameters are captured above the sphere in the plume. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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23 pages, 10346 KiB

Open AccessArticle

Numerical Investigation of Aerodynamic Drag and Pressure Waves in Hyperloop Systems

by Thi Thanh Giang Le, Kyeong Sik Jang, Kwan-Sup Lee and Jaiyoung Ryu

Mathematics 2020, 8(11), 1973; https://doi.org/10.3390/math8111973 - 6 Nov 2020

Cited by 43 | Viewed by 5993

Abstract

Hyperloop is a new, alternative, very high-speed mode of transport wherein Hyperloop pods (or capsules) transport cargo and passengers at very high speeds in a near-vacuum tube. Such high-speed operations, however, cause a large aerodynamic drag. This study investigates the effects of pod [...] Read more.

Hyperloop is a new, alternative, very high-speed mode of transport wherein Hyperloop pods (or capsules) transport cargo and passengers at very high speeds in a near-vacuum tube. Such high-speed operations, however, cause a large aerodynamic drag. This study investigates the effects of pod speed, blockage ratio (BR), tube pressure, and pod length on the drag and drag coefficient of a Hyperloop. To study the compressibility of air when the pod is operating in a tube, the effect of pressure waves in terms of propagation speed and magnitude are investigated based on normal shockwave theories. To represent the pod motion and propagation of pressure waves, unsteady simulation using the moving-mesh method was applied under the sheer stress transport k–ω turbulence model. Numerical simulations were performed for different pod speeds from 100 to 350 m/s. The results indicate that the drag coefficient increases with increase in BR, pod speed, and pod length. In the Hyperloop system, the compression wave propagation speed is much higher than the speed of sound and the expansion wave propagation speed that experiences values around the speed of sound. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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22 pages, 4576 KiB

Open AccessArticle

Unsteady Stagnation Point Flow of Hybrid Nanofluid Past a Convectively Heated Stretching/Shrinking Sheet with Velocity Slip

by Nurul Amira Zainal, Roslinda Nazar, Kohilavani Naganthran and Ioan Pop

Mathematics 2020, 8(10), 1649; https://doi.org/10.3390/math8101649 - 24 Sep 2020

Cited by 57 | Viewed by 3632

Abstract

Unsteady stagnation point flow in hybrid nanofluid (Al₂O₃-Cu/H₂O) past a convectively heated stretching/shrinking sheet is examined. Apart from the conventional surface of the no-slip condition, the velocity slip condition is considered in this study. By incorporating verified [...] Read more.

Unsteady stagnation point flow in hybrid nanofluid (Al₂O₃-Cu/H₂O) past a convectively heated stretching/shrinking sheet is examined. Apart from the conventional surface of the no-slip condition, the velocity slip condition is considered in this study. By incorporating verified similarity transformations, the differential equations together with their partial derivatives are changed into ordinary differential equations. Throughout the MATLAB operating system, the simplified mathematical model is clarified by employing the bvp4c procedure. The above-proposed approach is capable of producing non-uniqueness solutions when adequate initial assumptions are provided. The findings revealed that the skin friction coefficient intensifies in conjunction with the local Nusselt number by adding up the nanoparticles volume fraction. The occurrence of velocity slip at the boundary reduces the coefficient of skin friction; however, an upward trend is exemplified in the rate of heat transfer. The results also signified that, unlike the parameter of velocity slip, the increment in the unsteady parameter conclusively increases the coefficient of skin friction, and an upsurge attribution in the heat transfer rate is observed resulting from the increment of Biot number. The findings are evidenced to have dual solutions, which inevitably contribute to stability analysis, hence validating the feasibility of the first solution. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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19 pages, 12809 KiB

Open AccessArticle

Effects of Entropy Generation, Thermal Radiation and Moving-Wall Direction on Mixed Convective Flow of Nanofluid in an Enclosure

by Sivasankaran Sivanandam, Ali J. Chamkha, Fouad O. M. Mallawi, Metib S. Alghamdi and Aisha M. Alqahtani

Mathematics 2020, 8(9), 1471; https://doi.org/10.3390/math8091471 - 1 Sep 2020

Cited by 14 | Viewed by 2569

Abstract

A numeric investigation is executed to understand the impact of moving-wall direction, thermal radiation, entropy generation and nanofluid volume fraction on combined convection and energy transfer of nanoliquids in a differential heated box. The top wall of the enclosed box is assumed to [...] Read more.

