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The thermal analysis of hybrid nano-fluids is a significant research area with diverse applications in industries such as paint, electronics, and mechanical engineering. Existing literature provides limited solutions to the governing equations for the flow of these fluids. Modeling and deriving non-similar solutions for these equations pose interesting and challenging mathematical problems. This study focuses on investigating heat transfer in the flow of two types of nano-fluids, specifically Al2O3/H2O micropolar nano-fluid and Al2O3 + Ag/H2O hybrid nano-fluid, near an isothermal sphere. Conservation laws are employed to formulate the mathematical problem, and by normalizing the variables, the governing equations are converted into a set of dimensionless partial differential equations. Non-similar solutions are then obtained using numerical methods. A comparative analysis is carried out to assess the influence of various parameters on different profiles and engineering quantities for both types of nano-fluids. Both linear and rotational velocities fall down near the surface of sphere with rising microstructure in hybrid nanofluid. The micro-rotation parameter rises the temperature profile while reduces the Nusselt number of both traditional Al2O3/water based nanofluid as well as hybrid nanofluid.

The analysis of heat excess through non-Newtonian fluid is noticed proper and promising arena for biomedical technological applications than Newtonian liquid. Herein, a type of non-Newtonian (Reiner–Rivlin) nanomaterial has been scrutinized with gyrotactic microorganisms around an isothermal sphere with surface of zero mass flux. Being mindful of that the model of Buongiorno’s nanofluid is utilized. Non-dimensional formulas for the basic dimensional partial differential equation system are obtained using appropriate non-similarity transformations. MATLAB function bvp4c code is adopted to get the non-similar solutions. The computations disclosed that the velocity of Reiner–Rivlin nanofluid and its gradient accelerate away from the surface of sphere to infinity and reduces in proximity to the sphere surface for higher values of Reiner–Rivlin factor K. The rate of surface density transport, rate of surface heat transport, and gradient of velocity diminish as the stream wise coordinate boosts. As the problem is subjected to special circumstances, there is a remarkable agreement between the acquired and prior numerical calculations resulting from the comparison.

AbstractA theoretical study is carried out to explore the heat and mass transfer features in the flow of two types nanofluid, i.e., Al2O3/H2O micropolar nanofluid and (Al2O3+Ag)/H2O hybrid nanofluid, adjacent to a solid sphere. Furthermore, simultaneous effects of linear thermal radiation and chemical reaction are also explored on the heat and mass transfer characteristics. In mathematical formulation basic conservation laws are utilized along with the Soret and Doufer effects as a coupling agents in the energy and mass concentration equations. The normalized variables are used to transform the governing system of non-linear partial differential equations into the system of dimensionless partial differential equations. Stream function reduces the number of dependent variables and the resulting system is solved numerically by Keller box method. Several plots are presented to explore the impact of involved parameters on the different profiles and engineering interest quantities for both types of fluids. The temperature profile and Nusselt number are large in magnitude for hybrid nanofluid as compared to Al2O3/water based nanofluid. The heat transfer and mass transfer rates rises with thermal radiation parameter and chemical reaction parameter, respectively. [ABSTRACT FROM AUTHOR]

The goal of this study is to examine the influences of couple stress, the ratio of buoyancy forces, and thermal and solutal relaxation time constraints on two state line sheet flow of nonlinear movement flow of couple stress micropolar nanofluid past the isothermal ball. The mathematical modeling for the flow problem has been made with appropriate likeness change with nondimensional variables. The goal of nonlinear state line value problems were implied in combined high classifies nonlinear ordinary degree of difference equations. The equations were calculated by using the technique bvp4c from Matlab software for numerous quantities of main constraints. Influences of constraints on velocities, thermal, and solutal profiles are displayed from side to side of the charts. The junction test has been continued; for spots larger than apposite web spots, the meticulousness is not affected. Moreover, a comparison with previous papers reachable in the literature has been reported as well as an excellent agreement is presented. The results show that enhancing thermal relaxation time parameter Fo agree to reduce micro-inertia density that improve velocity profile f'(η), temperature distribution θ (η), Nusselt number −θ'(η) and Sherwood number −ϑ′(0), and reduce concentration distribution ϑ(η), coefficient of skin friction and wall couple stress −f″ (0) and h′(0) respectively. [ABSTRACT FROM AUTHOR]

