Flow over rotating spheres has serious applications in fiber coating, rotating machinery design and parts, and projectile missions. Therefore, the current study investigates the flow and entropy generation of a radiative magneto-hybrid nanofluid flow over rotating sphere. Further, the temperature-sensorial water properties in studying water-based nanofluids are ignored, though this erratic behavior of water characteristics influences the physical characteristics of the corresponding hybrid nanofluid. Hence the current framework is one of the foremost projects to introduce the variable nature of water properties in the water-hosted hybrid nanofluid flow analysis. The investigation is accomplished in the account of the modified Buongiorno model (MBM). The flow separation in this kind of flow geometry is controlled by using non-erratic slot-mass transfer. The mathematical representations of the physical principles of the flow are solved using (i) congenial transformation (ii) quasilinearization (iii) methods of finite differences to form a block matrix system, and (iv) Varga's iterative algorithm. Some of the major outcomes are: The flow separation can be significantly delayed using slot suctions, precisely appointing them downstream with higher strengths; heat transport performance drastically subsidies for high viscous dissipation (Ec), entropy produces at higher rate for viscous heating (Br) and high angular speed (Ω).

This research study discusses the flow of a magnetohydrodynamic Casson fluid under the influence of Soret, Dufour, and thermal radiation. Nonlinear partial differential equation (PDE) of governing equations is transformed into a dimensionless version of the modified PDEs presented in terms of dimensionless parameters. The solution of coupled PDEs is obtained by the finite difference method with a combination of the quasilinearization technique. The effects of various dimensionless parameters are shown graphically, such as buoyancy force (lambda), concentration buoyancy force (lambda(C)) , Casson parameter (beta), magnetic parameter (H), thermal radiation (Rd), Darcy parameter (K-0), Forchheimer (fr), Dufour (D-f), Soret (Sor), Brownian motion (Nb), thermopohersis (Nt), and Lewis number (Le). Prevention of heat transfer in the industrial system is critical, the velocity behavior (F), thermal variation (theta), and concentration profile (phi) are more prominent in the roles of coal, gas, and solar thermal collectors.

In this article, we consider the discretization of nonlocal coupled parabolic problem within the framework of the virtual element method. The presence of nonlocal coefficients not only makes the computation of the Jacobian more expensive in Newton’s method, but also destroys the sparsity of the Jacobian. In order to resolve this problem, an equivalent formulation that has very simple Jacobian is proposed. We derive the error estimates in the L^2 and H^1 norms. To further reduce the computational complexity, a linearized scheme without compromising the rate of convergence in different norms is proposed. Finally, the theoretical results are justified through numerical experiments over arbitrary polygonal meshes.

The present work focus on water boundary layer flow over an exponential permeable stretching sheet in the presence of suction/injection with variable viscosity and prandtl number. The nonlinear partial differential equations governing flow and thermal fields are presented in non-dimensional form using suitable non-similar transformation. Finally non dimensional partial differential the equations are solved by the implicit finite difference method in combination with the Quasi-linearization technique. The numerical results for skin-friction and local Nusselt number are shown graphically to display effects of physical parameters.

Purpose The purpose of the present study is to analyze the mixed convection water boundary layer flows over moving vertical plate with variable viscosity and Prandtl number. The non-linear partial differential equation governing the flow and thermal fields are presented in non-dimensional form by using appropriate transformation. The quasi-linearization technique in combination with implicit finite difference scheme has been adopted to solve the nonlinear-coupled partial differential equation. The numerical results are displayed graphically to illustrate the influence of various non-dimensional physical parameters on velocity and temperature. Further, the numerical results for local skin-friction coefficient and local Nusselt number are also reported. The present findings are compared with previously reported results, and these comparisons are found to be in excellent agreement. Design/methodology/approach The nonlinear partial differential equations governing the flow and thermal fields have been solved numerically using the implicit finite difference scheme in combination with the quasi-linearization technique. The numerical results are presented in terms of skin friction and heat transfer rate which are useful in determining the surface heat requirements for stabilizing the laminar boundary layer flow over a moving plate in water. Findings The effect of the ratio of free-stream velocity to the composite reference velocity is significant on the velocity profile. Near the wall region, as ratio of free stream velocity to composite reference velocity increases form 0.1 to 0.5, the velocity overshoot gets enhanced from 3 per cent to 41 per cent. The influence of buoyancy parameter and ration of free stream velocity to composite reference velocity on temperature profile is comparatively less than on velocity profiles. The increase in the skin friction coefficient is dependent on the increase in the value of ratio of free stream velocity to composite reference velocity if the buoyancy parameter λ is fixed and vice versa and increases in ΔT results in a decrease in N and Pr. Originality/value The present investigation is to deal with the solution of steady laminar water boundary layer flows over a moving plate with temperature-dependent viscosity and Prandtl number applicable for water using practical data. The fluid considered here is water, as it is one of the most common working fluids found in engineering applications.

