A mathematical model of MHD nanofluid flow having gyrotactic microorganisms with thermal radiation and chemical reaction effects

In this article, we have examined three-dimensional unsteady MHD boundary layer flow of viscous nanofluid having gyrotactic microorganisms through a stretching porous cylinder. Simultaneous effects of nonlinear thermal radiation and chemical reaction are taken into account. Moreover, the effects of velocity slip and thermal slip are also considered. The governing flow problem is modelled by means of similarity transformation variables with their relevant boundary conditions. The obtained reduced highly nonlinear coupled ordinary differential equations are solved numerically by means of nonlinear shooting technique. The effects of all the governing parameters are discussed for velocity profile, temperature profile, nanoparticle concentration profile and motile microorganisms’ density function presented with the help of tables and graphs. The numerical comparison is also presented with the existing published results as a special case of our study. It is found that velocity of the fluid diminishes for large values of magnetic parameter and porosity parameter. Radiation effects show an increment in the temperature profile, whereas thermal slip parameter shows converse effect. Furthermore, it is also observed that chemical reaction parameter significantly enhances the nanoparticle concentration profile. The present study is also applicable in bio-nano-polymer process and in different industrial process.

     

Effects of slip on sheet-driven flow and heat transfer of a non-Newtonian fluid past a stretching sheet

The flow and heat transfer of an electrically conducting non-Newtonian fluid due to a stretching surface subject to partial slip is considered. The constitutive equation of the non-Newtonian fluid is modeled by that for a third grade fluid. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHFcase) in presence of a uniform heat source or sink. Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations. The important finding in this communication is the combined effects of the partial slip, magnetic field, heat source (sink) parameter and the third grade fluid parameters on the velocity, skin friction coefficient and the temperature field. It is interesting to find that slip decreases the momentum boundary layer thickness and increases the thermal boundary layer thickness, whereas the third grade fluid parameter has an opposite effect on the thermal and velocity boundary layers.     

An optimal solution for magnetohydrodynamic nanofluid flow over a stretching surface with constant heat flux and zero nanoparticles flux

This article examines the hydromagnetic three-dimensional flow of viscous nanoliquid. A bidirectional linear stretching surface has been used to create the flow. Novel features regarding Brownian motion and thermophoresis have been studied by employing Buongiorno model to examine the slip velocity of nanoparticle. Viscous liquid is electrically conducting subject to uniform applied magnetic field. Problem formulation in boundary-layer region is performed for low magnetic Reynolds number. Simultaneous effects of constant heat flux and zero nanoparticles flux conditions are utilized at boundary. Appropriate transformations correspond to the strongly nonlinear ordinary differential expressions. The resulting nonlinear systems have been solved through the optimal homotopy analysis method. Graphs have been sketched in order to analyze that how the temperature and concentration profiles are affected by various physical parameters. Further the coefficients of skin-friction and heat transfer rate have been numerically computed and discussed. Our findings show that the temperature distribution has a direct relationship with the magnetic parameter. Moreover, the temperature distribution and thermal boundary-layer thickness are higher for hydromagnetic flow in comparison with the hydrodynamic flow.

     

Near-wall effects in micro scale Couette flow and heat transfer in the Maxwell-slip regimes

The effects of modified transport characteristics within an extremely thin layer adjacent to the fluid–solid interfaces are investigated for fully developed laminar micro-scale Couette flows with slip boundary conditions. The wall-adjacent layer effects are incorporated into the continuum-based mathematical model by imposing variable viscosity and thermal conductivity values close to the channel walls, for solving the momentum and energy conservation equations. Analytical expressions for the velocity profiles are derived and are subsequently utilized to obtain the temperature variations within the parallel plate channel, as a function of the significant system parameters. It is revealed that the variations in effective viscosity and thermal conductivity values within the wall-adjacent layer have profound influences on the fluid flow and the heat transfer characteristics within the channel, with an interesting interplay with the wall slip boundary conditions. These effects cannot otherwise be accurately captured by employing classical continuum based models for microscale Couette flows that do not take into account the alterations in effective transport properties within the wall adjacent layers.     

