Non-similar Solutions Research Articles

Scrutinization of 2D and mixed convection flow of generalized Newtonian fluid with nanoparticles and magnetic field

In this research paper we focus on presenting the local non-similar solutions for two-dimensional steady and mixed convective flow of electrically conducting Carreau nanofluid under the influence of Brownian motion and thermophoresis effects. This research paper also presents the new mass flux boundary condition of nanoparticles. The governing partial differential equations are converted into ordinary differential equations by using the local non-similar transformations called the Sparrow–Quack–Boerner local non-similar method. A numerical approach is used to investigate the local non-similar solutions of the entire problem. Outcomes for the skin friction and rate of heat and mass transfer have been obtained and discussed for parametric variations of the buoyancy parameter ξ, magnetic parameter M, Weissenberg number We, viscosity ratio parameter β*, Brownian motion parameter Nb, thermophoresis parameter Nt, Prandtl number Pr, and Schmidt number Sc. From the obtained results a considerable increase in the skin friction coefficient is observed with an increase in buoyancy parameter ξ. Also, ξ shows relative growth for the momentum and concentration profiles.

Mixed Convective Flow of Micropolar Nanofluid across a Horizontal Cylinder in Saturated Porous Medium

The micropolar nanofluids are the potential liquids that enhance the thermophysical features and ability of heat transportation instead of base liquids. Alumina and Titania nanoparticles are mixed in a micropolar fluid. The impact of convective boundary condition is also examined with assisting and opposing flows of both nanofluids. The main objective of this study is to investigate mixed convective flow and heat transfer of micropolar nanofluids across a cylinder in a saturated porous medium. Non-similar variables are used to make the governing equations dimensionless. The local similar and non-similar solutions are obtained by using the Runge-Kutta-Fehlberg method of seventh order. The impacts of various embedded variables on the flow and heat transfer of micropolar nanofluids are investigated and interpreted graphically. It is demonstrated that the skin friction and heat transfer rates depend on solid volume fraction of nanoparticles, Biot number, mixed convection, and material parameters.

Transport phenomena in MHD mixed convective nanofluid flow

Purpose This study aims to investigate the unsteady magnetohydrodynamic mixed convective nanofluid flow by using Buongiorno two-phase model to achieve an appropriate mechanism to improve the efficiency of solar energy systems by mitigating the energy losses. Design/methodology/approach The transport phenomena occurring in this physical problem are modelled using nonlinear partial differential equations and are non-dimensionalised by using non-similar transformations. The quasilinearisation technique is used to solve the resulting system with the help of a finite difference scheme. Findings The study reveals that the effect of the applied transverse magnetic parameter is to increase the temperature profile and to reduce the wall heat transfer rate. The Brownian diffusion and thermophoresis parameters that characterise the nanofluids contribute to the reduction in wall heat transfer rate. The presence of nanoparticles in the fluid gives rise to critical values for the thermophoresis parameter describing the behaviour of the wall heat and mass transfer rates. Wall heating and cooling are analysed by considering the percentage increase or percentage decrease in the heat and mass transfer rates in the presence of nanoparticles in the fluid. Research limitations/implications The investigation on wall cooling/heating leads to the analysis of control parameters applicable to the industrial design of thermal systems for energy storage, energy harvesting and cooling applications. Practical implications The analysis of the control parameters is of practical value to the solar industry. Social implications In countries, such as South Africa, daily power cuts are a reality. Any research into improving the quality of energy obtained from alternate sources is a national necessity. Originality/value From the literature survey in the present study, it is found that no similar work has been reported in the open literature that analyses the time-dependent mixed convection flow along the exponentially stretching surface in the presence of the effects of a magnetic field, nanoparticles and non-similar solutions.