A numeric investigation is executed to understand the impact of moving-wall direction, thermal radiation, entropy generation and nanofluid volume fraction on combined convection and energy transfer of nanoliquids in a differential heated box. The top wall of the enclosed box is assumed to move either to the left or the right direction which affects the stream inside the box. The horizontal barriers are engaged to be adiabatic. The derived mathematical model is solved by the control volume technique. The results are presented graphically to know the impact of the dissimilar ways of moving wall, Richardson number, Bejan number, thermal radiation, cup mixing and average temperatures. It is concluded that the stream and the thermal distribution are intensely affected by the moving-wall direction. It is established that the thermal radiation enhances the convection energy transport inside the enclosure. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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18 pages, 3339 KiB

Open AccessArticle

Radiative MHD Sutterby Nanofluid Flow Past a Moving Sheet: Scaling Group Analysis

by Mohammed M. Fayyadh, Kohilavani Naganthran, Md Faisal Md Basir, Ishak Hashim and Rozaini Roslan

Mathematics 2020, 8(9), 1430; https://doi.org/10.3390/math8091430 - 26 Aug 2020

Cited by 23 | Viewed by 3466

Abstract

The present theoretical work endeavors to solve the Sutterby nanofluid flow and heat transfer problem over a permeable moving sheet, together with the presence of thermal radiation and magnetohydrodynamics (MHD). The fluid flow and heat transfer features near the stagnation region are considered. [...] Read more.

The present theoretical work endeavors to solve the Sutterby nanofluid flow and heat transfer problem over a permeable moving sheet, together with the presence of thermal radiation and magnetohydrodynamics (MHD). The fluid flow and heat transfer features near the stagnation region are considered. A new form of similarity transformations is introduced through scaling group analysis to simplify the governing boundary layer equations, which then eases the computational process in the MATLAB bvp4c function. The variation in the values of the governing parameters yields two different numerical solutions. One of the solutions is stable and physically reliable, while the other solution is unstable and is associated with flow separation. An increased effect of the thermal radiation improves the rate of convective heat transfer past the permeable shrinking sheet. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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21 pages, 3839 KiB

Open AccessArticle

A Numerical Approach for the Heat Transfer Flow of Carboxymethyl Cellulose-Water Based Casson Nanofluid from a Solid Sphere Generated by Mixed Convection under the Influence of Lorentz Force

by Firas A. Alwawi, Hamzeh T. Alkasasbeh, Ahmed M. Rashad and Ruwaidiah Idris

Mathematics 2020, 8(7), 1094; https://doi.org/10.3390/math8071094 - 4 Jul 2020

Cited by 48 | Viewed by 3456

Abstract

The heat transfer of a carboxymethyl cellulose aqueous solution (CMC-water) based Casson nanofluid, flowing under the impact of a variable-strength magnetic field in mixed convection around a solid sphere, has been examined in this work. Aluminum (Al), copper (Cu), and silver (Ag) nanoparticles [...] Read more.

The heat transfer of a carboxymethyl cellulose aqueous solution (CMC-water) based Casson nanofluid, flowing under the impact of a variable-strength magnetic field in mixed convection around a solid sphere, has been examined in this work. Aluminum (Al), copper (Cu), and silver (Ag) nanoparticles were employed to support the heat transfer characteristics of the host fluid. A numerical approach called the Keller-box method (KBM) was used to solve the governing system for the present problem, and also to examine and analyze the numerical and graphic results obtained by the MATLAB program, verifying their accuracy through comparing them with the prior literature. The results demonstrate that a Al–CMC-water nanoliquid is superior in terms of heat transfer rate and skin friction. The velocity of CMC-water is higher with Ag compared to Al–CMC-water, and Ag–CMC-water possesses the lowest temperature. Growing mixed parameter values result in a rising skin friction, velocity and Nusselt number or decline in temperature. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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17 pages, 2147 KiB

Open AccessArticle

Simulation of Natural Convection in a Concentric Hexagonal Annulus Using the Lattice Boltzmann Method Combined with the Smoothed Profile Method

by Suresh Alapati

Mathematics 2020, 8(6), 1043; https://doi.org/10.3390/math8061043 - 26 Jun 2020

Cited by 12 | Viewed by 3909

Abstract

This research work presents results obtained from the simulation of natural convection inside a concentric hexagonal annulus by using the lattice Boltzmann method (LBM). The fluid flow (pressure and velocity fields) inside the annulus is evaluated by LBM and a finite difference method [...] Read more.