Heat and mass transfer phenomena in the flow of Casson fluid over an infinite oscillating plate in the presence of first-order chemical reaction and slip effect

The objective of this article is to study the physics of slip effect at the boundary of a vertical plate in starting the flow of Casson fluid with the combined effect of radiative heat and mass transfer in the presence of first-order chemical reaction. The problem has been modeled in terms of partial differential equations along with appropriate initial and boundary conditions. The dimensionless governing equations have been solved by means of the Laplace transform technique. Exact solutions have been obtained for velocity, temperature and concentration profiles. The obtained velocity has been computed in tabular form for steady and transient velocities. The physics of velocity profile has been studied for various physical parameters through numerical computation and displayed in graphs. From obtained solutions, the well-known published results in the open literature have been recovered and displayed in graphs and tables.

     

Dusty flow with different water based nanoparticles along a paraboloid revolution: thermal analysis

The thermal analysis on hydromagnetic two-dimensional flow of dusty nano fluid along an upper horizontal surface of paraboloid revolution have been scrutinized. The governing flow are derived under the assumptions of Boussinesq’s boundary layer approximation theory. The effects of Cattaneo-Christov heat flux, variable thermal conductivity, joule heating and viscous dissipation are incorporated in the energy equation. The governing PDE’s for the flow and energy transfer for both the phases are transformed into ODE’S by employing the suitable similarity transformations. The final dimensionless governing coupled ordinary differential equations are resolved with the aid of bvp5c procedure in computational Matlab software. The effects of dimensionless governing controlled flow parameters on velocity, micropolar velocity, and temperature profiles for both the phases are reported and discussed elaborately through plots and tables. The emerging three nanoparticles namely gold, silver and platinum (\(Au,Ag\) and \(Pt\)) are considered throughout graphical analysis along with \(H_{2} O\) is used as a base liquid. It is revealed that the flow velocity declined for strengthen of the applied magneticfield. It is worthy note that the larger values of thermal relaxation parameter \(\gamma\) declines the fluid temperature for both phases. Also, the rate of heat transfer is an increasing function to the escalating values of variable thermal conductivity \(\varepsilon\), while it is reverse trend for the thermal relaxation parameter \(\gamma\). The observations exhibit the prominent features in the field of an advanced bio-medical and thermal engineering.

     

Nonlinear convective flow with variable thermal conductivity and Cattaneo-Christov heat flux

An analysis is introduced to investigate the salient features of nonlinear convective flow of thixotropic fluid in the version of Cattaneo-Christov heat flux theory. The stagnation point flow is present. The flow phenomenon is by an impermeable stretching sheet. The energy expression is modeled through the theory of Cattaneo-Christov heat flux. Characteristics of heat transfer phenomenon are described within the frame of variable thermal conductivity. Suitable variables reduced to the nonlinear partial differential expressions to the ordinary differential expressions. Series solutions of resulting systems are acquired within the frame of homotopy theory. Convergence analysis is achieved and suitable values are determined by capturing the so-called ℏ−curves. Graphical results for velocity and temperature are displayed and argued for sundry physical variables. Expression of skin friction coefficient is calculated through numerical values. Higher values of mixed convection parameter, Prandtl number, and thermal relaxation time lead to decay the temperature and layer thickness.

     

The optimal solution for the flow of a fourth-grade fluid with partial slip

The steady flow of a non-Newtonian fluid when slippage between the plate and the fluid occurs is considered. The constitutive equations of the fluid are modeled for a fourth-grade non-Newtonian fluid with partial slip; they give rise to nonlinear boundary value problems. Analytical solutions are obtained using powerful analytic techniques for solving nonlinear problems, homotopy perturbation and optimal homotopy asymptotic methods. The results obtained are compared with the numerical results and it is shown that solutions exist for all values of the non-Newtonian parameters. The solutions valid for the no-slip condition for all values of the non-Newtonian parameters can be derived as special cases of the present analysis. Finally the solutions are discussed using a graphical approach.     

