Unveiling the Performance of Cu-Water Nanofluid Flow with Melting Heat Transfer, MHD, and Thermal Radiation over a Stretching/Shrinking Sheet.

The use of melting heat transfer (MHT) and nanofluids for electronics cooling and energy storage efficiency has gained the attention of numerous researchers. This study investigates the effects of MHD, mixed convection, thermal radiation, stretching, and shrinking on the heat transfer characteristics of a Cu-water-based nanofluid over a stretching/shrinking sheet with MHT effects. The governing equations are transformed into nonlinear ordinary differential equations and solved numerically using the Keller Box method. To the best of our knowledge, this comprehensive analysis, encompassing all of these factors, including the utilization of a robust numerical method, in a single study, has not been previously reported in the literature. Our findings demonstrate that an increase in the melting parameter leads to an enhanced rate of heat transfer, while an increase in the stretching/shrinking parameter results in a decrease in the rate of heat transfer. Additionally, we present a comprehensive analysis of the influences of all of the mentioned driving parameters. The results are presented through graphical and tabulated representations and compared with existing literature.

Unsteady natural convection flow of blood Casson nanofluid (Au) in a cylinder: nano-cryosurgery applications.

Nano-cryosurgery is one of the effective ways to treat cancerous cells with minimum harm to healthy adjacent cells. Clinical experimental research consumes time and cost. Thus, developing a mathematical simulation model is useful for time and cost-saving, especially in designing the experiment. Investigating the Casson nanofluid's unsteady flow in an artery with the convective effect is the goal of the current investigation. The nanofluid is considered to flow in the blood arteries. Therefore, the slip velocity effect is concerned. Blood is a base fluid with gold (Au) nanoparticles dispersed in the base fluid. The resultant governing equations are solved by utilising the Laplace transform regarding the time and the finite Hankel transform regarding the radial coordinate. The resulting analytical answers for velocity and temperature are then displayed and visually described. It is found that the temperature enhancement occurred by arising nanoparticles volume fraction and time parameter. The blood velocity increases as the slip velocity, time parameter, thermal Grashof number, and nanoparticles volume fraction increase. Whereas the velocity decreases with the Casson parameter. Thus, by adding Au nanoparticles, the tissue thermal conductivity enhanced which has the consequence of freezing the tissue in nano-cryosurgery treatment significantly.

Mixed convection flow of an electrically conducting viscoelastic fluid past a vertical nonlinearly stretching sheet.

The study of hydromagnetic mixed convection flow of viscoelastic fluid caused by a vertical stretched surface is presented in this paper. According to this theory, the stretching velocity varies as a power function of the displacement from the slot. The conservation of energy equation includes thermal radiation and viscous dissipation to support the mechanical operations of the heat transfer mechanism. Through the use of an adequate and sufficient similarity transformation for a nonlinearly stretching sheet, the boundary layer equations governing the flow issue are converted into a set of ordinary differential equations. The Keller box technique is then used to numerically solve the altered equations. To comprehend the physical circumstances of stretching sheets for variations of the governing parameters, numerical simulations are made. The influence and characteristic behaviours of physical parameters were portrayed graphically for the velocity field and temperature distributions. The research shows that the impact of the applied magnetic parameter is to improve the distribution of the viscoelastic fluid temperature and reduce the temperature gradient at the border. Temperature distribution and the associated thermal layer are shown to have improved because of radiative and viscous dissipation characteristics. Radiation causes additional heat to be produced in liquid, raising the fluid's temperature. It was also found that higher velocities are noticed in viscoelastic fluid as compared with Newtonian fluid (i.e., when K = 0).

Soret and Dufour effects on MHD squeezing flow of Jeffrey fluid in horizontal channel with thermal radiation.

The fluid flow with chemical reaction is one of well-known research areas in the field of computational fluid dynamic. It is potentially useful in the modelling of flow on a nuclear reactor. Motivated by the implementation of the flow in the industrial application, the aim of this study is to explore the time-dependent squeeze flow of magnetohydrodynamic Jeffrey fluid over permeable medium in the influences of Soret and Dufour, heat source/sink and chemical reaction. The presence of joule heating, joule dissipation and radiative heat transfer are analyzed. The flow is induced due to compress of two surfaces. Conversion of partial differential equations (PDEs) into ordinary differential equations (ODEs) is accomplished by imposing similarity variables. Then, the governing equations are resolved using Keller-box approach. The present outcomes are compared with previously outcomes in the literature to validate the precision of present outcomes. Both outcomes are shown in close agreement. The tabular and graphical results demonstrate that wall shear stress and velocity profile accelerate with the surfaces moving towards one another. Moreover, the concentration, temperature and velocity profiles decreasing for the increment of Hartmann numbers and Jeffrey fluid parameters. The impacts of heat generation/absorption, joule dissipation and Dufour numbers enhance the heat transfer rate and temperature profile. In contrast, the temperature profile drops and the heat transfer rate boosts when thermal radiation increases. The concentration profile decelerates, and the mass transfer rate elevates with raise in Soret number. Also, the mass transfer rate rises for destructive chemical reaction and contrary result is noted for convective chemical reaction.

