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The present study explores incompressible, steady power law nanoliquid comprising gyrotactic microorganisms flow across parallel plates with energy transfer. In which only one plate is moving concerning another at a time. Nonlinear partial differential equations have been used to model the problem. Using Liao's transformation, the framework of PDEs is simplified to a system of Ordinary Differential Equations (ODEs). The problem is numerically solved using the parametric continuation method (PCM). The obtained results are compared to the boundary value solver (bvp4c) method for validity reasons. It has been observed that both the results are in best settlement with each other. The temperature, velocity, concentration and microorganism profile trend versus several physical constraints are presented graphically and briefly discussed. The velocity profile shows positive response versus the rising values of buoyancy convection parameters. While the velocity reduces with the increasing effect of magnetic field, because magnetic impact generates Lorentz force, which reduces the fluid velocity.

The purpose of this paper is to identify an effective statistical distribution for examining COVID-19 mortality rates in Canada and Netherlands in order to model the distribution of COVID-19. The modified Kies Frechet (MKIF) model is an advanced three parameter lifetime distribution that was developed by incorporating the Frechet and modified Kies families. In particular with respect to current distributions, the latest one has very versatile probability functions: increasing, decreasing, and inverted U shapes are observed for the hazard rate functions, indicating that the capability of adaptability of the model. A straight forward linear representation of PDF, moment generating functions, Probability weighted moments and hazard rate functions are among the enticing features of this novel distribution. We used three different estimation methodologies to estimate the pertinent parameters of MKIF model like least squares estimators (LSEs), maximum likelihood estimators (MLEs) and weighted least squares estimators (WLSEs). The efficiency of these estimators is assessed using a thorough Monte Carlo simulation analysis. We evaluated the newest model for a variety of data sets to examine how effectively it handled data modeling. The real implementation demonstrates that the proposed model outperforms competing models and can be selected as a superior model for developing a statistical model for COVID-19 data and other similar data sets.

Installing obstructions in the path of the flowing stream in a Convergent-Divergent channel results in complex Reynolds Navier-Stokes (NS)-equations and an exact solution in this direction is not feasible. The current work is a notable attempt in this direction, reporting the evaluation of hydrodynamic forces faced by case-wise installation of infinite cylinders as obstacles to a continuous fluid stream. To be more specific, Newtonian fluid is initiated from inlet with bound of parabolic profile. The outlet of CD channel is carried with Neumann condition. The optimized path of installation of obstacle includes: (i) Two circular shaped cylinders are installed in left and right vicinity of CD throat (ii) Circular cylinder in right vicinity of CD throat is optimized with square cylinder (iii) Circular cylinder in left vicinity of CD throat is optimized with square cylinder. The flow field is mathematically modelled in terms of steady Reynolds NS-equations. The comparative solution is reported by using finite element method along rectangular and triangular elements as a hybrid meshing scheme. The obtained outcomes are shared by means of contour and line graphs. Optimized values of hydrodynamic forces subject to installed obstacles in convergent-divergent channel are reported by doing line integration around the outer surfaces of obstructions. It is observed that in each case, the obstructions in left vicinity of CD throat admits inciting values of drag coefficient as compared to obstruction rooted in right vicinity of CD throat.

The current research explores incremental effect of thermal radiation on heat transfer improvement corresponds to Darcy–Forchheimer (DF) flow of carbon nanotubes along a stretched rotating surface using RSM. Casson carbon nanotubes’ constructed model in boundary layer flow is being investigated with implications of both single-walled CNTs and multi-walled CNTs. Water and Ethylene glycol are considered a basic fluid. The heat transfer rate is scrutinized via convective condition. Outcomes are observed and evaluated for both SWCNTs and MWCNTs. The Runge–Kutta Fehlberg technique of shooting is utilized to numerically solve transformed nonlinear ordinary differential system. The output parameters of interest are presumed to depend on governing input variables. In addition, sensitivity study is incorporated. It is noted that sensitivity of SFC via SWCNT-Water becomes higher by increasing values of permeability number. Additionaly, sensitivity of SFC via SWCNT-water towards the permeability number is higher than the solid volume fraction for medium and higher permeability levels. It is also noted that sensitivity of SFC (SWCNT-Ethylene-glycol) towards volume fraction is higher for increasing permeability as well as inertia coefficient. Additionally, the sensitivity of LNN towards the Solid volume fraction is higher than the radiation and Biot number for all levels of Biot number. The findings will provide initial direction for future device manufacturing.