On stratified variable thermal conductivity stretched flow of Walter-B material subject to non-Fourier flux theory

The objective here is to examine the characteristics of non-Fourier flux theory in flow induced by a nonlinear stretched surface. Constitutive expression for an incompressible Walter-B liquid is taken into account. Consideration of thermal stratification and variable thermal conductivity characterizes the heat transfer process. The concept of boundary layer is adopted for the formulation purpose. Modern methodology for the computational process is implemented. Surface drag force is computed and discussed. Salient features of significant variables on the physical quantities are reported graphically. It is explored that velocity is enhanced for a larger ratio of rate constants. The increasing values of thermal relaxation factor correspond to less temperature.

     

Impacts of gold nanoparticles on MHD mixed convection Poiseuille flow of nanofluid passing through a porous medium in the presence of thermal radiation,thermal diffusion and chemical reaction

Impacts of gold nanoparticles on MHD Poiseuille flow of nanofluid in a porous medium are studied. Mixed convection is induced due to external pressure gradient and buoyancy force. Additional effects of thermal radiation, chemical reaction and thermal diffusion are also considered. Gold nanoparticles of cylindrical shape are considered in kerosene oil taken as conventional base fluid. However, for comparison, four other types of nanoparticles (silver, copper, alumina and magnetite) are also considered. The problem is modeled in terms of partial differential equations with suitable boundary conditions and then computed by perturbation technique. Exact expressions for velocity and temperature are obtained. Graphical results are mapped in order to tackle the physics of the embedded parameters. This study mainly focuses on gold nanoparticles; however, for the sake of comparison, four other types of nanoparticles namely silver, copper, alumina and magnetite are analyzed for the heat transfer rate. The obtained results show that metals have higher rate of heat transfer than metal oxides. Gold nanoparticles have the highest rate of heat transfer followed by alumina and magnetite. Porosity and magnetic field have opposite effects on velocity.

     

Analysis of magnetohydrodynamic flow and heat transfer of Cu–water nanofluid between parallel plates for different shapes of nanoparticles

The present study analyzes the heat transfer in the flow of copper–water nanofluids between parallel plates. For effective thermal conductivity of nanofluids, Hamilton and Crosser's model has been utilized to examine the flow by considering different shape factors. By employing the suitable similarity transformations, the equations governing the flow are transformed into a set of nonlinear ordinary differential equations. The resulting set of equations is solved numerically with the help of Runge–Kutta–Fehlberg numerical scheme. The graphical simulation presents the analysis of variations, in velocity and temperature profiles, for emerging parameters. A comprehensive discussion also accompanies the graphical results. Moreover, the effects of relevant parameters, on skin friction coefficient and Nusselt number, are highlighted graphically. It is noticed that the velocity field is an increasing function of all the parameters involved. Furthermore, the temperature of the fluid is maximum for the platelet-shaped particles followed by the cylinder- and brick-shaped particles.

     

Thermophoresis and MHD mixed convection three-dimensional flow of viscoelastic fluid with Soret and Dufour effects

Heat and mass transfer effects in three-dimensional mixed convection flow of viscoelastic fluid over a stretching surface with convective boundary conditions are investigated. The fluid is electrically conducting in the presence of constant applied magnetic field. Conservation laws of energy and concentration are based upon the Soret and Dufour effects. First order chemical reaction effects are also taken into account. By using the similarity transformations, the governing boundary layer equations are reduced into the ordinary differential equations. The transformed boundary layer equations are computed for the series solutions. Dimensionless velocity, temperature, and concentration distributions are shown graphically for different values of involved parameters. Numerical values of local Nusselt and Sherwood numbers are computed and analyzed. It is found that the behaviors of viscoelastic, mixed convection, and concentration buoyancy parameters on the Nusselt and Sherwood numbers are similar. However, the Nusselt and Sherwood numbers have qualitative opposite effects for Biot number, thermophoretic parameter, and Soret-Dufour parameters.