This research work presents results obtained from the simulation of natural convection inside a concentric hexagonal annulus by using the lattice Boltzmann method (LBM). The fluid flow (pressure and velocity fields) inside the annulus is evaluated by LBM and a finite difference method (FDM) is used to get the temperature filed. The isothermal and no-slip boundary conditions (BC) on the hexagonal edges are treated with a smooth profile method (SPM). At first, for validating the present simulation technique, a standard benchmarking problem of natural convection inside a cold square cavity with a hot circular cylinder is simulated. Later, natural convection simulations inside the hexagonal annulus are carried out for different values of the aspect ratio, AR (ratio of the inner and outer hexagon sizes), and the Rayleigh number, Ra. The simulation results are presented in terms of isotherms (temperature contours), streamlines, temperature, and velocity distributions inside the annulus. The results show that the fluid flow intensity and the size and number of vortex pairs formed inside the annulus strongly depend on AR and Ra values. Based on the concentric isotherms and weak fluid flow intensity at the low Ra, it is observed that the heat transfer inside the annulus is dominated by the conduction mode. However, multiple circulation zones and distorted isotherms are observed at the high Ra due to the strong convective flow. To further access the accuracy and robustness of the present scheme, the present simulation results are compared with the results given by the commercial software, ANSYS-Fluent^®. For all combinations of AR and Ra values, the simulation results of streamlines and isotherms patterns, and temperature and velocity distributions inside the annulus are in very good agreement with those of the Fluent software. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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13 pages, 2782 KiB

Open AccessArticle

Lie Group Analysis of Unsteady Flow of Kerosene/Cobalt Ferrofluid Past A Radiated Stretching Surface with Navier Slip and Convective Heating

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

Mathematics 2020, 8(5), 826; https://doi.org/10.3390/math8050826 - 19 May 2020

Cited by 21 | Viewed by 2701

Abstract

In this work, we identified the characteristics of unsteady magnetohydrodynamic (MHD) flow of ferrofluid past a radiated stretching surface. Cobalt–kerosene ferrofluid is considered and the impacts of Navier slip and convective heating are additionally considered. The mathematical model which describes the problem was [...] Read more.

In this work, we identified the characteristics of unsteady magnetohydrodynamic (MHD) flow of ferrofluid past a radiated stretching surface. Cobalt–kerosene ferrofluid is considered and the impacts of Navier slip and convective heating are additionally considered. The mathematical model which describes the problem was built from some partial differential equations and then converted to self-similar equations with the assistance of the Lie group method; after that, the mathematical model was solved numerically with the aid of Runge–Kutta–Fehlberg method. Graphical representations were used to exemplify the impact of influential parameters on dimensionless velocity and temperature profiles; the obtained results for the skin friction coefficient and Nusselt number were also examined graphically. It was demonstrated that the magnetic field, Navier slip, and solid volume fraction of ferroparticles tended to reduce the dimensionless velocity, while the radiation parameter and Biot number had no effects on the dimensionless velocity. Moreover, the magnetic field and solid volume fraction increase skin friction whereas Navier slip reduces the skin friction. Furthermore, the Navier slip and magnetic field reduce the Nusselt number, whereas solid volume fraction of ferroparticles, convective heating, and radiation parameters help in increasing the Nusselt number. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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23 pages, 6640 KiB

Open AccessArticle

Unsteady Three-Dimensional MHD Non-Axisymmetric Homann Stagnation Point Flow of a Hybrid Nanofluid with Stability Analysis

by Nurul Amira Zainal, Roslinda Nazar, Kohilavani Naganthran and Ioan Pop

Mathematics 2020, 8(5), 784; https://doi.org/10.3390/math8050784 - 13 May 2020

Cited by 61 | Viewed by 4138

Abstract

The hybrid nanofluid under the influence of magnetohydrodynamics (MHD) is a new interest in the industrial sector due to its applications, such as in solar water heating and scraped surface heat exchangers. Thus, the present study accentuates the analysis of an unsteady three-dimensional [...] Read more.