Introduction Nanofluids are considered a better alternative to conventional fluids in many industrial situations and unfolding new opportunities for various applications owing to the optical and thermal properties of additive nanosized materials. Objectives In this study, the thermal and hydraulic characteristics of a Casson-based (sodium alginate) multiwall carbon nanotube (MWCNT) nanofluid were computationally investigated inside a wavy square enclosure containing a circular-shaped obstacle. The square enclosure comprised two cooled vertical walls and a wavy adiabatic top wall. The central part of the bottom wall comprised a heated wavy structure, and the remaining parts exhibited a flat and adiabatic structure. Methods The Navier–Stokes (N–S) equations and boundary conditions were established using the non-Newtonian Casson fluid model and Rosseland thermal radiation. The present problem was numerically simulated using the Galerkin finite element method for three types of obstacles, namely, adiabatic, hot, and cold. The impacts of Casson parameter (0.001 ≤β ≤0.1), Rayleigh number (103≤ Ra ≤106), nanoparticle volume fraction (0.01 ≤φ ≤0.1) and radiation parameter (1≤ Rd ≤ 4) are analysed. A numerical code validation was performed using the available benchmark results. Results The characteristics of the convective radiation heat transport were clearly analyzed through the stream function and isotherm plots. For all types of obstacles, the mean Nusselt number along the heated wavy wall increased with the Casson parameter, MWCNT volume fraction, Rayleigh number, and radiation parameter. Conclusion: The heat and flow characteristics of a Casson-based MWCNT nanofluid inside a wavy square enclosure were investigated. The mean Nusselt number was higher (lower) in the presence of cold (hot) obstacles.

In this paper, the Haar technique is applied to both nonlinear and linear eight-order boundary value problems. The eight-order derivative in the boundary value problem is approximated using Haar functions in this technique and the integration process is used to obtain the expression of the lower order derivative and the approximate solution of the unknown function. For the verification of validation and convergence of the proposed technique, three linear and two nonlinear examples are taken from the literature. The results are also compared with other methods available in the literature. Maximum absolute and root mean square errors at various collocation and Gauss points are contrasted with the exact solution. The convergence rate is also measured, which is almost equivalent to 2, using different numbers of collocation points.

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

The main goal of current research is to analyze the influences of Arrhenius activation energy in heat and mass transfer of second grade nanofluid flow. The governing equations are modeled with non–linear thermal radiation, elastic deformation of second grade nanofluid, Brownian motion, thermophoresis and Arrhenius activation energy. The momentum slip, active and passive controls of nanoparticles are assumed in the boundary. Similarity transformations, Runge-Kutta of order four and shooting methods are used to solve the governing equations. Graphical results are presented. It is found that the concentration profile augments with activation energy and decreases with exponential fitted rate. The local Nusselt number is increased with activation energy and decreased with elastic deformation and exponential fitted rate.

The current study offers the numerical findings on uniformly heated Y-shaped fin embedded in a rectangular domain having height 0.6 m and 2 m length. The no-slip condition is carried at left, upper and lower wall of channel. These walls are taken cold as well. Both the Neumann and adiabatic conditions are implemented at the right wall. The cold Newtonian fluid enters with the parabolic velocity profile from inlet of channel. The problem is controlled mathematically in terms of partial differential equations. The finite element method is adopted along with hybrid meshing scheme to report acceptable solution. The velocity, pressure and temperature distributions around the heated Y-shaped fin are offered graphically. The temperature findings are reported for the cold, adiabatic and heated tips of Y-shaped fin. The line integration around Y-shaped fin is performed to evaluate the lift and drag coefficients values. The lift coefficient and drag coefficient corresponds the lift force and drag force experienced by the heated Y-shaped fin. It is observed that for the higher values of Reynolds number the lift coefficient shows an inciting values while the drag coefficient shows decline nature.

It is well consensus among researchers that the non-Newtonian fluid equipped in closed enclosures brings complicated mathematical models and hence one cannot examine flow field truly. The present attempt is first numerical investigation to evaluate the buoyantly convective non-Newtonian fluid flow equipped in hexagonal shaped cavity. The uniformly heated T-shaped fin is embedded at lower wall of hexagonal cavity. The top wall of hexagonal cavity is manifested with an adiabatic condition. The bottom wall of enclosure is taken uniformly heated. Both left and right walls are kept cold. The buoyantly convective Casson fluid flow around uniformly heated T-shaped fin is mathematically controlled by way of system of partial differential equations. The finite element method is adopted to report numerical solution. The hexagonal enclosure as a computational domain is discretized by means of both triangular and rectangular elements. The velocity and temperature distribution around uniformly heated T-shaped are examined towards Rayleigh number. The line graph study is also executed to report the dimensionless Casson fluid and Casson temperature along center and vertical line directions of T-shaped fin. The impact of Rayleigh number on heat transfer rate along the surface of heated fin is also examined and offered by means of line graph. It is observed that the heat transfer rate enhances along the surface of T-shaped fin when we increase the Rayleigh number.

