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.

     

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.

     

Exact and numerical solutions for unsteady heat and mass transfer problem of Jeffrey fluid with MHD and Newtonian heating effects

The combined heat and mass transfer of unsteady magnetohydrodynamic free convection flow of Jeffrey fluid past an oscillating vertical plate generated by thermal radiation and Newtonian heating is investigated. The incompressible fluid is electrically conducting in the presence of a uniform magnetic field which acts in a direction perpendicular to the flow. Mathematical formulation of the problem is modeled in terms of partial differential equations with some physical conditions. Some suitable non-dimensional variables are introduced to transform the system of equations. The dimensionless governing equations are solved analytically for exact solutions using the Laplace transform technique. Numerical solutions of velocity are obtained via finite difference scheme. Graphical results for velocity, temperature and concentration fields for various pertinent parameters such as material parameter of Jeffrey fluid \(\lambda_{1}\), dimensionless parameter of Jeffrey fluid \(\lambda\), Newtonian heating parameter \(\xi\), phase angle \(\omega t\), Grashof number \(Gr\), modified Grashof number \(Gm\), Hartmann number or magnetic parameter \(Ha\), Prandtl number \(Pr\), radiation parameter \(Rd\), Schimdt number \(Sc\) and dimensionless time \(t\) are displayed and discussed in detail. This study showed that the magnetic field resists the fluid flow due to the Lorentz force, whereas the thermal radiation and Newtonian heating parameters lead to the enhancement of velocity and temperature fields. Present results are also compared with the existing published work, and an excellent agreement is found.

     

Laminar fluid flow and heat transfer in an internally corrugated tube by means of the method of fundamental solutions and radial basis functions

The paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube. Cross-section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations, e.g. inside arteries. The boundary value problem is described by two partial differential equations (one for fluid flow problem and one for heat transfer problem) and appropriate boundary conditions. During solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically. In the paper the Nusselt number and the product of friction factor and Reynolds number are presented for some selected geometrical parameters (the number and amplitude of corrugations). It is shown that for a given number of corrugations a minimal value of the product of friction factor and Reynolds number can be found. As it was expected the Nusselt number increases with increasing amplitude and number of corrugations.     

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.

     

Boundary-layer flow of the power-law fluid over a moving wedge: a linear stability analysis

We study the stability of the two-dimensional boundary-layer flow of a power-law (Ostwald de Waele) non-Newtonian fluid over a moving wedge. The mainstream velocity is assumed to have a power of distance from the leading boundary layer, such that the system admits to the self-similar solutions. We discuss the problem in question for both shear-thickening and shear-thinning fluids which lead to a non-uniqueness (double solutions) in the base flow solutions. We then address an issue of the stability of the non-unique solutions. A linear eigenvalue analysis of the double solution reveals that the basic flow represented by the first solution is always stable, and this flow is practically encountered. The system becomes unstable to the second solutions which have the mode-two perturbations with larger boundary-layer thickness. The first and second solutions form a tongue-like structure in the solution space. Furthermore, the modification of the viscosity for the power-law fluids reveals that the system predicts an infinite viscosity in the confinement of the boundary-layer region. Extensive comparisons of the solutions with the existing models with Newtonian fluid are made, and a physical explanation behind these solutions is proposed.

     

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.

     

Numerical solution of shear-thinning and shear-thickening boundary-layer flow for Carreau fluid over a moving wedge

This paper investigates the linear stability of the flow in the two-dimensional boundary-layer flow of the Carreau fluid over a wedge. The corresponding rheology is analysed using the non-Newtonian Carreau fluid. Both mainstream and wedge velocities are approximated in terms of the power of distance from the leading edge of the boundary layer. These forms exhibit a class of similarity flows for the Carreau fluid. The governing equations are derived from the theory of a non-Newtonian fluid which are converted into an ordinary differential equation. We use the Chebyshev collocation and shooting techniques for the solution of governing equations. Numerical results show that the viscosity modification due to Carreau fluid makes the boundary layer thickness thinner. Numerical results predict an additional solution for the same set of parameters. Thus, a further aim was to assess the stability of dual solutions as to which of the solutions can be realized. This leads to an eigenvalue problem in which the positive eigenvalues are important and intriguing. The results from eigenvalues form tongue-like structures which are rather new. The presence of the tongue means that flow becomes unstable beyond the critical value when the velocity ratio is increased from the first solution.

     

Remarks on the cellular automaton global synchronisation problem: deterministic versus stochastic models

In the global synchronisation problem, one is asked to find a cellular automaton which has the property that every initial condition evolves into a homogeneous blinking state. We study this simple inverse problem for the case of one-dimensional systems with periodic boundary conditions. Two paradoxical observations are made: (a) despite the apparent simplicity of finding rules with good statistical results, there exist no perfect deterministic solutions to this problem, (b) if we allow the use of randomness in the local rule, constructing “perfect” stochastic solutions is easy. For the stochastic case, we give some rules for which the mean time of synchronisation varies quadratically with the number of cells and ask if this result can be improved. To explore more deeply the deterministic rules, we code our problem as a SAT problem and use SAT solvers to find rules that synchronise a large set of initial conditions (in appendix).