Numerical study of unsteady boundary layer flow and heat transmission of a nano-fluid past a continuously stretching sheet immersed in porous media

PurposeThis paper aims to incorporate a numerical investigation of unsteady flow of nano-fluid near the boundary wall and heat transmission over a flat sheet in porous media under the assumption that sheet is continuously stretching. A comprehensive study is accomplished to probe the impacts of different physical parameters on heat transmission and fluid flow.Design/methodology/approachThe problem is designed mathematically by using Navier–Stokes equations and corresponding equations are transmuted into non-dimensional form, which are reduced to local non-similarity equations. Implicit finite difference scheme is used to attain a non-similar solution for values of similarity variable ranging in [0,1]FindingsThe reduced Nusselt number is a decreasing function of each dimensionless number, while the reduced Sherwood number is an increasing function of each considered value of parameters Pr, Sc, Nb and Nt.Originality valueMotivated by the industrial, technological and scientific importance, the objective of this study is to discuss the unsteady flow generated by linear stretching of the surface in Nano-fluid in a porous medium. However, the investigation of unsteady boundary layer flow and heat transfer of a Nano-fluid past a stretching surface immersed in a porous medium has not been discussed yet. Furthermore, finite difference scheme is used to solve the problem, and authenticate results are achieved which are in good agreement with published work.

Propagation of strong cylindrical shock wave in a self-gravitating rotational axisymmetric mixture of small solid particles and perfect gas with density varying exponentially

The non-similarity solution for the propagation of a strong cylindrical shock wave in a self-gravitating and rotational axisymmetric dusty gas, having variable azimuthal and axial fluid velocities is obtained. The dusty gas is assumed to be a mixture of small solid particles of micro size and perfect gas. The equilibrium flow conditions are assumed to be maintained. The density of the mixture and the fluid velocities in the ambient medium are assumed to be varying and obeying an exponential law. The shock wave moves with variable velocity and the total energy of the wave is non-constant. The effects of variation of the mass concentration of solid particles in the mixture, the ratio of the density of solid particles to the initial density of the gas, and the gravitational parameter on the flow variables in the region behind the shock are investigated at a given time in both the rotating and non-rotating cases. Our analysis reveals that an increase in gravitational parameter or in the ratio of the density of solid particles to the initial density of the gas surprisingly the shock strength increases and remarkable differences are found in the distribution of flow variables in both the rotating and non-rotating cases. An increase in time also, increases the shock strength in the case of isothermal flow, but to decrease the shock strength in the case of adiabatic flow in general. Further, it is investigated that the consideration of isothermal flow increases the shock strength. A comparison is made between the solutions in isothermal and adiabatic flow cases with or without gravitational effects in both the rotating and non-rotating medium. The obtained solutions are applicable for arbitrary values of time.

Non-similar solutions of the boundary layers equations with favorable and adverse pressure gradients, isothermal wall and slip boundary conditions: Application to Falkner–Skan gaseous flow

Abstract This article presents the non-similar solutions of the steady incompressible two-dimensional boundary layers equations of momentum and energy for Falkner–Skan gaseous flow with favorable and adverse pressure gradient under slip boundary conditions at a relatively low Mach number. By considering the slip boundary conditions at the fluid–wallinterface for the velocity and temperature, the non-dimensional boundary layer equations are solved numerically by using the centered implicit finite difference Keller Box methods. The results of the effects of different dimensionless parameters involved in the problem, namely the modified boundary layer Knudsen number, i.e., the slip parameter K and the shape parameter β on the local hydrodynamic and heat transfer characteristics of the flow are investigated and discussed. The results of these characteristics obtained from non-similar solutions are compared to those obtained from local similarity approach often used by several authors in the last decades. We show that for small shape parameter β and moderate or large values of the slip parameter, the self-similarity of the Falkner–Skan flows is generally lost because of slip boundary conditions. Therefore, the use of the local similarity approach may produce substantial errors for hydrodynamic and thermal characteristics of the flow as compared to non-similar solution. But in contrast, when β becomes larger than 2/3 up to 1, the flow regains its self-similarity character, in this case, the local similarity approach remains valid and may be used. While for β > 1 up to 2, the solutions of the boundary layer equations again become non-similar. Furthermore, a special attention was made to evaluate the separation point under rarefaction, the results show that the effect of slip parameter leads to decrease the separation point β s and it is concluded that separation of the boundary layer should be delayed in the slip flow regime.