Buoyancy‐determined steady, laminar nonlinear convection flow of micropolar nanofluid with Soret effect, viscous dissipation over a cylinder were evaluated numerically using method bvp4c from matlab software for assorted quantities of leading parameters. Mathematical modeling for the flow problem has been completed with fitting resemblance change and dimensionless variable. A variable similarity solution is offered that bases on various quantities of leading parameters. Influences of these parameters on θ″(0), −h (0), ϑ (0), −ϕ (0) ′′ ′ are tested and displayed with the charts and graphs. The convergence test has been continued; for quantity of spots larger than suitable mesh number of points. The correctness is unchanged, but it takes time. Also, a comparison with prior study reachable in the literature has been offered with very good conformity is got. The findings show that the existence of Soret number agrees to enhance concentration profile ϕ ζ( ), Wall couple stress coefficient −h′(0), θ″(0), and Nusselt number –ϑ′(0). But the presence of Eckert Number Ec, allows declining velocity, microrotation, temperature, and concentration profiles. [ABSTRACT FROM AUTHOR]

Buoyancy-driven convective heat and mass transfer in boundary layer flow of a viscoelastic Jeffrey fluid from a permeable isothermal sphere embedded in a porous medium is studied. Thermal radiation flux and heat generation/absorption are also incorporated in the model. A non-Darcy drag force model is employed to simulate the effects of linear porous media drag and second order Forchheimer drag. The Rosseland diffusion algebraic approximation is utilized to simulate thermal radiation effects. The non-dimensionalized boundary layer equations are solved using implicit, finite-difference scheme. The influence of Darcy number (Da), Deborah number (De), ratio of relaxation to retardation times (λ), radiation parameter (F), Forchheimer inertial parameter (Λ) and heat generation/absorption parameter (Δ), on normalized velocity, temperature, concentration, skin friction, heat and mass transfer rates are also studied. The present study has applications in the storage of nuclear waste materials.

In view of astrophysical applications, we obtain very large scale numerical solutions to the Lane-Emden equations for spherical polytropes of index m  > 5 and for the isothermal sphere ( m = ∞ ), by considering ϕ − 1 , the inverse function of the gravitational potential. Since the domain of ϕ − 1 is bounded, and the asymptotic behavior of its end-point singularities is known, highly accurate solutions can be obtained by spectral collocation methods. This leads to solutions for ϕ that extend accurately to extremely large radii, well beyond those achieved by traditional numerical schemes on [0, ∞). As a reference, we include a table of values for the isothermal sphere (Lane–Emden function of the 2nd kind) spanning r = 10 − 4 through r = 10 150 with at least nine significant figures. The corresponding semi-analytical asymptotic solution as r  → ∞ is ϕ a ( r ) = ln r 2 2 + 1.1779636 ± 1 × 10 − 7 r 1 / 2 cos ( 7 2 ln r + 1.0623271 ± 1 × 10 − 7 ) . [ABSTRACT FROM AUTHOR]

The flow and heat transfer of Casson fluid from a permeable isothermal sphere in the presence of slip condition in a non-Darcy porous medium is analyzed. The sphere surface is maintained at a constant temperature. The boundary layer conservation equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite-difference scheme. Increasing the velocity slip parameter is found to decrease the velocity and boundary layer thickness and increases the temperature and the boundary layer thickness. The velocity decreases with the increase the non-Darcy parameter and is found to increase the temperature. The velocity increases with the increase the Casson fluid parameter and is found to decrease the temperature. The Skin-friction coefficient and the local Nusselt number is found to decrease with the increase in velocity and thermal slip parameters respectively.

The gravitational instability, responsible for the formation of the structure of the Universe, occurs below energy thresholds and above spatial scales of a self-gravitating expanding region, when thermal energy can no longer counterbalance self-gravity. I argue that at sufficiently-large scales, dark energy may restore thermal stability. This stability re-entrance of an isothermal sphere defines a turnaround radius, which dictates the maximum allowed size of any structure generated by gravitational instability. On the opposite limit of high energies and small scales, I will show that an ideal, quantum or classical, self-gravitating gas is subject to a high-energy relativistic gravothermal instability. It occurs at sufficiently-high energy and small radii, when thermal energy cannot support its own gravitational attraction. Applications of the phenomenon include neutron stars and core-collapse supernovae. I also extend the original Oppenheimer–Volkov calculation of the maximum mass limit of ideal neutron cores to the non-zero temperature regime, relevant to the whole cooling stage from a hot proto-neutron star down to the final cold state.