A simple deterministic epidemic model for tuberculosis is addressed in this article. The impact of effective contact rate, treatment rate, and incomplete treatment versus efficient treatment is investigated. We also analyze the asymptotic behavior, spread, and possible eradication of the TB infection. It is observed that the disease transmission dynamics is characterized by the basic reproduction ratio $\Re _{0}$ ℜ 0 ; if $\Re _{0} ℜ 0 < 1 , there is only a disease-free equilibrium which is both locally and globally asymptotically stable. Moreover, for $\Re _{0}>1$ ℜ 0 > 1 , a unique positive endemic equilibrium exists which is globally asymptotically stable. The global stability of the equilibria is shown via Lyapunov function. It is also obtained that incomplete treatment of TB causes increase in disease infection while efficient treatment results in a reduction in TB. Finally, for the estimated parameters, some numerical simulations are performed to verify the analytical results. These numerical results indicate that decrease in the effective contact rate λ and increase in the treatment rate γ play a significant role in the TB infection control.

A modified fractional model for the magnetohydrodynamic (MHD) flow of a fluid is developed utilizing Atangana–Baleanu fractional derivative (ABFD). Natural convection and wall oscillation instigate the flow over a vertical plate positioned in a porous medium. The partial differential equations (PDEs) are transmuted to ordinary differential equations (ODEs). The Laplace transform method with its inversion is employed to accomplish the exact solutions of momentum and heat equations. The final solution is expressed in terms of gamma function, modified Bessel function, and Mittag-Leffler function. The previous definitions Caputo fractional and Riemann–Liouville are rarely used by the researchers now due to their limitations. The newly introduced ABFD has got significance nowadays due to its nonlocal and nonsingular kernel. This work focuses on the oscillating boundary conditions for the viscous model in terms of ABFD. The influence of involved parameters is interpreted through plots. The velocity profile is an increasing function of fractional parameter and jumps for a higher Grashof number due to buoyancy push. Furthermore, the Atangana–Baleanu (AB) model is compared with the ordinary derivative model for limiting case and analyzed in detail. It is noted that the ordinary fluid flows faster compared to the fractional fluid.

In this article, we study different types of fractional-order logistic models in the frame of Caputo type fractional operators generated by conformable derivatives (Caputo CFDs). We present the existence and uniqueness theorems to solutions of these models and discuss their stability by perturbing the equilibrium points. Finally, we furniture our results by illustrative numerical examples for the studied models.

In this paper we apply the fractional integrals with arbitrary order depending on the fractional operators of Riemann type (ABR) and Caputo type (ABC) with kernels of Mittag Lefller in three parameters $E_{\alpha,\mu}^\gamma(\lambda,t)$ for solving the time fractional parabolic nonlinear equation. We utilize these operators with homotopy analysis method (HAM) for constructing the new scheme for generating the successive approximations. This procedure are used successfully on two examples for finding the solutions. The effectiveness and accuracy are verified by clarifying the convergence region in the $h$-curves as well as by calculating the residual error and the results were accurate. Depending on this results, this treatment can be used to find the approximate solutions to many fractional differential equations.

In this attempt, nanoparticle enhanced PCM is employed to enhance solidification rate. To augment low thermal conductivity of H2O, CuO nanoparticles were dispersed in it. Also Brownian motion impact was involved in properties of NEPCM. Graphs are obtained via FEM and displayed the impacts of radiation, length of fin and nanoparticles' size. Outputs display that selecting dp = 40 nm leads to the fastest discharging. Discharging time reduces with increase of thermal radiation.

Forced convection from a heated rotationally oscillating circular cylinder placed in a uniform cross flow of constant properties fluid is investigated. The two-dimensional governing equations of flow motion and energy are solved numerically on non-uniform polar grids using a higher order compact (HOC) formulation. The flow and thermal fields are mainly influenced by Reynolds number, Re, Prandtl number, Pr, maximum angular velocity of the cylinder, α m , and the frequency ratio, f / f 0 , which represents the ratio between the oscillation frequency, f, and the natural vortex shedding frequency, f 0 . The numerical simulations are performed at Re = 200 , Pr = 0.5 - 1.0 , α m ∈ [ 0.5 , 4.0 ] and f / f 0 ∈ [ 0.5 , 3.0 ] . The resulting lock-on phenomena behind the cylinder is detected and thermal field is determined. Comparisons with previous numerical and experimental results verify the accuracy and the reliability of the present study. Variations in heat transfer coefficients within the lock-on ranges are investigated to build a connection between the heat transfer and the lock-on regimes.