Locally non-similar and thermally radiative Sisko fluid flow with magnetic and Joule heating effects

Abstract The locally non-similar solutions of Sisko fluid for boundary layer flow in the presence of thermal radiation and magnetic field have been presented. Further, heat transfer analysis is carried out in the presence of Joule heating. Three equations model for Sisko fluid flow and heat transfer over a moving surface is modeled while using Sparrow-Quack-Boerner local non-similarity method (LNSM). The numerical computation is performed to explore the impacts of different physical parameters which are responsible for controlling the flow and heat transfer. Specifically, the impact of magnetic field introducing a decreasing velocity profile and force balance boundary layer thickness and increasing the temperature profile as well as thermal boundary layer thickness, respectively for shear-thinning as well as shear-thickening fluids. However, the growing parallel values of material parameter ( A ) and streamwise coordinate ( ξ ) increases the velocity profile significantly for both pseudo plastic ( 0 n 1 ) fluid as well as dilatant ( n > 1 ) fluid. For the same parameters, an opposite trend is noticed for the temperature distribution. Additionally, with a fixed value of streamwise coordinate along with the increasing variation in Eckert number ( Ec ) and radiation parameter ( R d ) , the temperature profile is founded with enhancement conduct. Comparisons in the form of local skin friction (resistive forces) and the local Nusselt number (rate of heat transfer) are determined in excellent correlation with the previous published results in the limiting cases.

Bioconvection in Casson nanofluid flow with Gyrotactic microorganisms and variable surface heat flux

This paper presents a two-dimensional unsteady laminar boundary layer mixed convection flow heat and mass transfer along a vertical plate filled with Casson nanofluid located in a porous quiescent medium that contains both nanoparticles and gyrotactic microorganisms. This permeable vertical plate is assumed to be moving in the same direction as the free stream velocity. The flow is subject to a variable heat flux, a zero nanoparticle flux and a constant density of motile microorganisms on the surface. The free stream velocity is time-dependent resulting in a non-similar solution. The transport equations are solved using the bivariate spectral quasilinearization method. A grid independence test for the validity of the result is given. The significance of the inclusion of motile microorganisms to heat transfer processes is discussed. We show, inter alia, that introducing motile microorganisms into the flow reduces the skin friction coefficient and that the random motion of the nanoparticles improves the rate of transfer of the motile microorganisms.

Non-similar Solutions of MHD Mixed Convection over an Exponentially Stretching Surface: Influence of Non-uniform Heat Source or Sink

In this paper, an analysis of magnetohydrodynamic (MHD) mixed convection over an exponentially stretching surface in the presence of a non-uniform heat source/sink and suction/injection is presented. The governing boundary layer equations are transformed into a set of non-dimensional equations by using a group of non-similar transformations. The resulting highly non-linear coupled partial differential equations are solved by using the implicit finite difference method in combination with the quasilinearization technique. Numerical results for the velocity, temperature and concentration profiles, as well as the skin friction coefficient, wall heat transfer and mass transfer rates are computed and presented graphically for various parameters. The results indicate that the velocity profile reduces, while the temperature profile increases in presence of the effects of magnetic field and suction at the wall. The velocity ratio parameter increases the skin-friction coefficient and the Schmidt number decreases the wall mass transfer rate. The temperature profile increases for the positive values of Eckert number and space as well as temperature dependent heat source/sink parameters, while the opposite behavior is observed for negative values of same parameters.

Impact of enhancing diffusion on Carreau–Yasuda fluid flow over a rotating disk with slip conditions

The present study is devoted to acquire non-similar solutions for the behavior of slip conditions on the steady MHD Carreau–Yasuda fluid flow over a rotating disk. In order to examine the heat transfer phenomena, superior form of Fourier’s law is used and the conductivity of the fluid is assumed to be changeable. The nonlinear partial differential equations leading the flow and thermal field are written in the non-dimensional ordinary differential form by using suitable transformations. The non-dimensional set of coupled ordinary differential equations is solved using the RK method. The impact of various non-dimensional physical parameters on velocity and temperature fields is explored. The numerical results of resistant force in terms of the skin friction coefficient are revealed graphically for various physical parameters involved in the problem.