Galilean invariant vortex identification methods and proper orthogonal decomposition (POD) are used to capture the coherent structures (CS) in the flow field generated by rigid wing in main flapping motion. Experiments are conducted for a square rigid wing at a flapping frequency f = 1.5 Hz. Tests are performed in water as a fluid medium in hover mode. The main flapping mechanism executes asymmetric lower-upper stroke of 1:3 ratio single degree of freedom motion. Two dimensional particle image velocimetry (PIV) technique is used to generate the phase locked velocity field at each discrete flapping angle by illuminating the mid chord plane of the wing. Three different Galilean invariant methods namely λ 2 criterion, Q criterion and Δ criterion are used for vortex identification. These methods were found to be consistent in detecting swirling vortices or coherent structures (CS). Proper orthogonal decomposition is used to exhibit the most energetic modes of the flow field. The captured modes were identified to be in-connect with vortex identification methods. The combination of these tools were found to be more effective in comparison with velocity field data for achieving a deeper understanding of the complex flow produced by the flapping wing.

A comprehensive analysis based on the irreversibilities associated with the energy flow and entropy generation is highly essential for the optimization of thermal systems. Entropy generation during mixed convection process has been studied in entrapped triangular cavities for moving horizontal walls involving isothermally hot inclined walls and cold horizontal walls (case 1) or isothermally cold inclined walls and hot horizontal walls (case 2). Overall it is found that, Re = 100 may be preferred over Re → 0, Re = 1 and Re = 10 at Pr = 0.026 and 7.2, Gr = 103 − 105 within the cavities, irrespective of the cases. In addition to Re = 100, Re = 10 may be optimal for the upper cavity with case 1 and lower cavity with case 2 at Gr ≈ 105 (higher Gr regime) and Pr = 7.2 based on moderate heat transfer rates.

This paper presents a numerical investigation of the steady two-dimensional mixed convection flowalong a vertical semi-infinite stretching sheet of variable thickness. The effect of double diffusion on velocity, thermal and concentration fields in presence of power-law temperature and concentration distributions at wall along with surface mass transfer is considered. The nonlinear coupled partial differential equations governing the flow, thermal and concentration fields are first transformed into a nondimensional set of coupled nonlinear partial differential equations and solved numerically using an implicit finite-difference scheme in combination with the Newton's linearization technique to obtain nonsimilar solutions at each stream-wise location. Numerical results are presented to discuss the effects of various physical parameters on the velocity, temperature, and concentration fields. Furthermore, the numerical results for the local skin friction coefficient, local Nusselt number, and local Sherwood number are also reported. For a fixed buoyancy force, the skin friction coefficient and Nusselt number increase with Prandtl number. The increase in the Prandtl number causes about a 30% reduction in the thickness of the thermal boundary layer. The wall thickness parameter enhances the thickness of the momentum boundary layer and the velocity overshoot is observed up to 20% for wall thickness parameter alpha = 0.9. In contrast, the increase of power-law index parameter m from m = 0.5 to m = 1.5 reduces approximately 10% to 25% the momentum and thermal boundary layer thicknesses depending on the values of other parameters (C) 2017 Wiley Periodicals, Inc.