Interaction of multi-walled carbon nanotubes in mineral oil based Maxwell nanofluid.

The most pressing issue now is to improve the cooling process in an electrical power system. On the other hand, nanofluids are regarded as reliable coolants owing to their exceptional characteristics, which include excellent thermal conductivity, a faster heat transfer rate, and higher critical heat flux. Considering these fascinating properties of nanofluid, this research looks at the flow of mineral oil based Maxwell nanofluid with convective heat. Moreover, introducing heat radiation, viscous dissipation and Newtonian heating add to the novelty of the problem. The coupled partial differential equations supported by the accompanying boundary conditions are numerically solved using an implicit finite difference method. The simulations are carried out using MATLAB software, and the obtained results are illustrated graphically. It is observed that the velocity of fluid increases concernign the relaxation time parameter but decreases against fractional derivative.

Heat and mass transfer on MHD squeezing flow of Jeffrey nanofluid in horizontal channel through permeable medium.

The heat and mass transfer on time dependent hydrodynamic squeeze flow of Jeffrey nanofluid across two plates over permeable medium in the slip condition with heat generation/absorption, thermal radiation and chemical reaction are investigated. The impacts of Brownian motion and thermophoresis is examined in the Buongiorno's nanofluid model. Conversion of the governing partial differential equations to the ordinary differential equations is conducted via similarity transformation. The dimensionless equations are solved by imposing numerical method of Keller-box. The outputs are compared with previous reported works in the journals for the validation of the present outputs and found in proper agreement. The behavior of velocity, temperature, and nanoparticles concentration profiles by varying the pertinent parameters are examined. Findings portray that the acceleration of the velocity profile and the wall shear stress is due to the squeezing of plates. Furthermore, the velocity, temperature and concentration profile decline with boost in Hartmann number and ratio of relaxation to retardation times. It is discovered that the rate of heat transfer and temperature profile increase when viscous dissipation, thermophoresis and heat source/sink rises. In contrast, the increment of thermal radiation reduces the temperature and enhances the heat transfer rate. Besides, the mass transfer rate decelerates for increasing Brownian motion in nanofluid, while it elevates when chemical reaction and thermophoresis increases.

Shape effect on MHD flow of time fractional Ferro-Brinkman type nanofluid with ramped heating.

The colloidal suspension of nanometer-sized particles of Fe3O4 in traditional base fluids is referred to as Ferro-nanofluids. These fluids have many technological applications such as cell separation, drug delivery, magnetic resonance imaging, heat dissipation, damping, and dynamic sealing. Due to the massive applications of Ferro-nanofluids, the main objective of this study is to consider the MHD flow of water-based Ferro-nanofluid in the presence of thermal radiation, heat generation, and nanoparticle shape effect. The Caputo-Fabrizio time-fractional Brinkman type fluid model is utilized to demonstrate the proposed flow phenomenon with oscillating and ramped heating boundary conditions. The Laplace transform method is used to solve the model for both ramped and isothermal heating for exact solutions. The ramped and isothermal solutions are simultaneously plotted in the various figures to study the influence of pertinent flow parameters. The results revealed that the fractional parameter has a great impact on both temperature and velocity fields. In the case of ramped heating, both temperature and velocity fields decreasing with increasing fractional parameter. However, in the isothermal case, this trend reverses near the plate and gradually, ramped, and isothermal heating became alike away from the plate for the fractional parameter. Finally, the solutions for temperature and velocity fields are reduced to classical form and validated with already published results.

MHD radiative nanofluid flow induced by a nonlinear stretching sheet in a porous medium.

In this article, we numerically investigate the influence of thermal radiation and heat generation on the flow of an electrically conducting nanofluid past a nonlinear stretching sheet through a porous medium with frictional heating. The partial differential equations governing the flow problems are reduced to ordinary differential equations via similarity variables. The reduced equations are then solved numerically with the aid of Keller box method. The influence of physical parameters such as nanoparticle volume fraction ϕ, permeability parameter K, nonlinear stretching sheet parameter n, magnetic field parameter M, heat generation parameter Q and Eckert number Ec on the flow field, temperature distribution, skin friction and Nusselt number are studied and presented in graphical illustrations and tabular forms. The results obtained reveal that there is an enhancement in the rate of heat transfer with the rise in nanoparticle volume fraction and permeability parameter. The temperature distribution is also influenced with the presence of K, Q, R and ϕ. This shows that the solid volume fraction of nanoparticle can be used in controlling the behaviours of heat transfer and nanofluid flows.