Nanofluid radiation in a semi permeable duct is analyzed in existence of Lorentz forces. Lorentz forces impact on energy equation are involved. Last ODEs were solved via RK4 approach. Effects of suction, radiation, Eckert, Hartmann numbers on flow, heat and mass transfer have been described. Outputs reveal that as Ha augments, vertical velocity reduces but temperature enhances. Concentration of nanofluid reduces with augment of magnetic force.

In this paper, we present a numerical technique for solving fractional Sturm–Liouville problems with variable coefficients subject to mixed boundary conditions. The proposed algorithm is a spectral Galerkin method based on fractional-order Legendre functions. Tedious manipulation of the series appearing in the implementation of the method have been carried out to obtain a system of algebraic equations for the coefficients. Our findings demonstrate the possibility of having no eigenvalues, finite number of eigenvalues or infinite number of eigenvalues depending on the fractional order. The convergence and effectiveness of the present algorithm are demonstrated through several numerical examples.

In current attempt, nanoparticle Electrohydrodynamic transportation has been modeled numerically via control volume based finite element method. Mixture of Fe3O4 and Ethylene glycol is elected. Impact of radiation parameter $$ \text{(}Rd\text{)} $$ , voltage supplied $$ \text{(}\Delta \varphi \text{)} $$ , nanoparticle concentration, Permeability and Reynolds number have been displayed. Results display that permeability and thermal radiation can improve temperature gradient.

In this article, a complete structure of corrugated surface is established for heat transfer effects in the presence of uniform magnetic field. A natural convection phenomenon is presented for MHD flow filled in a porous corrugated cavity at various wavelengths and partially heated domain. The governing partial differential equations consist of continuity, momentum and energy equations along with the corrugated conditions at the surface. This system is properly nondimensionalized and then solved via finite element method (FEM). In order to obtain the high resolution near the surface of corrugation, mesh generation is improved at the various portions of the cavity. The flow patterns and temperature distribution within the entire domain of the cavity can be visualized through streamlines and isotherms, respectively. Computational experiment is performed for various values of wavelength number ( 0 ⩽ n ⩽ 15 ) , Rayleigh number ( 10 4 ⩽ Ra ⩽ 10 6 ) , Darcy number ( 10 - 5 ⩽ Da ⩽ 10 - 3 ) , and Hartmann number ( 10 ⩽ Ha ⩽ 10 3 ) to illustrate the effects on streamlines, isotherms, velocities and heat transfer rate. Heat transfer rate is increased due to increase in Rayleigh number and wavelength parameter. Darcy and Hartmann number does not have significant effects on the temperature distribution.

The free convective micropolar fluid over a shrinking sheet in presence of heat source/sink has been studied in this paper. The method of solution involves similarity transformation. The coupled non-linear partial differential equations representing momentum and concentration and non homogeneous heat equation are reduced into set of non-linear ordinary differential equations. The transformed equations are solved by applying Runge-Kutta method followed by shooting technique. The effect of pertinent parameters characterizing the flow has been presented through the graphs and then discussed. It is found that heat source has significant effect on velocity profile and causes a decrease in the boundary layer. Angular velocity profile increases with the increasing value of material parameter. Present study showed an excellent agreement with published literature.

This framework is established for alumina and copper-water nanofluid over a shrinking sheet with thermal radiation effect. To determine the better thermal performance of base fluid, model is constructed in the form of effective thermal conductivity and viscosity for nanoparticles. Thermal radiation effects are also incorporated to determine the significant influence of temperature in the restricted domain of the model. Partial differential equations are acquired from momentum and energy equations along with the boundary constraint at the surface. The nonlinear ODEs are further acquired from governing PDEs by using the similarity transformation. Dual nature's results are accomplished in the form of closed form exact solutions. The consequences of various physical parameters on the velocity and temperature profile are investigated along with the assist of tables and graphs. Additionally, temperature undershoot is noticed in some cases. Consequently, the heat absorption on surface arises due to heat transfer from the surface in some situations. In addition, the alumina and copper nanoparticles based fluids tender astounding antimicrobial actions to various gram class microscopic organisms. Some major results affirm that local Nusselt number enhance by increasing the values of thermal radiation parameter for Cu- and Al2O3-water. In addition, rate of heat transfer is higher for the first solution of both of Al2O3- and Cu-water.