The hybrid nanofluid under the influence of magnetohydrodynamics (MHD) is a new interest in the industrial sector due to its applications, such as in solar water heating and scraped surface heat exchangers. Thus, the present study accentuates the analysis of an unsteady three-dimensional MHD non-axisymmetric Homann stagnation point flow of a hybrid Al₂O₃-Cu/H₂O nanofluid with stability analysis. By employing suitable similarity transformations, the governing mathematical model in the form of the partial differential equations are simplified into a system of ordinary differential equations. The simplified mathematical model is then solved numerically by the Matlab solver bvp4c function. This solving approach was proficient in generating more than one solution when good initial guesses were provided. The numerical results presented significant influences on the rate of heat transfer and fluid flow characteristics of a hybrid nanofluid. The rate of heat transfer and the trend of the skin friction coefficient improve with the increment of the nanoparticles’ concentration and the magnetic parameter; however, they deteriorate when the unsteadiness parameter increases. In contrast, the ratio of the escalation of the ambient fluid strain rate to the plate was able to adjourn the boundary layer separation. The dual solutions (first and second solutions) are obtainable when the surface of the sheet shrunk. A stability analysis is carried out to justify the stability of the dual solutions, and hence the first solution is seen as physically reliable and stable, while the second solution is unstable. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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18 pages, 5873 KiB

Open AccessArticle

Numerical Investigation on the Swimming of Gyrotactic Microorganisms in Nanofluids through Porous Medium over a Stretched Surface

by Anwar Shahid, Hulin Huang, Muhammad Mubashir Bhatti, Lijun Zhang and Rahmat Ellahi

Mathematics 2020, 8(3), 380; https://doi.org/10.3390/math8030380 - 9 Mar 2020

Cited by 96 | Viewed by 5188

Abstract

In this article, the effects of swimming gyrotactic microorganisms for magnetohydrodynamics nanofluid using Darcy law are investigated. The numerical results of nonlinear coupled mathematical model are obtained by means of Successive Local Linearization Method. This technique is based on a simple notion of [...] Read more.

In this article, the effects of swimming gyrotactic microorganisms for magnetohydrodynamics nanofluid using Darcy law are investigated. The numerical results of nonlinear coupled mathematical model are obtained by means of Successive Local Linearization Method. This technique is based on a simple notion of the decoupling systems of equations utilizing the linearization of the unknown functions sequentially according to the order of classifying the system of governing equations. The linearized equations, that developed a sequence of linear differential equations along with variable coefficients, were solved by employing the Chebyshev spectral collocation method. The convergence speed of the SLLM technique can be willingly upgraded by successive applying over relaxation method. The comparison of current study with available published literature has been made for the validation of obtained results. It is found that the reported numerical method is in perfect accord with the said similar methods. The results are displayed through tables and graphs. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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20 pages, 4155 KiB

Open AccessArticle

Insights into the Stability of Mixed Convective Darcy–Forchheimer Flows of Cross Liquids from a Vertical Plate with Consideration of the Significant Impact of Velocity and Thermal Slip Conditions

by Umair Khan, Aurang Zaib, Ilyas Khan, Kottakkaran Sooppy Nisar and Dumitru Baleanu

Mathematics 2020, 8(1), 31; https://doi.org/10.3390/math8010031 - 24 Dec 2019

Cited by 11 | Viewed by 2387

Abstract

This paper reflects the effects of velocity and thermal slip conditions on the stagnation-point mixed convective flow of Cross liquid moving over a vertical plate entrenched in a Darcy–Forchheimer porous medium. A Cross liquid is a type of non-Newtonian liquid whose viscosity depends [...] Read more.

This paper reflects the effects of velocity and thermal slip conditions on the stagnation-point mixed convective flow of Cross liquid moving over a vertical plate entrenched in a Darcy–Forchheimer porous medium. A Cross liquid is a type of non-Newtonian liquid whose viscosity depends on the shear rate. The leading partial differential equations (PDEs) are altered to nonlinear ordinary differential equations (ODEs) via feasible similarity transformations. These transmuted equations are computed numerically through the bvp4c solver. The authority of sundry parameters on the temperature and velocity distributions is examined graphically. In addition, the characteristics of heat transfer are analyzed in the presence of the impact of drag forces. The outcomes reveal that the permeability parameter decelerates the drag forces and declines the rate of heat transfer in both forms of solutions. Moreover, it is found that the drag forces decline with the growing value of the Weissenberg parameter in the upper branch solutions, while a reverse trend is revealed in the lower branch solutions. However, the rate of heat transfer shows a diminishing behavior with an increasing value of the Weissenberg parameter. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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17 pages, 493 KiB