Mathematical modeling of radiotherapy cancer treatment using Caputo fractional derivative.

BACKGROUND: This paper presents a mathematical model that simulates a radiotherapy cancer treatment process. The model takes into consideration two important radiobiological factors, which are repair and repopulation of cells. The model was used to simulate the fractionated treatment process of six patients. The results gave the population changes in the cells and the final volumes of the normal and cancer cells. METHOD: The model was formulated by integrating the Caputo fractional derivative with the previous cancer treatment model. Thereafter, the linear-quadratic with the repopulation model was coupled into the model to account for the cells' population decay due to radiation. The treatment process was then simulated with numerical variables, numerical parameters, and radiation parameters. The numerical parameters which included the proliferation coefficients of the cells, competition coefficients of the cells, and the perturbation constant of the normal cells were obtained from previous literature. The radiation and numerical parameters were obtained from reported clinical data of six patients treated with radiotherapy. The patients had tumor volumes of 24.1cm3, 17.4cm3, 28.4cm3, 18.8cm3, 30.6cm3, and 12.6cm3 with fractionated doses of 2 Gy for the first two patients and 1.8 Gy for the other four. The initial tumor volumes were used to obtain initial populations of cells after which the treatment process was simulated in MATLAB. Subsequently, a global sensitivity analysis was done to corroborate the model with clinical data. Finally, 96 radiation protocols were simulated by using the biologically effective dose formula. These protocols were used to obtain a regression equation connecting the value of the Caputo fractional derivative with the fractionated dose. RESULTS: The final tumor volumes, from the results of the simulations, were 3.58cm3, 8.61cm3, 5.68cm3, 4.36cm3, 5.75cm3, and 6.12cm3, while those of the normal cells were 23.87cm3, 17.29cm3, 28.17cm3, 18.68cm3, 30.33cm3, and 12.55cm3. The sensitivity analysis showed that the most sensitive model factors were the value of the Caputo fractional derivative and the proliferation coefficient of the cancer cells. Lastly, the obtained regression equation accounted for 99.14% of the prediction. CONCLUSION: The model can simulate a cancer treatment process and predict the results of other radiation protocols.

Numerical simulation of normal and cancer cells' populations with fractional derivative under radiotherapy.

Background This paper presents a numerical simulation of normal and cancer cells' population dynamics during radiotherapy. The model used for the simulation was the improved cancer treatment model with radiotherapy. The model simulated the population changes during a fractionated cancer treatment process. The results gave the final populations of the cells, which provided the final volumes of the tumor and normal cells. Method The improved model was obtained by integrating the previous cancer treatment model with the Caputo fractional derivative. In addition, the cells' population decay due to radiation was accounted for by coupling the linear-quadratic model into the improved model. The simulation of the treatment process was done with numerical variables, numerical parameters, and radiation parameters. The numerical variables include the populations of the cells and the time of treatment. The numerical parameters were the model factors which included the proliferation rates of cells, competition coefficients of cells, and perturbation constant for normal cells. The radiation parameters were clinical data based on the treatment procedure. The numerical parameters were obtained from the previous literature while the numerical variables and radiation parameters, which were clinical data, were obtained from reported data of four cancer patients treated with radiotherapy. The four cancer patients had tumor volumes of 28.4 cm3, 18.8 cm3, 30.6 cm3, and 12.6 cm3 and were treated with different treatment plans and a fractionated dose of 1.8 Gy each. The initial populations of cells were obtained by using the tumor volumes. The computer simulations were done with MATLAB. Results The final volumes of the tumors, from the results of the simulations, were 5.67 cm3, 4.36 cm3, 5.74 cm3, and 6.15 cm3 while the normal cells' volumes were 28.17 cm3, 18.68 cm3, 30.34 cm3, and 12.54 cm3. The powers of the derivatives were 0.16774, 0.16557, 0.16835, and 0.16. A variance-based sensitivity analysis was done to corroborate the model with the clinical data. The result showed that the most sensitive factors were the power of the derivative and the cancer cells' proliferation rate. Conclusion The model provided information concerning the status of treatments and can also predict outcomes of other treatment plans.