Forced convection from a heated non-rotating circular cylinder of radius a performing a circular motion of radius A ∗ and placed in a uniform cross flow of constant properties fluid is investigated numerically. The two-dimensional governing equations of flow motion and energy are solved numerically using Fourier spectral analysis together with finite difference approximations to determine the flow field characteristics and the heat transfer parameters. The flow and thermal fields are mainly influenced by Reynolds numbers, Re , Prandtl number, Pr , amplitude of circular motion, Ar = A ∗ / a , and the frequency ratio, Fr = f ∗ / f 0 ∗ , which represents the ratio between the frequency of circular motion, f ∗ , and the natural vortex shedding frequency, f 0 ∗ . The ranges considered for these parameters are 60 ⩽ Re ⩽ 180 , 0.1 ⩽ Ar ⩽ 1.0 and 0.5 ⩽ Fr ⩽ 3.0 , while the Prandtl number is kept constant at 0.7. The study, in general, showed that the heat transfer rate increases appreciably in the high range of Re , Ar and Fr . Comparisons with previous numerical and experimental results verify the accuracy and the validity of the present study.

The aim of this work is to investigate the dual nature solutions of the fluid flow due to stretching/shrinking permeable surface saturated in a porous medium. Heat transfer analysis has been carried out in the presence of convective boundary conditions. Compatible transform are utilized to rehabilitate the system of nonlinear partial differential equations into the nonlinear ordinary differential equations. Closed form dual solutions are obtained for each unknown velocity and temperature profiles in terms of Gamma functions. Graphical interpretation of the possible dual solutions of dimensionless velocity, temperature, skin-friction coefficient, local Nusselt number as well as for stream lines and isotherms are analyzed under the influence of different physical parameters. It is finally concluded that all the obtained results against each velocity profile depicts the both increasing and decreasing behavior according the upper and lower solution. However, temperature profile shows the same behavior of two different solutions for each parameter. Isotherms behavior in the entire domain of the model shows the significant variation for three different fluids (air, water and kerosene oil).

This study deals the simultaneous effects of inclined magnetic field and prescribed surface temperature (PST) on boundary layer flow of nanofluid over a stretching sheet. In order to make this mechanism more feasible, we have further considered the velocity slip and thermal radiation effects. Moreover, this perusal is made to consider the two kinds of nanofluid namely: Cu -water and A l 2 O 3 -water. Inclined magnetic field is utilized to accompanying an aligned angle that varies from 0 to π / 2 . The exact solutions are acquired from the transformed non-dimensional momentum and energy equations in the form of confluent hypergeometric function. Lorentz forces and aligned magnetic field depicts the significant effects on nanofluid. We found that, due to the increase in the aligned angle provides the enhancement in local skin friction coefficient and a reduction in the local Nusselt number. The combined impacts of inclined magnetic field with other emerging parameters such as velocity slip, thermal radiation and nanoparticles volume fraction on velocity, temperature, local Nusselt number and skin friction coefficient are examined. Flow behavior of nanofluid is also determined via stream lines pattern.

This article investigates heat transfer enhancement in free convection flow of Maxwell nanofluids with carbon nanotubes (CNTs) over a vertically static plate with constant wall temperature. Two kinds of CNTs i.e. single walls carbon nanotubes (SWCNTs) and multiple walls carbon nanotubes (MWCNTs) are suspended in four different types of base liquids (Kerosene oil, Engine oil, water and ethylene glycol). Kerosene oil-based nanofluids are given a special consideration due to their higher thermal conductivities, unique properties and applications. The problem is modelled in terms of PDE’s with initial and boundary conditions. Some relevant non-dimensional variables are inserted in order to transmute the governing problem into dimensionless form. The resulting problem is solved via Laplace transform technique and exact solutions for velocity, shear stress and temperature are acquired. These solutions are significantly controlled by the variations of parameters including the relaxation time, Prandtl number, Grashof number and nanoparticles volume fraction. Velocity and temperature increases with elevation in Grashof number while Shear stress minimizes with increasing Maxwell parameter. A comparison between SWCNTs and MWCNTs in each case is made. Moreover, a graph showing the comparison amongst four different types of nanofluids for both CNTs is also plotted.

The literature reveals that numerous real-life phenomena in the subjects of physics and engineering which are governed by highly nonlinear Partial differential equations (PDEs) with unknown analytical solutions. More precisely, the (PDEs) arise in a wide variety of physical problems such as; by way of example not exhaustive enumeration, fluid dynamics, engineering mathematics, electrostatics, plasma physics, solid mechanics, chemistry, quantum field theory, bio-mathematics, etc. Therefore, such (PDEs) have received a huge attention from mathematicians, physicists, and engineers for the sake of approximating their analytical solutions. We aimed in this special issue to publish articles focusing on recent advanced numerical studies on Differential Equations related to physics and engineering. The well-developed analysis of existing numerical algorithms in terms of efficiency, applicability, convergence, stability and accuracy is of importance. A discussion of nontrivial numerical examples is encouraged.