Open AccessEditor’s ChoiceArticle

Significance of Double Stratification in Stagnation Point Flow of Third-Grade Fluid towards a Radiative Stretching Cylinder

by Anum Shafiq, Ilyas Khan, Ghulam Rasool, Asiful H. Seikh and El-Sayed M. Sherif

Mathematics 2019, 7(11), 1103; https://doi.org/10.3390/math7111103 - 14 Nov 2019

Cited by 38 | Viewed by 3914

Abstract

The present article is devoted to examine the significance of double stratification in third grade stagnation point flow towards a radiative stretching cylinder. The stagnation point is discussed categorically. Analysis is scrutinized in the presence of Thermophoresis, Brownian diffusion, double stratification and heat [...] Read more.

The present article is devoted to examine the significance of double stratification in third grade stagnation point flow towards a radiative stretching cylinder. The stagnation point is discussed categorically. Analysis is scrutinized in the presence of Thermophoresis, Brownian diffusion, double stratification and heat source/sink. Suitable typical transformations are used to drive the system of ordinary differential equation. The governing system is subjected to optimal homotopy analysis method (OHAM) for convergent series solutions. The impact of pertinent fluid parameters on the velocity field, temperature distribution and concentration of the nanoparticles is shown graphically. Numerical data is compiled in tabulare form for skin friction, Nusselt and Sherwood numbers to analyze the variation caused by the present model and to see the impact for industrial and engineering point of view. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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15 pages, 493 KiB

Open AccessArticle

On the MHD Casson Axisymmetric Marangoni Forced Convective Flow of Nanofluids

by Anum Shafiq, Islam Zari, Ghulam Rasool, Iskander Tlili and Tahir Saeed Khan

Mathematics 2019, 7(11), 1087; https://doi.org/10.3390/math7111087 - 11 Nov 2019

Cited by 62 | Viewed by 3852

Abstract

The proposed investigation concerns the impact of inclined magnetohydrodynamics (MHD) in a Casson axisymmetric Marangoni forced convective flow of nanofluids. Axisymmetric Marangoni convective flow has been driven by concentration and temperature gradients due to an infinite disk. Brownian motion appears due to concentration [...] Read more.

The proposed investigation concerns the impact of inclined magnetohydrodynamics (MHD) in a Casson axisymmetric Marangoni forced convective flow of nanofluids. Axisymmetric Marangoni convective flow has been driven by concentration and temperature gradients due to an infinite disk. Brownian motion appears due to concentration of the nanosize metallic particles in a typical base fluid. Thermophoretic attribute and heat source are considered. The analysis of flow pattern is perceived in the presence of certain distinct fluid parameters. Using appropriate transformations, the system of Partial Differential Equations (PDEs) is reduced into non-linear Ordinary Differential Equations (ODEs). Numerical solution of this problem is achieved invoking Runge–Kutta fourth-order algorithm. To observe the effect of inclined MHD in axisymmetric Marangoni convective flow, some suitable boundary conditions are incorporated. To figure out the impact of heat/mass phenomena on flow behavior, different physical and flow parameters are addressed for velocity, concentration and temperature profiles with the aid of tables and graphs. The results indicate that Casson fluid parameter and angle of inclination of MHD are reducing factors for fluid movement; however, stronger Marangoni effect is sufficient to improve the velocity profile. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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23 pages, 1967 KiB

Open AccessArticle

Three-Dimensional Hydro-Magnetic Flow Arising in a Long Porous Slider and a Circular Porous Slider with Velocity Slip

by Naeem Faraz, Yasir Khan, Amna Anjum and Muhammad Kahshan

Mathematics 2019, 7(8), 748; https://doi.org/10.3390/math7080748 - 16 Aug 2019

Cited by 7 | Viewed by 3217

Abstract

The current research explores the injection of a viscous fluid through a moving flat plate with a transverse uniform magneto-hydrodynamic (MHD) flow field to reduce sliding drag. Two cases of velocity slip between the slider and the ground are studied: a long slider [...] Read more.

