There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The present work aims to examine the thermal efficacy of half-sinusoidal nonuniform heating at different spatial frequencies for a porous natural convection system using Cu–Al2O3/water hybrid nanofluid and magnetic field. The system is presented utilizing a classical square enclosure heated nonuniformly at the bottom wall, and the sidewalls are allowed to exchange heat with the surroundings. The Brinkman–Forchheimer–Darcy model is adopted catering other additional terms for buoyant force and magnetic field. The governing equations are transformed into nondimensional forms and then solved numerically using a finite volume-based computing code. The importance and fundamental flow physics are explored in terms of the pertinent parameters such as the amplitude (I) and spatial frequency (f) of half-sinusoidal heating, Darcy–Rayleigh number (Ram), volume fraction of hybrid nanoparticles (
 $$ \phi $$
 ), and Hartmann number (Ha). The flow structure and heat transfer characteristics are analyzed and presented utilizing heatlines, streamlines and isotherms and average Nusselt number. The results show that the use of half-sinusoidal nonuniform heating along with hybrid nanofluid can be a viable method for enhancement and control of the overall thermal performance. The study indicates that half-sinusoidal heating could be a promising technique for better heat transfer even in the presence of flow dampening effects like porous media and magnetic fields.

Magneto-hydrodynamic thermal convection of Cu–Al 2 O 3 /water hybrid nanofluid saturated with porous media subjected to half-sinusoidal nonuniform heating

The intent of this study is to demonstrate an approach for augmenting heat transfer through porous media subjected to nonuniform heating during the magnetohydrodynamic flow of a hybrid nanofluid of Cu–Al2O3/water. The efficacy of such a heating technique is examined utilizing a classical flow geometry consisting of a square cavity. The heating is made at the bottom following a half-sinusoidal function of different frequencies, along with the presence of a uniform magnetic field. The thermal conditions of the cavity, particularly at the bottom wall, drive thermo-hydrodynamics and associated heat transfer. Furthermore, the addition of different types of nanoparticles to the base liquid in order to boost the thermal performance of conventional fluids and mono-nanofluids is a current technique. The coupled nonlinear governing equations are solved numerically in dimensionless forms adapting the finite volume approach, the Brinkman–Forchheimer–Darcy model, local thermal equilibrium and single-phase model. The study is conducted for wide ranges of parametric impacts to analyze global heat transfer performance. The results of this study reveal that the multi-frequency spatial heating during hybrid nanofluid flow can be utilized as a powerful means to improve the thermal performance of a system operating under different ranges of parameters, even with the presence of porous media and magnetic fields. In addition to different heating frequencies, the variations in amplitude (I) and superposed uniform temperature (
 $$\theta_{\text{os}}$$
 ) to half-sinusoidal heating are also examined thoughtfully in the analysis for different concentrations of Cu–Al2O3 nanoparticles. Compared to the base liquid, the hybrid nanofluid can contribute toward higher heat transfer.

Effects of half-sinusoidal nonuniform heating during MHD thermal convection in Cu–Al 2 O 3 /water hybrid nanofluid saturated with porous media

Background Emerging applications in nanomaterials processing are increasingly featuring multiple physical phenomena including magnetic body forces, chemical reactions and high temperature behavior. Stimulated by developing a deeper insight of nanoscale fluid dynamics in such manufacturing systems, in the current article, we study the magnetic nanofluid dynamics along a nonlinear porous stretching sheet with Arrhenius chemical kinetics and wall transpiration. Appropriate similarity transformations are employed to simplify the governing flow problem. Methods The emerging momentum, thermal energy and nanoparticle concentration ordinary differential conservation equations are solved numerically with a hybrid technique combining Successive Linearization and Chebyshev Spectral Collocation. A parametric study of the impacts of magnetic parameter, porous media parameter, Brownian motion parameter, parameters for thermophoresis, radiation, Arrhenius function, suction/injection (transpiration) and nonlinear stretching in addition to Schmidt number on velocity, temperature and nanoparticle (concentration) distribution is conducted. A detail numerical comparison is presented with different numerical and analytical techniques as a specific case of the current investigation. Findings Increasing chemical reaction constant parameter significantly decreases nanoparticle concentration magnitudes and results in a thickening of the nanoparticle concentration boundary layer. Enhancing the values of activation energy parameter significantly increases the nanoparticle concentration magnitudes. Increasing thermophoresis parameter elevates both temperature and nanoparticle concentration. Increasing radiation parameter increases temperature and thermal boundary layer thickness. Enlarging Brownian motion parameter (smaller nanoparticles) and Schmidt number both depress the nanoparticle concentration.

Nonlinear nanofluid fluid flow under the consequences of Lorentz forces and Arrhenius kinetics through a permeable surface: A robust spectral approach

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

In this article, a comprehensive numerical investigation of the natural convection in a hydro-dynamically as well as thermally anisotropic porous enclosure with insulated side walls is presented. T...