This work scrutinizes the hybrid nanofluids flow past a flat surface with radiation together with the subsistence of aligned magnetic field. Three distinct hybrid nanofluids such as aluminium oxide-silicon dioxide/water ( A l 2 O 3 − S i O 2 / H 2 O ), aluminium oxide-titanium dioxide/water ( A l 2 O 3 − T i O 2 / H 2 O ) and titanium dioxide-silicon dioxide/water ( T i O 2 − S i O 2 / H 2 O ) are used in this study through Marangoni boundary condition. The partial differential equations controlling the heat transfer problem were upgraded to ordinary differential equations utilizing appropriate similarity transformation. Analytical result of the ordinary differential equations is achieved using Laplace technique. The precision of the analytical solution of flow function is validated by numerical solutions. The physical parameters achieve on the flow field and temperature field are brought about by graph and tables and are deliberated in detail. The important findings are the surface velocity and rate of heat transfer is little higher for A l 2 O 3 − S i O 2 / H 2 O and T i O 2 − S i O 2 / H 2 O respectively. The permeable parameter has a direct relation with flow field and indirect relation with temperature field.

The current study is carried out to investigate the natural convection of enclosures contains a horizontal/vertical heated baffle at its center with inlet and outlet ports on vertical walls. The vertical walls of the cavity are cold and the horizontal walls are insulated. The working medium of fluid is air with the Prandtl number 0.71. The governing equations of the problem are converted into the vorticity−stream function formulation. The finite-difference method is adopted to solve these equations. The calculations were performed for various Rayleigh numbers $$(10^3 - 10^6)$$ and Reynolds numbers (10, 100, 500), and the effects of the different aspect ratios ($$A=0.25, 0.5, 1.0, 2.0$$) are tested with the most efficient configured cavity with the baffle. The numerical results are illustrated graphically in the form of streamlines, isotherms, and local and averaged Nusselt numbers. The results show that the BB configuration cavity with both orientations of the heated baffle performs better in optimizing the heat inside the cavity. The ventilated cavity with vertical baffle transfers heat more rapidly than the horizontal baffle.

In this paper, we steady natural convection in a discrete heat transfer and the effect of Rayleigh number in a two- dimensional square enclosure with an inner square block. The present problem taken by three different cases were containing on the partially heated which left vertical wall maintained at high temperature Th(Th > Tc). The right vertical wall considered at Tc and other horizontal walls and the inner square block should be thermally insulated. The mathematical model considered as mass, momentum and energy equation solved to using the simple algorithm by Finite Volume Method (FVM) on a flounder grid system consisting of modified Alternating Direction Implicit Method (ADIM) which are used to solve vorticity-stream function formulation.This result found that the streamlines and isotherms achieved a maximum for the distribution of partial heater and also increasing Rayleigh number.

The paper contains extended study on hydrodynamic forces. The uniformly heated rectangular shaped obstacle is installed in the path of ongoing cold fluid towards partially heated channel as a computational domain. The developed flow narrating mathematical system is solved by using LBB-stable finite element pair. The velocity and temperature distributions are offered in terms of contour plots. The hydrodynamic forces are evaluated by performing line integration around the outer surface of rectangular obstacle. It is noticed that the drag coefficient is inclination dependent subject to cold fluid flow in a partially heated channel.

In this paper, the rotatory non-Newtonian fluid flow subject to rigid disk is debated. The Casson fluid is used as a non-Newtonian fluid model and the fluid is equipped above the rigid disk. The rotatory flow field is interacted with an applied magnetic field. The heat transfer aspects are evaluated in the presence of heat source/sink. Further, the nanosized particles are suspended in the flow regime. The physical statement is mathematically controlled in terms of partial differential equations. The numerical method named shooting method is utilized in this analysis. The involved flow controlling parameters includes the Casson fluid parameter, magnetic field parameter, heat generation parameter, heat absorption parameter, thermophoresis parameter, Brownian motion parameter and Lewis number. The impact of these parameters are examined on the Casson fluid velocities, temperature and concentration. It is noticed that the both radial and tangential velocities are supressed in the magnetized rotatory flow field as compared to non-magnetized rotatory flow field. The Casson fluid temperature enhances towards higher values of thermophoresis and heat generation parameters. The results are compared with an existing work which yields the surety of adopted computational algorithm. Keywords: Casson fluid model, Rotating rigid disk, Heat transfer, Heat generation/absorption, Nanoparticles

In this paper, we apply a hybrid method due to coupling the Laplace transform with the Adomian decomposition method (LADM) for solving nonlinear fractional differential equations that appear in the model of Keller-Segel equations with one dimension. We explain the adopted method is with several examples. It turns out that the reliability of LADM and the reductions in computations show that LADM is widely applicable. We also compare our results with the results of homotopy decomposition method (HDM).