The current research explores the injection of a viscous fluid through a moving flat plate with a transverse uniform magneto-hydrodynamic (MHD) flow field to reduce sliding drag. Two cases of velocity slip between the slider and the ground are studied: a long slider and a circular slider. Solving the porous slider problem is applicable to fluid-cushioned porous sliders, which are useful in reducing the frictional resistance of moving bodies. By using a similarity transformation, three dimensional Navier–Stokes equations are converted into coupled nonlinear ordinary differential equations. The resulting nonlinear boundary value problem was solved analytically using the homotopy analysis method (HAM). The HAM provided a fast convergent series solution, showing that this method is efficient, accurate, and has many advantages over the other existing methods. Solutions were obtained for the different values of Reynolds numbers (R), velocity slip, and magnetic fields. It was found that surface slip and Reynolds number had substantial influence on the lift and drag of the long and the circular sliders. Moreover, the effects of the applied magnetic field on the velocity components, load-carrying capacity, and friction force are discussed in detail with the aid of graphs and tables. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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10 pages, 1641 KiB

Open AccessArticle

Optimal Design of Isothermal Sloshing Vessels by Entropy Generation Minimization Method

by Mohammad Yaghoub Abdollahzadeh Jamalabadi

Mathematics 2019, 7(5), 380; https://doi.org/10.3390/math7050380 - 26 Apr 2019

Cited by 7 | Viewed by 2589

Abstract

In this manuscript, the optimal design of geometry for a forced sloshing in a rigid container based on the entropy generation minimization (EGM) method is presented. The geometry of the vessel considered here is two dimensional rectangular. Incompressible inviscid fluid undergoes horizontal harmonic [...] Read more.

In this manuscript, the optimal design of geometry for a forced sloshing in a rigid container based on the entropy generation minimization (EGM) method is presented. The geometry of the vessel considered here is two dimensional rectangular. Incompressible inviscid fluid undergoes horizontal harmonic motion by interaction with a rigid tank. The analytical solution of a fluid stream function is obtained and benchmarked by Finite element results. A parameter study of the aspect ratio, amplitude, and frequency of the horizontal harmonic motion is performed. As well, an analytical solution for the total entropy generation in the volume is presented and discussed. The total entropy generation is compared with the results of the Reynolds-Averaged Navier–Stokes (RANS) solver and the Volume-of-Fluid (VOF) method). Then, the effect of parameters is studied on the total entropy generated by sway motion. Finally, the results show that, based on the excitation frequency, an optimal design of the tank could be found. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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11 pages, 2040 KiB

Open AccessArticle

MHD Flow of Nanofluid with Homogeneous-Heterogeneous Reactions in a Porous Medium under the Influence of Second-Order Velocity Slip

by Fahd Almutairi, S.M. Khaled and Abdelhalim Ebaid

Mathematics 2019, 7(3), 220; https://doi.org/10.3390/math7030220 - 26 Feb 2019

Cited by 27 | Viewed by 3223

Abstract

The influence of second-order velocity slip on the MHD flow of nanofluid in a porous medium under the effects of homogeneous-heterogeneous reactions has been analyzed. The governing flow equation is exactly solved and compared with those in the literature for the skin friction [...] Read more.

The influence of second-order velocity slip on the MHD flow of nanofluid in a porous medium under the effects of homogeneous-heterogeneous reactions has been analyzed. The governing flow equation is exactly solved and compared with those in the literature for the skin friction coefficient in the absence of the second slip, where great differences have been observed. In addition, the effects of the permanent parameters on the skin friction coefficient, the velocity, and the concentration have been discussed in the presence of the second slip. As an important result, the behavior of the skin friction coefficient at various values of the porosity and volume fraction is changed from increasing (in the absence of the second slip) to decreasing (in the presence of the second slip), which confirms the importance of the second slip in modeling the boundary layer flow of nanofluids. In addition, five kinds of nanofluids have been investigated for the velocity profiles and it is found that the Ag-water nanofluid is the lowest. For only the heterogeneous reaction, the concentration equation has been exactly solved, while the numerical solution is applied in the general case. Accordingly, a reduction in the concentration occurs with the strengthening of the heterogenous reaction and also with the increase in the Schmidt parameter. Moreover, the Ag-water nanofluid is of lower concentration than the Cu-water nanofluid. This is also true for the general case, when both of the homogenous and heterogenous reactions take place. Full article

(This article belongs to the Special Issue Computational Fluid Dynamics 2020)

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