A comprehensive investigation has been carried out to analyze the fluid and heat flow during mixed convection in entrapped triangular cavities for heat disposal (case 1: hot inclined and cold horizontal wall) and heat recovery (case 2: cold inclined and hot horizontal walls) to/from the industrial fluids via heatline approach. Galerkin finite element method is employed to solve the governing equations and the results are illustrated for various Grashof numbers (10(3) <= Gr <= 10(5)), Prandtl numbers (Pr = 0.026 and 7.2) and Reynolds numbers (1 <= Re <= 100). At Re = 1, the symmetric fluid circulation cells occur whereas, the asymmetric fluid circulation cells occur at Re >= 10 and Gr = 10(5) in both the cases and both Pr. The conduction dominant heat transfer is observed at Pr = 0.026 and Gr = 10(5) involving all Re. In contrast to Pr = 0.026, the heat flow field is highly influenced by the fluid flow field at Pr = 7.2 for all Re and Gr = 10(5). In the upper cavity, the average Nusselt number of the top wall ((Nu(t)) over bar) is higher at Re -> 0, Gr = 10(5) for the case 1, and at Re = 100, Gr = 10(3) for the case 2 involving Pr = 7.2. In the lower cavity, the average Nusselt number of the bottom wall ((Nu(b)) over bar) is higher at Re = 100, Gr = 10(3) for the case 1, and at Re -> 0, Gr = 10(5) for the case 2 involving Pr = 7.2. (C) 2017 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

This paper is devoted to obtain non-similar solutions for the effect of viscous dissipation on the steady double diffusive mixed convection flow over a vertical exponentially permeable stretching surface. The non-linear partial differential equations governing the flow, thermal and species concentration fields are written in the non-dimensional form by using suitable group of transformations. The final non dimensional set of coupled partial differential equations is solved using the implicit finite difference method in combination with the Newton's linearization technique. The effects of various non dimensional physical parameters on velocity, temperature and species concentration fields are discussed. The presence of the suction/injection at the surface expedites the mass transfer phenomena. The numerical results in terms of the skin friction coefficient, the rate of heat transfer in terms of local Nusselt number and mass transfer rate in terms of Sherwood number shown graphically for various physical parameter involved in the problem. The present results are compared with previously published work, and these comparisons are found to be in excellent agreement. (C) 2017 Elsevier Ltd. All rights reserved.

Finite element simulation of the mixed convection within porous square cavities for Darcy–Brinkman–Forchheimer model has been carried out in the present work. The penalty optimization based Galerkin finite element method is used to solve the partial differential equations of heat and fluid flow. Bejan’s heatline concept has been employed to visualize the heat flow within the closed cavities based on the motion of the horizontal wall(s) (cases 1a–1d) or vertical wall(s) (cases 2a–2c) involving isothermally hot bottom wall, cold side walls and insulated top wall for various fluids with Prandtl number, Prm=0.026, 0.7 and 7.2, Reynolds number, Re=10–100 and Grashof number, Gr=103–105. The higher permeability at Dam≥10−3 leads to the enhanced buoyancy convection for all the cases. Although the direction of the motion of wall(s) significantly influences the fluid flow field within the enclosure, due to the decoupling between the fluid and thermal fields at the low Pem (Pem=0.26 and 2.6), conductive heat transfer occurs as seen from the end-to-end heatlines. It is also found that the overall heat transfer rates at the bottom wall (Nub¯) are identical for the cases 1a–1d and cases 2a–2c at Prm=0.026, irrespective of Gr and Re at Dam=10−2. At Prm=0.7 and 7.2, the convection dominant heat transfer occurs for all the cases for Gr=105, Re=10 and 100 and Dam=10−2. The strong convective circulation cells are observed at Prm=0.7 and 7.2 for all the cases. The plume shaped isotherms are also observed along the centerline at Prm=7.2, Re=10, Gr=105 and Dam=10−2 for all the cases. At Prm=7.2, Gr=105, Dam=10−2 and Re=100, the multiple convective heatline cells are observed for the cases 1a–1d. It is observed that, the strengths of fluid and heat circulation cells are less at Re=100 compared to Re=10 for all the cases due to weak buoyancy force at the high Re. In order to achieve the high heat transfer rate at the bottom wall (Nub¯) for the mixed convection involving various moving walls, case 2b (a case of the vertically moving wall) is preferred at Prm=0.7, Re=100, Gr=105 and Dam=10−2. At the high Prm (Prm=7.2, Re=100, Gr=105 and Dam=10−2), case 2a (a case of the vertically moving wall) is preferred based on the maximum heat transfer rate at the bottom wall (Nub¯).