In many industrial processes, the cooling process can be improved by varying the flow geometry or changing the additives in the working fluid. The present work concentrates on the flow of γ Al2O3 –Water/Ethylene Glycol over a Gailitis and Lielausis device with an effective Prandtl number for the first time. The thermal transport aspects of electro-MHD boundary layer flow of γ Al2O3 nanofluids over a stretchable Riga plate are studied in two dimensions. The wall parallel Lorentz force is produced due to an external electric field by Riga plate to control the nanofluid flow. Mathematical models are developed with an effective Prandtl number. The no-slip and the prescribed surface temperature boundary conditions are assumed. Results are discussed using numerical results obtained by fourth order RK method with shooting technique. Special case analytical solutions are presented for both momentum and energy equations. The increasing behaviour in velocity profile and decreasing behaviours in temperature, skin friction and Nusselt number are observed with increasing modified Hartmann number. The higher modified Hartmann number leads to a sudden enhancement in the velocity profile of the nanofluid in the presence of effective Pr near the riga plate wall.

In the present article, the Newtonian fluid flow problem with buoyancy and thermal radiation in non-linear form is investigated by considering the entropy generation. The boundary is mathematically modeled with slip conditions (velocity and thermal) of order two. The parameters for the slip conditions of order two are calculated from the slip parameters of order one. The governing PDE's and the corresponding BC's are derived and transformed into ODE's by suitable transformation. Fourth order RK with shooting method is utilized to solve the transformed non-dimensional governing equations along with the boundary conditions. The efficiency of the present type of thermal fluidic system can be increased by reducing the generation of entropy. In view of these, the influences of slip conditions of order two on the entropy generation number are discussed. It is found that the increase of second order velocity and thermal slip effects reduces the entropy generation. Furthermore, the significant results are discussed with buoyancy effects. The present research work suggests to increase the second order slip factors in order to achieve the higher efficiency in the present type of thermal fluidic system. Keywords: Entropy generation, Buoyancy effects, Heat transfer, Newtonian fluid, Non-linear thermal radiation, Slip conditions of order two

Purpose The current mathematical model is developed to scrutinize the consequence of bioconvective cross diffusion flow of magnetized viscous nanofluid past multiple geometries (cone, wedge and plate) with convective boundary conditions. Together the nanoparticles and motile microorganism are incorporated into the dimensionless nonlinear differential expressions. The behavior of Cattaneo-Christov heat and mass flux is accounted for energy and concentration expressions. The influence of activation energy and thermal radiation are considered. The mathematical model is reduced into an ordinary one by using adequate similarity transformation. Buongiorno model is utilized for nanofluid (nanoliquids) analysis. Methodology/approach The renovated dimensionless self-similarity systems are then solved numerically by utilizing shooting technique built-in function bvp4c solver with the help of commercial software Matlab. The obtained results are verified and an outstanding agreement has been found. Engineering quantities of interest are observed physically. Findings The features of various emerging parameters against velocity distribution, thermal distribution, and solutal field of species, microorganism concentration as well as skin friction coefficient, gradient of temperature, local Sherwood number and density number of motile microorganisms are interpreted and deliberated in tabulated and graphical form. Results The results indicate that velocity field is raises via larger Grashof number. The resultant velocity is decline via larger magnetic parameter. Larger estimation of thermal Biot number increases the heat transfer. Larger thermal relaxation parameter reduces the temperature of fluid. The concentration of nanoparticles is declines via concentration relaxation parameter. The microorganism's field is declines by varying the variations of Peclet number.

The present analysis concentrates to examine the influence of both thermal and solutal stratification on magneto-hydrodynamics (MHD) nanofluid flow along an exponentially stretching sheet. Moreover, simultaneous effects of mixed convection and viscous dissipation are also analyzed to determine the thermal conductivity within the restricted domain. Energy and concentration equation consist of two important slip mechanisms, namely: the Brownian motion of nanoparticles and the thermophoresis due to concentration difference. By the mean of compatible similarity transformed, a system of PDEs is converted into the system of nonlinear ODEs. The resulting nonlinear ODEs are successfully solved via the implicit finite difference method (FDM). Obtained numerical solutions are plotted for each profile for different and converging values of including parameters. To validate the results, numerical values of Nusselt number are compared with the existing literature for a particular case. Obtained results present the significant impact of each parameter on temperature and concentration. Nanofluid flow behaviour is also observed via velocity profile.