This article investigates the nonlinear, steady boundary layer flow and heat transfer of an incompressible Eyring-Powell non-Newtonian fluid from an isothermal sphere with Biot number effects. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The influence of a number of emerging dimensionless parameters, namely the Eyring-Powell rheological fluid parameter (ε), the local non-Newtonian parameter based on length scale (δ), Prandtl number (Pr), Biot number (γ) and dimensionless tangential coordinate (ξ) on velocity and temperature evolution in the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. It is found that the velocity and heat transfer rate (Nusselt number) decrease with increasing (ε), whereas temperature and skin friction increase. An increasing (δ) is observed to enhance velocity, local skin friction and heat transfer rate but reduces the temperature. An increase (γ) is seen to increase velocity, temperature, local skin friction and Nusselt number. The study is relevant to chemical materials processing applications. [ABSTRACT FROM AUTHOR]

Buoyancy-driven convective heat and mass transfer in boundary layer flow of a viscoelastic Jeffrey fluid from a permeable isothermal sphere embedded in a porous medium is studied. Thermal radiation flux and heat generation/absorption are also incorporated in the model. A non-Darcy drag force model is employed to simulate the effects of linear porous media drag and second order Forchheimer drag. The Rosseland diffusion algebraic approximation is utilized to simulate thermal radiation effects. The non-dimensionalized boundary layer equations are solved using implicit, finite-difference scheme. The influence of Darcy number ( Da ), Deborah number ( De ), ratio of relaxation to retardation times ( λ ) , radiation parameter ( F ), Forchheimer inertial parameter ( Λ ) and heat generation/absorption parameter ( Δ ) , on normalized velocity, temperature, concentration, skin friction, heat and mass transfer rates are also studied. The present study has applications in the storage of nuclear waste materials. [ABSTRACT FROM AUTHOR]

Chandra's academic life had several phases each culminating in a monograph describing that subject. I shall deal with aspects of his work in the two earliest phases. I shall describe the overall structure of statistical mechanics of gravitating systems, the relevance of isothermal sphere in the mean-field approximation and issues related to collisional relaxation and dynamical friction in self-gravitating system of particles. There are several curious features in the history of these topics which I comment upon. [ABSTRACT FROM AUTHOR]

Abstract: This study aims to reveal the temporal and spatial heat transfer performance and the transition process from a steady axisymmetric separated flow to a chaotic flow of an isothermally heated sphere placed in a uniform, downwardly directed flow by experimentation and transient 3D numerical simulation. Experiments were carried out using air with a Grashof number of 3.3×105 and with several Reynolds numbers up to 1900. Four types of flow patterns appear showing that the chaotic flow around the sphere is a spatiotemporal chaos by the calculation of a set of Lyapunov exponents from the calculated time series of temperatures. [Copyright &y& Elsevier]

Abstract: The full Navier–Stokes equations and the energy equation for laminar natural convection heat transfer over an isothermal sphere have been discretized using the finite control volume formulation and solved by employing the SIMPLEC method. Transient and “steady-state” results have been obtained for a wide range of high Grashof numbers (105 ⩽ Gr ⩽109) and a wide range of Prandtl numbers (Pr =0.02, 0.7, 7 and 100). Main results are listed below. A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The upward movement of the plume cap is slowed as the Prandtl number increases. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. The time at which the “steady-state” is reached, increases with the Prandtl number. The presence of a vortex in the wake of the sphere has been predicted and has been clearly delineated as a function of both Grashof and Prandtl numbers. The overall Nusselt numbers and total drag coefficients for the range of Grashof and Prandtl numbers investigated are presented and they are in very good agreement with studies available in the literature. [Copyright &y& Elsevier]

In this article, it is found an analytical expression for temporal retard using the isothermal sphere models with nucleus (ΔtEIN). This expression is applied or assigned to any quasar system of lensing by galaxies particularly the temporal retard is applied to the SSO 0218+357 system, using the observational values measured in the mentioned system and setting the cosmological parameters move acceptable in the literature. Because of this the temporal retard just, remains in function of the nuclear radius. Theoretically when the nuclear radius is zero, the temporal retard with nucleus (ΔtEIN), must coincide with the temporal retard of the Singular Isothermal Sphere model (ΔtEIN) in feet must be accod with the value of the temporal retard measured observationally in the mentioned system (10.1 days). It is about analyzing the sensitivity of the temporal retard with respect to the nuclear radius of the deflector galaxy. [ABSTRACT FROM AUTHOR]

The steady mixed convection boundary layer flow of a micropolar fluid about a sphere with a constant surface temperature is considered for both the assisting and opposing flow cases. The transformed conservation equations of the non-similar boundary layers are solved numerically using a very efficient finite-difference method known as the Keller-box scheme. Numerical results are presented for different values of the material and mixed convection parameters K and λ, respectively, and with the Prandtl number Pr=0.7 and 7. It is found that assisting flow (λ>0) delays separation of the boundary layer and can, if the assisting flow is strong enough, suppress it completely. The opposing flow (λ<0), on the other hand, brings the separation point nearer to the lower stagnation point of the sphere and for sufficiently strong opposing flows there will not be a boundary layer on the sphere. Some results were given in the form of tables. Such tables are very important and they can serve as a reference against which other exact or approximate solutions can be compared in the future. [Copyright &y& Elsevier]