The strategic application of entropy generation concept for optimization of the natural convection process has been studied in the present work. The wall, AB is isothermally heated and walls, BC and DA are cooled in presence of adiabatic wall CD. The numerical results are presented in terms of isotherms (theta), streamlines (psi) and entropy generation maps for heat transfer (S-theta) and fluid flow (S-psi) for various modified Prandtl numbers (Pr-m = 0.015 and 1000), modified Darcy numbers (Da(m) = 10(-5)-10(-2)) and modified Rayleigh numbers (Ra-m = 10(3) and 10(6)). The maximum value of the entropy generation due to heat transfer (S-theta,S-max) is observed at hot and cold junction points (A and B), due to high temperature gradient, irrespective of Da(m), Pr-m and inclination angles (phi). The active regions of S-theta occur at top portion of the walls, BC and DA at high Da(m) (Da(m) = 10(-2)) and high Pr-m (Pr-m =1000). The maximum value of entropy generation due to fluid flow (S-psi,S-max) is found at various locations on the walls of the cavity whereas significant S-psi is also observed in the interior regions due to the friction between counter rotating circulation cells. The heat transfer rate along the wall AB ((Nu) over bar (AB)) and the total entropy generation (S-total) are found to be constant for the conduction dominant regime (10(-5) <= Da(m) <= 10(-3)) whereas non linear increasing trends are observed for convection dominant regime (10(-3) <= Da(m) <= 10(-2)). The inclination angle ranges, 30 degrees <= phi <= 50 degrees and 25 degrees <= phi <= 75 degrees are optimal tilt angles for Pr-m = 0.015 and 1000, respectively based on minimum entropy generation and reasonable heat transfer rate at high Da(m) (Da(m) = 10(-2)) with Ra-m = 10(6). (C) 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Entropy generation during the mixed convection process have been studied in a square enclosure for various moving horizontal (cases 1a–1d) or vertical wall(s) (cases 2a–2c) where the bottom wall of the cavity is isothermally hot, side walls are cold, and the top wall is adiabatic. Simulations have been performed for Prandtl number Pr=0.026 and 7.2, Reynolds number Re=10−100, and Grashof number Gr=103−105. Results show that, in the case of the horizontally moving wall(s) (cases 1a–1d), the overall heat transfer rate Nub¯ and total entropy generation (Stotal) are identical for cases 1a–1d and the cup-mixing temperature (θcup) is high for case 1b at Pr=0.026, Re=100, and Gr=105. Similarly, in the case of the vertically moving wall(s) (cases 2a–2c), Nub¯ and Stotal are identical for cases 2a–2c with the maximum θcup occurring for the case 2a. At Pr=7.2, Gr=105, and Re=10, case 1a and case 1c are preferable for horizontally moving wall(s) and either of case 2a–2c is preferable for vertically moving wall(s). At Pr=7.2, Gr=105, and Re=100, case 1d may be preferable for the horizontally moving wall(s) and case 2a may be preferable for the vertically moving wall(s).

Finite element simulations were carried out to analyze entropy generation during mixed convection inside square enclosures with an isothermally hot bottom wall, adiabatic top wall, and isothermally cold side walls (case 1) or linearly heated side walls (case 2), or linearly heated left wall with isothermally cold right wall (case 3) for Pr = 0.015–7.2, Re = 1–100, and Gr = 103–105. Local entropy maps are studied in detail, and the dominance of thermal (Sθ,l) and frictional (Sψ,l) irreversibility is studied using Bejan number maps. In addition, variation in total entropy generation (Stotal), average Bejan number (Beav), and average Nusselt number at the bottom wall with Gr are analyzed to correlate irreversibility and the overall heat transfer rate of the system or process. It is found that, for Pr = 0.015 and 7.2, Re = 100 may be the optimal level for higher convective heat transport with minimum entropy generation in all the cases for Gr = 103–105.