     

Homogenous–heterogeneous reactions in MHD flow of Powell–Eyring fluid over a stretching sheet with Newtonian heating

This article addresses the effects of homogenous–heterogeneous reactions on electrically conducting boundary layer fluid flow and heat transfer characteristics over a stretching sheet with Newtonian heating are examined. Using similarity transformations, the governing equations are transformed into nonlinear ordinary differential equations. The constricted ordinary differential equations are solved computationally by shooting technique. The impact of pertinent physical parameters on the velocity, concentration and temperature profiles is discussed and explored via figures and tables. It is clear from figures that the velocity profile reduces for large values of fluid parameter B and Hartmann number H. Skin friction coefficient decreases for large values of Hartmann number H and fluid parameter B. Also, heat transfer rate monotonically enhances with conjugate parameter of Newtonian heating γ and Prandtl number Pr.

     

Rotating flow of viscoelastic fluid with nonlinear thermal radiation: a numerical study

This work is concerned with the numerical solution for rotating viscoelastic flow developed by an exponentially stretching impermeable surface. Temperature at the sheet is also assumed to vary exponentially. Energy equation involves the novel nonlinear radiation heat flux term. Suitable transformations are utilized to nondimensionalize the relevant boundary layer equations. Numerical solutions are developed by means of standard shooting approach. The results demonstrate that both rotation and viscoelasticity serve to reduce the hydrodynamic boundary layer thickness. Temperature function has a special S-shaped profile when the difference between wall and ambient temperatures is sufficiently large. Heat transfer coefficient at the surface diminishes when rotation parameter is increased. Current numerical computations are consistent with those of the existing studies in the literature.

     

Analytical solution for suction and injection flow of a viscoplastic Casson fluid past a stretching surface in the presence of viscous dissipation

In this study, steady two-dimensional flow of a viscoplastic Casson fluid past a stretching surface is considered under the effects of thermal radiation and viscous dissipation. Both suction and injection flows situations are considered. The partial differential governing equations are transformed into ordinary differential equations and solved analytical. Analytical solutions for velocity and temperature are obtained in terms of hypergeometric function and discussed graphically. Moreover, numerical results are also obtained by Runge–Kutta–Fehlberg fourth–fifth-order (RKF45) method and compared with the analytical results. The results showed that the injection and suction parameter can be used to control the direction and strength of flow. The effects of Casson parameter on the temperature and velocity are quite opposite. The effects of thermal radiation on the temperature are much more stronger in case of injection. The heat transfer coefficient shows higher value for Casson fluid while for Newtonian fluid is the lowest.

     

Molecular dynamics-based prediction of boundary slip of fluids in nanochannels

Computational modeling and simulation can provide an effective predictive capability for flow properties of the confined fluids in micro/nanoscales. In this paper, considering the boundary slip at the fluid–solid interface, the motion property of fluids confined in parallel-plate nanochannels are investigated to couple the atomistic regime to continuum. The corrected second-order slip boundary condition is used to solve the Navier–Stokes equations for confined fluids. Molecular dynamics simulations for Poiseuille flows are performed to study the influences of the strength of the solid–fluid coupling, the fluid temperature, and the density of the solid wall on the velocity slip at the fluid boundary. For weak solid–fluid coupling strength, high temperature of the confined fluid and high density of the solid wall, the large velocity slip at the fluid boundary can be obviously observed. The effectiveness of the corrected second-order slip boundary condition is demonstrated by comparing the velocity profiles of Poiseuille flows from MD simulations with that from continuum.     