Local non-similar solutions of convective flow of Carreau fluid in the presence of MHD and radiative heat transfer

In this research article we studied a realistic methodology to inspect the non-similar solutions for the two-dimensional steady Carreau fluid flow in the presence of applied magnetic field and mixed convection within the sight of infinite shear rate viscosity. With the help of local non-similar method, we presented the nonlinear PDEs for the flow and heat transfer analysis. The leading PDEs are converted into an system of nonlinear ODEs by using the local non-similarity method. The final resulting non-dimensional set of coupled nonlinear PDEs is then solved with the help of bvp4c function in MATLAB. This investigation discover numerous physical aspects of flow and heat transfer. Major outcomes in the form of velocity enhancement and temperature reduction for the higher values of buoyancy parameter $$\left( \xi \right)$$ are observed. On the other hand, for increasing values of $$N_{\mathrm{R}}$$ the temperature of the fluid increases, while for larger values of suction/injection parameter the temperature of the fluid reduces. Parallel variation of buoyancy parameter and Weissenberg number shows a slight difference regarding local similar and local non-similar solution while computing the local skin friction number. The enhancement in buoyancy parameter causes enhancement in local skin friction as well as local Nusselt number. Additionally this investigation is validated through a comparison with previous results and found a good correlation with the previous results.

A Linear Threshold for Uniqueness of Solutions to Random Jigsaw Puzzles

We consider a problem introduced by Mossel and Ross (‘Shotgun assembly of labeled graphs’, arXiv:1504.07682). Suppose a random n × n jigsaw puzzle is constructed by independently and uniformly choosing the shape of each ‘jig’ from q possibilities. We are given the shuffled pieces. Then, depending on q, what is the probability that we can reassemble the puzzle uniquely? We say that two solutions of a puzzle are similar if they only differ by a global rotation of the puzzle, permutation of duplicate pieces, and rotation of rotationally symmetric pieces. In this paper, we show that, with high probability, such a puzzle has at least two non-similar solutions when 2 ⩽ q ⩽ 2e−1/2n, all solutions are similar when q ⩾ (2+ϵ)n, and the solution is unique when q = ω(n).

The magnetohydrodynamic wall jets: Techniques for rendering similar and perturbative non-similar solutions

In the present work, it is given extensive analytic analysis on the magnetohydrodynamic (MHD) wall jets for the 1st time in the history. It is hopeful that this work provides a careful ground for further studies in this field. The main findings are summarized as: If u(x,0)=v(x,0)=u(x,y→∞)=0, then there will be no similarity solution of either algebraically or exponentially decaying type for a jet discharged over a plane surface in the presence of a transverse magnetic field. The proof is given via various analytic approaches together with a numeric validation. In particular, the conserved vector associated with the multiplier Λ=Λ(x,y,ψ,ψx,ψy)does not yield a conserved quantity accounting the above boundary conditions for the MHD wall jet. However, similarity solutions may exist in the context of Glauert type e-jets withu(x,0)∝x−12,v(x,0)∝x−34,u(x,y→∞)=0being connected to f′(0)=β,f(0)=α,f′(η→∞)=0 in similarity terms. The cases with α=0,β≠0 and β=0,α≠0are considered in the present paper. For the 1st case, it is found that a certain magnetic parameter is connected to dual solutions being representative of stretchable/shrinkable wall. It should be mentioned that for this case, some primary analysis have been recently performed by the present author in [27] and here, this part has been included mainly for the consistency purpose and the focus has been given to the stability analysis of the associated dual solutions. Spatial stability analysis revealed that the solutions associated with the stretchable wall are physically stable whilst the solutions associated with the shrinkable wall are stable only for very small magnetic parameters. For the 2nd case, it is given an explicit proof that upon the assumptions off′(η)≥0,f(η→∞)>0 and f″(0)>0 the Glauert type solutions exist if only−f(η→∞)<α<0, in particular implying that no suction assumption is allowed. As a consequence of the present findings, the MHD wall jet with u(x,0)=v(x,0)=u(x,y→∞)=0 was treated as non-similar flows. A perturbative treatment was accounted for the stream function, eventually leading to physically-acceptable solutions for this case. The following Table shows an overview of the original contributions of the present work to the existing literature. u(x,0)=v(x,0)=u(x,y→∞)=0 (The Original MHD Wall Jet)u(x,0)=0,v(x,0)∝x−34, u(x,y→∞)=0u(x,0)∝x−12,v(x,0)=0, u(x,y→∞)=0✓Proofs on which that No similarity solutions of the form of a-jet/e-jet exist.✓Similarity solutions: Explicit proof on which that only Injection is allowed upon setting some normal assumptions.✓Similarity Solutions: dual solutions exist associated with stretchable/shrinkable wall [27].✓Presenting Non-similar perturbative solution.✓Presenting an extensive stability analysis.