Mixed convection in closed cavities are important for various processing industries especially those associated with conservation of energy. Finite element based simulations are carried out for two cases based on the motion of the horizontal wall(s) (cases la Id) or vertical wall(s) (cases 2a-2c). Heat flow distribution within the cavity enclosed by isothermally hot bottom wall, cold side walls and insulated top wall is analyzed for various fluids with Prandtl number, Pr = 0.026 and 7.2, Reynolds number, Re = 10-100 and Grashof number, Gr = 10(3)-10(5). The direction of motion of wall(s) plays a significant role on the fluid flow field at Pr = 0.026, Gr = 10(3) and Re = 10 due to dominant forced convection for both horizontally (cases 1a 1d) or vertically (cases 2a-2c) moving wall(s). At Pr = 7.2, Gr = 10(5) and Re = 100, multiple convective heatline cells are observed for cases 1a-1d. It is found that, the strength of fluid or heatline circulation cells is less at Re = 100 compared to Re = 10 for cases 2a-2c due to weak buoyancy force at high Re. Energy transfer rates are assessed via local and average Nusselt numbers for cases la id and 2a-2c. (C) 2014 Elsevier Ltd. All rights reserved.

Computational studies on entropy generation during laminar mixed convection in porous square enclosures have been carried out based on Darcy-Brinkman-Forchheimer model using the penalty finite element method. Finite element simulations are performed for the isothermally hot bottom wall, adiabatic top wall, and isothermally cold side walls (case 1) or linearly heated side walls (case 2) or linearly heated left wall with isothermally cold right wall (case 3) for a wide range of modified Darcy numbers (10(-5)Da(m)10(-2)), Grashof numbers (Gr=10(3)-10(5)), and modified Prandtl numbers (Pr-m=0.026 and 7.2). Further, the effects of Da(m) on the total entropy generation (S-total), average Bejan number (Be-av), and average Nusselt number (Nu(b))($) over bar are discussed. It is found that Re=100 is preferred over Re=1 based on larger heat transfer rate with minimum entropy generation for Pr-m=0.026 and 7.2, 10(-5)Da(m)10(-2) at Gr=10(5) for all the cases.

The main objective of the present paper is to obtain non-similar solutions numerically for steady two dimensional double diffusive mixed convection boundary layer flows along a vertical semi-infinite permeable surface under the influence of convective boundary condition along with surface mass transfer. The nonlinear partial differential equations governing the flow, temperature, and species concentration fields are expressed in non-dimensional form using suitable non-similar transformations. The final non-dimensional set of coupled nonlinear partial differential equations is solved by using an implicit finite difference scheme in combination with quasi-linearization technique. The effects of various governing parameters involved on the velocity, temperature and species concentration profiles are discussed in the present paper. The results show that the streamwise co-ordinate ξ significantly influences the flow, thermal, and concentration fields which indicate the importance of non-similar solutions. Results indicate that buoyancy parameter (Ri) and the ratio of buoyancy forces parameter (N) enhance the skin friction coefficient and decrease the heat transfer coefficient. Also, it is observed that the increase of suction parameter (A=1) causes the decrease in the magnitude of temperature and concentration profiles from their values for injection parameter (A=−1). In the present investigation, dual solutions are also obtained under similarity assumptions and compared with previously published work.

Finite element based numerical simulation has been carried out for analysis of heat flow visualization and entropy generation during natural convection within inclined square cavities with hot wall (DA), cold wall (BC) and adiabatic walls (AB and CD). The numerical results are presented in terms of isotherms (θ), streamlines (ψ), heatlines (Π), entropy generation due to heat transfer irreversibility (Sθ) and fluid friction irreversibility (Sψ). Further, detailed discussion on variation of the total entropy generation (Stotal), average Bejan number (Beav) and average Nusselt number (Nu¯), with Rayleigh number (Ra) is also presented. It is found that, large heat transfer rate Nu¯DA with less entropy generation (Stotal) occurs for φ = 15° cavities at convection dominant mode (Ra = 105) irrespective of Pr. Thus, inclined square cavities with φ = 15° may be used for all thermal processing operations involving various fluids (Pr = 0.025 and 998).