Natural convection in a square cavity filled with an anisotropic porous medium due to sinusoidal heat flux on horizontal walls

In the present study, MHD boundary layer flow with heat and mass transfer of a nanofluid with viscous dissipation over a non-linear stretching sheet embedded in a porous medium is studied. The governing non-linear partial differential equations are converted into coupled non-linear ordinary differential equations which are solved by hybrid technique consisting of finite element method with symbolic computation. The effect of Viscous dissipation, magnetic influence parameter, permeability parameter, Brownian motion, Eckert number, Thermophoresis parameter, Prandtl number parameter, Dufour Lewis number and nanofluid Lewis number are studied for velocity $(f'(\eta))$, temperature $(\theta(\eta))$, concentration of salt ($S(\eta)$) and concentration of nanoparticles $(\gamma(\eta))$ and results are shown graphically. Applications of such type of problems are found in the electromagnetic control of complex magnetic nanofluid materials relevant to energy and biomedical systems.

Numerical simulation of MHD boundary layer flow and heat transfer over a nonlinear stretching sheet in the porous medium with viscous dissipation using hybrid approach

Cryosurgery has been consistently used as an effective treatment to eradicate irregular tumor tissues. During this process, many difficulties occur such as intense cooling may also damage the neighboring normal tissues due to the release of large amount of cold from the cooling probe. In order to protect the normal tissues in the vicinity of target tumor tissues, coolant was released in a regulated manner accompanied with the nanoparticle to regulate the size and orientation of ice balls formed together with improved probe capacity. The phase change occurs in the target tumor tissues during cryosurgery treatment. The effective heat capacity method is used for simulation of phase change in bio-heat transfer equation to take into account the latent heat of phase transition. The bio-heat transfer equation is solved by using element free Galerkin method (EFGM) to simulate the phase change problem of biological tissues subject to nano cryosurgery. In this study, Murshed model with cylindrical nanoparticles is used for the high thermal conductivity of nanofluids as compared to Leong Model with the spherical nanoparticle. The important effects of the interfacial layer at the mushy region (i.e. liquid to the solid interface), size and concentration of nanoparticles are shown on the freezing process. This type of problem has applications in biomedical treatment such as drug delivery. Application of cryosurgery in bio-fluids used for drug delivery in cancer therapy can be made more efficient in the presence of nanoparticles (such as Iron oxide ($Fe_{3}O_{4}$), alumina ($Al_{2}O_{3}$) and gold ($Au$)).

Study of bioheat transfer phase change during cryosurgery for an irregular tumor tissue using EFGM

Abstract In this article, a comprehensive study of convective flow over the obstacles (i.e., heaters and freezer) inside the cavity filled with Cu–water nanofluid-saturated porous medium is made. The flow is induced due to heated vertical walls and a cooled top wall, as well as partial attachment of obstacles with the bottom wall. A non-Darcy model that includes Darcy and Brinkman terms has been adopted. The coupled governing equations are solved numerically by the finite element method using the penalty approach. The influence of different controlling parameters, Rayleigh number (Ra), nanoparticle volume fraction ( ), Darcy number (Da), Prandtl number (Pr), aspect ratio (Ar ), and porosity (ε) on the dynamics of flow and heat transfer rate have been addressed. The flow dynamics as well as heat transfer rate have been analyzed with help of streamfunction and temperature contours. Results show that the flow structure (i.e. streamlines) is controlled by nipple shape bicellular structure and the velocity profile includes the point of inflection. The variation of the Nusselt number as a function of the volume fraction of the nanoparticle is almost linear. In contrast to pure fluid, at 10% volume fraction of nanoparticle, the enhancement in heat transfer rate is found up to 30%.

Buoyancy-induced convective flow and heat transfer over obstacles embedded inside a Cu–water nanofluid-saturated porous cavity

Abstract In the present article, a comprehensive numerical investigation of the magneto-convection in an electrically conducting isotropic, and hydro-dynamically as well as a thermally anisotropic porous cavity, is presented. The motion of the flow is due to the applied nonuniform heat flux on the bottom wall of the cavity and the insulation of other walls. The non-Darcy model has been adopted that includes Darcy and Brinkman terms in the momentum equations. The coupled governing equations are solved numerically by using the finite volume method (FVM). The impact of periodicity parameter (N), Hartmann number (Ha), and anisotropy parameters on the dynamics of flow as well as heat transfer rate have been investigated. From rigorous numerical experiments, it has been found that the structure of streamlines is unicellular except for the inner kernel in some situations. In general, the profile of local heat transfer rate possesses points of singularity for sinusoidal heat flux with periodicity parameter (N), and the profile of local Nusselt number is not symmetric for even values of N. The heat transfer rate is highly affected by a relatively large value of thermal conductivity ratio ( ) as well as Ha, and the absolute difference between the maximum and minimum temperature of the system increases as a function of as well as Ha, whereas the strength of flow decreases gradually with increasing Ha.

Harish Chandra

Papers

Natural convection in a square cavity filled with an anisotropic porous medium due to sinusoidal heat flux on horizontal walls

Numerical simulation of MHD boundary layer flow and heat transfer over a nonlinear stretching sheet in the porous medium with viscous dissipation using hybrid approach

Study of bioheat transfer phase change during cryosurgery for an irregular tumor tissue using EFGM

Buoyancy-induced convective flow and heat transfer over obstacles embedded inside a Cu–water nanofluid-saturated porous cavity

Magneto-convection in an anisotropic porous cavity due to nonuniform heat flux at bottom wall