Soret and Dufour effects on unsteady mixed convection slip flow of Casson fluid over a nonlinearly stretching sheet with convective boundary condition.

Unsteady mixed convection flow of Casson fluid towards a nonlinearly stretching sheet with the slip and convective boundary conditions is analyzed in this work. The effects of Soret Dufour, viscous dissipation and heat generation/absorption are also investigated. After using some suitable transformations, the unsteady nonlinear problem is solved by using Keller-box method. Numerical solutions for wall shear stress and high temperature transfer rate are calculated and compared with previously published work, an excellent arrangement is followed. It is noticed that fluid velocity reduces for both local unsteadiness and Casson parameters. It is likewise noticed that the influence of a Dufour number of dimensionless temperature is more prominent as compared to species concentration. Furthermore, the temperature was found to be increased in the case of nonlinear thermal radiation.

Exact solutions for unsteady free convection flow over an oscillating plate due to non-coaxial rotation.

BACKGROUND: Non-coaxial rotation has wide applications in engineering devices, e.g. in food processing such as mixer machines and stirrers with a two-axis kneader, in cooling turbine blades, jet engines, pumps and vacuum cleaners, in designing thermal syphon tubes, and in geophysical flows. Therefore, this study aims to investigate unsteady free convection flow of viscous fluid due to non-coaxial rotation and fluid at infinity over an oscillating vertical plate with constant wall temperature. METHODS: The governing equations are modelled by a sudden coincidence of the axes of a disk and the fluid at infinity rotating with uniform angular velocity, together with initial and boundary conditions. Some suitable non-dimensional variables are introduced. The Laplace transform method is used to obtain the exact solutions of the corresponding non-dimensional momentum and energy equations with conditions. Solutions of the velocity for cosine and sine oscillations as well as for temperature fields are obtained and displayed graphically for different values of time (t ), the Grashof number (Gr), the Prandtl number ([Formula: see text]), and the phase angle ([Formula: see text]). Skin friction and the Nusselt number are also evaluated. RESULTS: The exact solutions are obtained and in limiting cases, the present solutions are found to be identical to the published results. Further, the obtained exact solutions also validated by comparing with results obtained by using Gaver-Stehfest algorithm. CONCLUSION: The interested physical property such as velocity, temperature, skin friction and Nusselt number are affected by the embedded parameters time (t), the Grashof number (Gr), the Prandtl number ([Formula: see text]), and the phase angle ([Formula: see text]).

MHD Natural Convection Flow of Casson Nanofluid over Nonlinearly Stretching Sheet Through Porous Medium with Chemical Reaction and Thermal Radiation.

In the present work, the effects of chemical reaction on hydromagnetic natural convection flow of Casson nanofluid induced due to nonlinearly stretching sheet immersed in a porous medium under the influence of thermal radiation and convective boundary condition are performed numerically. Moreover, the effects of velocity slip at stretching sheet wall are also examined in this study. The highly nonlinear-coupled governing equations are converted to nonlinear ordinary differential equations via similarity transformations. The transformed governing equations are then solved numerically using the Keller box method and graphical results for velocity, temperature, and nanoparticle concentration as well as wall shear stress, heat, and mass transfer rate are achieved through MATLAB software. Numerical results for the wall shear stress and heat transfer rate are presented in tabular form and compared with previously published work. Comparison reveals that the results are in good agreement. Findings of this work demonstrate that Casson fluids are better to control the temperature and nanoparticle concentration as compared to Newtonian fluid when the sheet is stretched in a nonlinear way. Also, the presence of suspended nanoparticles effectively promotes the heat transfer mechanism in the base fluid.

Unsteady MHD Mixed Convection Slip Flow of Casson Fluid over Nonlinearly Stretching Sheet Embedded in a Porous Medium with Chemical Reaction, Thermal Radiation, Heat Generation/Absorption and Convective Boundary Conditions.

Numerical results are presented for the effect of first order chemical reaction and thermal radiation on mixed convection flow of Casson fluid in the presence of magnetic field. The flow is generated due to unsteady nonlinearly stretching sheet placed inside a porous medium. Convective conditions on wall temperature and wall concentration are also employed in the investigation. The governing partial differential equations are converted to ordinary differential equations using suitable transformations and then solved numerically via Keller-box method. It is noticed that fluid velocity rises with increase in radiation parameter in the case of assisting flow and is opposite in the case of opposing fluid while radiation parameter has no effect on fluid velocity in the forced convection. It is also seen that fluid velocity and concentration enhances in the case of generative chemical reaction whereas both profiles reduces in the case of destructive chemical reaction. Further, increase in local unsteadiness parameter reduces fluid velocity, temperature and concentration. Over all the effects of physical parameters on fluid velocity, temperature and concentration distribution as well as on the wall shear stress, heat and mass transfer rates are discussed in detail.