Thermal radiation and slip effects on MHD stagnation point flow of non-Newtonian nanofluid over a convective stretching surface

The present analysis examines the combine effects of thermal radiation and velocity slip along a convectively nonlinear stretching surface. Moreover, MHD effects are also considered near the stagnation point flow of Casson nanofluid. Slipped effects are considered with the porous medium to reduce the drag reduction at the surface of the sheet. Main structure of the system is based upon the system of partial differential equations attained in the form of momentum, energy, and concentration equations. To determine the similar solution system of PDEs is rehabilitated into the set of nonlinear ordinary differential equations (ODEs) by employing compatible similarity transformation. Important physical parameters are acquired through obtained differential equations. To determine the influence of emerging parameters, resulting set of ODE’s in term of unknown function of velocity, temperature, and concentration are successfully solved via Keller’s box-scheme. All the obtained unknown functions are discussed in detail after plotting the results against each physical parameter. To analyze the behavior at the surface: skin friction, local Nusselt and Sherwood numbers are also illustrated against the velocity ratio parameter A, Brownian motion Nb, thermophoresis Nt, and thermal radiation parameters R. Results obtained from the set of equations described that skin friction is decreasing function of A, and local Nusselt and Sherwood number demonstrate the significant influenced by Brownian motion Nb, thermophoresis Nt, and radiation parameters R.

     

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v_θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.     

Numerical analysis of the micro-Couette flow using a non-Newton–Fourier model with enhanced wall boundary conditions

Non-equilibrium effects exist extensively in microfluidic flows, and the accurate simulation of the Knudsen layer behind them is rather challenging for the linear Newton–Fourier model. In this paper, a high-order reduced model (nonlinear coupled constitutive relations) from Eu’s generalized hydrodynamic equations is applied for the investigation of the micro-Couette flows of diatomic nitrogen and monatomic argon as well as Maxwell and hard-sphere molecules using the MacCormack scheme. In order to simulate the confined flows accurately, a set of enhanced wall boundary conditions based on this model are derived with respect to the degree of non-equilibrium. Both the 1st-order Maxwell–Smoluchowski model and the Langmuir slip model are also investigated. For a large range of Knudsen numbers, the results show that the enhanced boundary conditions make a significant improvement in the prediction of flow profiles, especially the temperature profile. The reason behind that is analyzed in detail. The numerical predictions obtained from the high-order model in conjunction with the enhanced boundary conditions are also compared with DSMC, regularized 13 moment equations, Burnett-type equations as well as Navier–Stokes solutions, which highlight its excellent capability in describing the underlying mechanism of the Knudsen layer in the Couette flow.     

Velocity and temperature slip effects on squeezing flow of nanofluid between parallel disks in the presence of mixed convection

Velocity and temperature slip effects on squeezing flow of nanofluid between parallel disks in the presence of mixed convection is considered. Equations that govern the flow are transformed into a set of differential equations with the help of transformations. For the purpose of solution, homotopy analysis method is used. The BVPh2.0 package is utilized for the said purpose. Deviations in the velocity, temperature and the concentration profiles are depicted graphically. Mathematical expressions for skin friction coefficient, Nusselt and the Sherwood numbers are derived and the variations in these numbers are portrayed graphically. From the results obtained, we observed that the coefficient of skin friction increases with increase in Hartmann number M for the suction flow (A > 0), while in the blowing flow (A < 0) a fall is seen with increasing M. However, for rising values of velocity parameter β the effect of skin friction coefficient is opposite to that accounted for M. Variations in thermophoresis parameter N _T and thermal slip parameter γ give rise in Nusselt number for both the suction and injection at wall. For both the suction and injection at wall, Sherwood number gets a rise with growing values of Brownian motion parameter N _B, while a drop is seen in Sherwood number for increasing values of thermophoresis parameter N _T. For the sake of comparison, the same problem is also solved by employing a numerical scheme called Runge–Kutta–Fehlberg (RKF) method. Results thus obtained are compared with existing ones and are found to be in agreement.