Computational Modelling and Solutions for Mixed Convection Boundary Layer Flows of Nanofluid from a Non-Isothermal Wedge

Buoyancy-driven (“Falkner-Skan”) mixed convection flow of a nanofluid from a nonisothermal wedge has been investigated numerically using an implicit finite difference scheme. The model used for nanofluid includes the effects of Brownian motion, buoyancy ratio and thermophoresis. A non-similarity solution is developed which depends on the Prandtl number (Pr), Schmidt number (Sc), Brownian motion number (Nb), buoyancy ratio parameter (Nr), thermophoresis number (Nt), mixed convection parameter (λ) and pressure gradient parameter (m). The variation of the skin friction, the reduced heat transfer rate and reduced mass transfer rate with Nb, Nr and Nt for various values of Pr, m and Sc are also tabulated.

Minimization of Entropy Generation in MHD Mixed Convection Flow with Energy Dissipation and Joule Heating: Utilization of Sparrow-Quack-Boerner Local Non-Similarity Method

This article aims to present the non-similar solution of MHD mixed convection flow using the Sparrow-Quack-Boerner local non-similarity method. Entropy analysis is also performed in the presence of energy dissipation and Joule heating. The buoyancy parameter is chosen as the non-similarity variable and the equations are derived up to the second level of truncation. The dependency of dimensionless velocity profile, temperature distribution, Bejan and entropy generation number on physical parameters has been discussed. As far as the knowledge of the authors is concerned, no attempt has been made on the entropy analysis of MHD mixed convection flow by the local non-similarity method.

Numerical simulation of unsteady triple diffusive mixed convection in NaCl-water and Sucrose-water solutions

The consequences of the presence of triple diffusive phenomenon on mixed convection from a continuously moving vertical plate are investigated in this paper under variations in time. The behavior of aqueous solutions, such as NaCl-water and Sucrose-water, are modelled and analyzed via a coupled system of nonlinear partial differential equations which are transformed into a dimensionless coupled system of nonlinear partial differential equations with the aid of non-similar transformations. The Quasi-linearization technique in association with the implicit finite difference scheme is employed to obtain non-similar solutions. The graphs acquired for momentum, thermal and species concentration fields involving the effects of the Richardson number, velocity ratio parameter, ratio of buoyancy parameter and Schmidt number of both solutal components have been depicted in a manner which assists in providing a physical interpretation of the results. The graphical interpretations for the skin-friction coefficient, Nusselt number and Sherwood number for both the aqueous solutions are also scrutinized and are discussed with proper physical explanations.