Energy Transfer in Mixed Convection MHD Flow of Nanofluid Containing Different Shapes of Nanoparticles in a Channel Filled with Saturated Porous Medium.

Energy transfer in mixed convection unsteady magnetohydrodynamic (MHD) flow of an incompressible nanofluid inside a channel filled with saturated porous medium is investigated. The channel with non-uniform walls temperature is taken in a vertical direction under the influence of a transverse magnetic field. Based on the physical boundary conditions, three different flow situations are discussed. The problem is modelled in terms of partial differential equations with physical boundary conditions. Four different shapes of nanoparticles of equal volume fraction are used in conventional base fluids, ethylene glycol (EG) (C 2 H 6 O 2 ) and water (H 2 O). Solutions for velocity and temperature are obtained discussed graphically in various plots. It is found that viscosity and thermal conductivity are the most prominent parameters responsible for different results of velocity and temperature. Due to higher viscosity and thermal conductivity, C 2 H 6 O 2 is regarded as better convectional base fluid compared to H 2 O.

Heat Transfer in MHD Mixed Convection Flow of a Ferrofluid along a Vertical Channel.

This study investigated heat transfer in magnetohydrodynamic (MHD) mixed convection flow of ferrofluid along a vertical channel. The channel with non-uniform wall temperatures was taken in a vertical direction with transverse magnetic field. Water with nanoparticles of magnetite (Fe3O4) was selected as a conventional base fluid. In addition, non-magnetic (Al2O3) aluminium oxide nanoparticles were also used. Comparison between magnetic and magnetite nanoparticles were also conducted. Fluid motion was originated due to buoyancy force together with applied pressure gradient. The problem was modelled in terms of partial differential equations with physical boundary conditions. Analytical solutions were obtained for velocity and temperature. Graphical results were plotted and discussed. It was found that temperature and velocity of ferrofluids depend strongly on viscosity and thermal conductivity together with magnetic field. The results of the present study when compared concurred with published work.

Unsteady MHD Thin Film Flow of an Oldroyd-B Fluid over an Oscillating Inclined Belt.

This paper studies the unsteady magnetohydrodynamics (MHD) thin film flow of an incompressible Oldroyd-B fluid over an oscillating inclined belt making a certain angle with the horizontal. The problem is modeled in terms of non-linear partial differential equations with some physical initial and boundary conditions. This problem is solved for the exact analytic solutions using two efficient techniques namely the Optimal Homotopy Asymptotic Method (OHAM) and Homotopy Perturbation Method (HPM). Both of these solutions are presented graphically and compared. This comparison is also shown in tabular form. An excellent agreement is observed. The effects of various physical parameters on velocity have also been studied graphically.

Formulation and application of optimal homotopty asymptotic method to coupled differential-difference equations.

In this paper we applied a new analytic approximate technique Optimal Homotopy Asymptotic Method (OHAM) for treatment of coupled differential-difference equations (DDEs). To see the efficiency and reliability of the method, we consider Relativistic Toda coupled nonlinear differential-difference equation. It provides us a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier and explicit.

Heat transfer analysis of MHD thin film flow of an unsteady second grade fluid past a vertical oscillating belt.

This article aims to study the thin film layer flowing on a vertical oscillating belt. The flow is considered to satisfy the constitutive equation of unsteady second grade fluid. The governing equation for velocity and temperature fields with subjected initial and boundary conditions are solved by two analytical techniques namely Adomian Decomposition Method (ADM) and Optimal Homotopy Asymptotic Method (OHAM). The comparisons of ADM and OHAM solutions for velocity and temperature fields are shown numerically and graphically for both the lift and drainage problems. It is found that both these solutions are identical. In order to understand the physical behavior of the embedded parameters such as Stock number, frequency parameter, magnetic parameter, Brinkman number and Prandtl number, the analytical results are plotted graphically and discussed.

Thin film flow in MHD third grade fluid on a vertical belt with temperature dependent viscosity.

In this work, we have carried out the influence of temperature dependent viscosity on thin film flow of a magnetohydrodynamic (MHD) third grade fluid past a vertical belt. The governing coupled non-linear differential equations with appropriate boundary conditions are solved analytically by using Adomian Decomposition Method (ADM). In order to make comparison, the governing problem has also been solved by using Optimal Homotopy Asymptotic Method (OHAM). The physical characteristics of the problem have been well discussed in graphs for several parameter of interest.