In this article, thermal performance of engine oil in the presence of both single and multiple wall carbon nanotubes (SWCNTs and MWCNTs) between two concentric cylinders is presented. Flow is driven with oscillatory pressure gradient and magneto-hydrodynamics (MHDs) effects are also introduced to control the random motion of the nanoparticles. Arrived broad, it is perceived that the inclusion of nanoparticles increases the thermal conductivity of working fluid significantly for both turbulent and laminar regimes. Fundamental momentum and energy equations are based upon partial differential equations (PDEs) that contain thermos-physical properties of both SWCNTs and MWCNTs. The solution has been evaluated for each mixture, namely: SWCNT-engine oil and MWCNT-engine oil. Results are determined for each velocity, temperature, pressure and stress gradient. Graphical results for the numerical values of the emerging parameters, namely: Hartmann number (M), the solid volume fraction of the nanoparticles (ϕ), Reynolds number (Reω), and the pulsation parameter based on the periodic pressure gradient are analyzed for pressure difference, frictional forces, velocity profile, temperature profile, crux, streamlines and vorticity phenomena. In addition, the assets of various parameters on the flow quantities of observation are investigated. Keywords: MHD, Pulsating flow, Nanofluids, Carbon nanotubes, Concentric cylinders

This article is devoted to both theoretical and numerical studies of eigenvalues of regular fractional $2\alpha $-order Sturm-Liouville problem where $\frac{1}{2}< \alpha \leq 1$. In this paper, we implement the reproducing kernel method RKM) to approximate the eigenvalues. To find the eigenvalues, we force the approximate solution produced by the RKM satisfy the boundary condition at $x=1$. The fractional derivative is described in the Caputo sense. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the eigenfunctions of the proposed problem. Uniformly convergence of the approximate eigenfunctions produced by the RKM to the exact eigenfunctions is proven.

Current study is devoted to examine the magneto-hydrodynamics (MHD) flow of water based Cu Nanoparticles with oscillatory pressure gradient between two concentric cylinders. Arrived broad, it is perceived that the inclusion of nanoparticles has increased considerably the heat transfer near the surface of both laminar and turbulent regimes. Mathematical model is constructed in the form of partial differential equations which contains the effective thermal conductivity and viscosity of base fluid and nanoparticles. Close form solution is attained corresponding to the momentum and energy equation and results are evaluated for velocity, temperature and pressure gradient in the restricted domain. Graphical results for numerical values of the flow control parameters: Hartmann number M , Reynolds number Re ω , the solid volume fraction of nanoparticles ϕ and the pulsation parameter based on the periodic pressure gradient have been presented for the pressure difference, frictional forces, velocity profile, temperature profile, and vorticity phenomena have been discussed. The assets of various parameters on the flow quantities of observation are investigated. To the same degree a concluding crux, the streamlines are examined and plotted. The results confirmed that the velocity and temperature may be controlled with the aid of the outside magnetic field and due to the growth in the nanoparticles and considerable enhancement in the heat transfer rate can be found by adding/removing the strength of magnetic field and nanoparticle volume fraction.

Microorganisms lives with us in our environment, touching infectious material on the surfaces by hand-mouth which causes infectious diseases and some of these diseases are rapidly spreading from person to person. These days the world facing COVID-19 pandemic disease. This article concerned with existence of results and stability analysis for a nabla discrete ABC-fractional order COVID-19. The nabla discrete ABC-fractional operator as more general and applicable in modeling of dynamical problems due to its non-singular kernel. For the existence and uniqueness theorems and Hyers-Ulam stability, we need to suppose some conditions which will play important role in the proof of our main results. At the end, an expressive example is given to provide an application for the nabla discrete ABC-fractional order COVID-19 model.

The transmission of influenza has been explained by analyzing a diffusive epidemic model. The Operating splitting based on finite difference (OSBFD), explicit formula based on meshless method (EFBMM), Operator splitting based on meshless method (OSBMM) are applied to obtain numerical solutions of equations under varied initial distribution of dense population. The specific role of diffusion and distribution has been accentuated in spread of ailment. It is also presented that how the transmission of disease is specifically reduced by the medicative and non-medicative innovations. The numerical solutions involved in stability of all the equilibria are also stated.