Influence of non-Darcy and MHD on free convection of non-Newtonian fluids over a vertical permeable plate in a porous medium with soret/dufour effects and thermal radiation

This study investigated the effects of non-Darcy (Forchheimer equation) and magnetohydrodynamic (MHD) forces on free convection of non-Newtonian fluids over a vertical permeable plate in a porous medium with Soret/Dufour effects and thermal radiation. The vertical plate surface has a uniform wall temperature and uniform wall concentration (UWT/UWC). Non-similar solutions for the transformed governing equations are obtained by Keller box method. Comparisons showed excellent agreement with numerical data reported in previous works. Numerical data entered in tables or plotted in figures for the dimensionless temperature profile, the dimensionless concentration profile, the local Nusselt number and the local Sherwood number are presented for the main parameters: the non-Darcy parameter ND, the magnetic field parameter M, the power-law index of the non-Newtonian fluid n, the blowing/suction parameter ξ and the thermal radiation parameter Rd. The numerical results show that increasing the non-Darcy parameter ND or the magnetic field parameter M tends to decrease both the local Nusselt number and the local Sherwood number. Additionally, increasing the thermal radiation parameter Rd increases both the local Nusselt number and the local Sherwood number. Generally, for the case of suction, both the local Nusselt number and the local Sherwood number increase. The opposite trend occurs in the case of blowing fluid.

FREE CONVECTION HEAT AND MASS TRANSFER OF A NANOFLUID PAST A HORIZONTAL CYLINDER EMBEDDED IN A NON-DARCY POROUS MEDIUM

In the present paper, we analyzed the laminar boundary layer flow and heat transfer from a horizontal cylinder in a nanofluid-saturated non-Darcy porous medium in the presence of thermal radiation. This is the first paper presenting non-similar solutions for such a regime.The boundary layer conservation equations,which are parabolic in nature,are normalized into non-similar form and then solved computationally with an efficient, implicit, stable Keller-box finite difference scheme. Non-Darcy effects are simulated via a second-order Forchheimer drag force term in the momentum boundary layer equation. The model used for the nanofluid incorporates the effects of Brownian motion, buoyancy ratio, and thermophoresis. A non-similarity solution is presented that depends on the Brownian motion number (Nb), buoyancy ratio (Nr), thermophoresis number (Nt), Forchheimer parameter (Λ), and radiation parameter (F). Velocity is reduced with increasing Forchheimer parameter, whereas temperature and nanoparticle concentration are both enhanced.The model finds applications in energy systems and thermal enhancement of industrial flow processes

Unsteady MHD Mixed Convection Flow with Slip of a Nanofluid in the Stagnation Region of an Impulsively Rotating Sphere with Effects of Thermal Radiation and Convective Boundary Conditions

Unsteady magnetohydrodynamic mixed convection flow of an electrically conducting nanofluid in a stagnation region of a rotating sphere is studied numerically in the present article. Slip and convective boundary conditions are imposed to surface of the sphere and the thermal radiation effects are taken into account. The nanofluid is simulated using Buongiorno’s nanofluid model and the nanofluid particle fraction on the boundary is considered to be passively rather than actively controlled. Non-similar solutions are applied on the governing equations and the MATLAB function bvp4c is used to solve the resulting system. Effects of the key-parameters such as slip parameter, Biot number, radiation parameter, rotation parameter, Lewis number and Brownian motion parameter on the fluid flow, temperature and nanoparticle volume fraction characteristics are examined. Details of the numerical solution and a comprehensive discussion with the physical meaning for the obtained results are performed. The results indicated that the increase in slip parameter enhances the velocity profiles, while it decreases the temperature distributions. Also, the increase in either slip parameter or Biot number causes an improvement in the rate of heat transfer.

Nonsimilarity boundary layer flow of copper–water nanofluid near stagnation point in a porous medium

AbstractThe effects of nanoparticle shape are first introduced to study the nonsimilar solutions of stagnation point boundary layer flow of water–copper nanofluid saturated in a porous medium. Two cases of solid matrix of porous medium, including glass balls and aluminum foam, are considered. By using a new empirical correlation for the heat capacitance, thermal conductivity, and thermal diffusivity of the nanofluid saturated in a porous medium, the governing equations of the problem are constructed and reduced by dimensionless variables and nonsimilar transformations, and the homotopy analysis method is adopted to solve the partial differential equations. The results indicate that the heat transfer is significantly enhanced with the increase of permeability of the porous medium on the surface of the stagnation point boundary layer flow. In addition, it is found that the empirical shape of the nanoparticle has an impact on the heat transfer.