     

A revised model to study the MHD nanofluid flow and heat transfer due to rotating disk: numerical solutions

Here our main interest is to present numerical simulations for magneto-nanofluid flow and heat transfer near a rotating disk. Buongiorno model, featuring the novel aspects of Brownian motion and thermophoresis, is accounted. Heat dissipation effect is preserved in the energy balance equation. We take into account more realistic wall condition which requires passive control of nanoparticle concentration at the disk. The traditional Von Karman relations have been invoked to attain self-similar differential system. Keller–Box method has been implemented to compute similarity solutions of the problem. Streamlines are prepared in both two and three dimensions for adequate flow visualization. The behavior of involved parameters on the flow fields is examined graphically. It is predicted that the torque required to maintain disk in steady rotation increases when magnetic field effects are enhanced. Fluid flow in the radial, azimuthal and vertical directions is opposed by the magnetic field strength. Thermophoresis effect enhances temperature and reduces heat flux from the disk. However, Brownian diffusion has a marginal influence on temperature distribution. Heat transfer coefficient is reduced due to the inclusion of heat dissipation terms. Present results are consistent with those of the available studies in a limiting situation.

     

Asymptotic and finite element approximations for heat transfer in rotating compressible flow over an infinite porous disk

The laminar boundary layer equations for the compressible flow due to the finite difference in rotation and temperature rates are solved for the case of uniform suction through the disk. The effects of viscous dissipation on the incompressible flow are taken into account for any rotation rate, whereas for a compressible fluid they are considered only for a disk rotating in a stationary fluid. For the general case, the governing equations are solved numerically using a standard finite element scheme. Series solutions are developed for those cases where the suction effect is dominant. Based on the above analytical and numerical solutions, a new asymptotic finite element scheme is presented. By using this scheme one can significantly improve the pointwise accuracy of the standard finite element scheme.     

Purely analytic solutions of magnetohydrodynamic swirling boundary layer flow over a porous rotating disk

The prime objective of the present study is to derive analytical expressions for the solution of steady, laminar, incompressible, viscous and electrically conducting fluid of the boundary layer flow due to a rotating disk subjected to a uniform suction and injection through the wall in the presence of a uniform transverse magnetic field. To serve this purpose, the recently popular homotopy analysis method is employed to obtain the exact solutions, in contrast to the numerically evaluated ones in the literature. It is shown here that such a technique is extremely powerful in gaining magnetohydrodynamic solutions in terms of the purely exponential and decaying functions if a special care is taken into account. This makes it possible to obtain explicitly analytic solutions particularly in coincident with the Ackroyd’s solutions in (Ackroyd, 1978) [1] and with the solutions in (Ariel, 2001) [2]. The method is further shown to be capable of overcoming the difficulties existed in calculating Ackroyd’s solutions for high values of injection. Using the homotopy analysis method, electrically conducting mean velocity profiles corresponding to a wide range of suction and injection velocities can be readily computed non-iteratively and analytically. Explicit formulas are also derived for some parameters of physical significance.     

Optimal redundancy resolution of a kinematically redundant manipulator for a cyclic task

This article proposes a method for the global optimization of redundancy over the whole task period in a kinematically redundant manipulator. The necessary conditions based on the calculus of variations for integral-type criteria result in a second-order differential equation. For a cyclic task, the boundary conditions for conservative joint motions are discussed. Then, we reformulate a two-point boundary value problem to an initial value adjustment problem and suggest a numerical search method based on the iterative optimization for providing a globally optimal solution using the gradient projection method. Since the initial joint velocity is parameterized with the number of redundancy, we only search parameter values in the parameterized space using the configuration error between the initial and final time. We show through numerical examples that multiple nonhomotopic extremal solutions satisfying periodic boundary conditions exist according to initial joint velocities for the same initial configuration. Finally, we discuss an algorithm for topological liftings of the paths and demonstrate the generality of the proposed method by considering the dynamics of a manipulator.     

Critical rotational speed,critical velocity of fluid flow and free vibration analysis of a spinning SWCNT conveying viscous fluid

In this article, the influences of rotational speed and velocity of viscous fluid flow on free vibration behavior of spinning single-walled carbon nanotubes (SWCNTs) are investigated using the modified couple stress theory (MCST). Taking attention to the first-order shear deformation theory, the modeled rotating SWCNT and its equations of motion are derived using Hamilton’s principle. The formulations include Coriolis, centrifugal and initial hoop tension effects due to rotation of the SWCNT. This system is conveying viscous fluid, and the related force is calculated by modified Navier–Stokes relation considering slip boundary condition and Knudsen number. The accuracy of the presented model is validated with some cases in the literatures. Novelty of this study is considering the effects of spinning, conveying viscous flow and MCST in addition to considering the various boundary conditions of the SWCNT. Generalized differential quadrature method is used to approximately discretize the model and to approximate the equations of motion. Then, influence of material length scale parameter, velocity of viscous fluid flow, angular velocity, length, length-to-radius ratio, radius-to-thickness ratio and boundary conditions on critical speed, critical velocity and natural frequency of the rotating SWCNT conveying viscous fluid flow are investigated.     

Dynamic stability of viscoelastic nanotubes conveying pulsating magnetic nanoflow under magnetic field

In this study, dynamic stability analysis of viscoelastic carbon nanotubes (CNTs) conveying pulsating magnetic nanoflow subjected to a longitudinal magnetic field is investigated. Based on Hamilton’s principle, the governing equations as well as boundary conditions, are extracted. The dynamic instability region and pulsation frequency of the CNTs are obtained through both the Galerkin technique and the Bolotin method. The effects of the nonlocal parameter gather with strain gradient parameter, Knudsen number, magnetic field, mass fluid ratio, fluid velocity, tension, gravity, viscoelastic characteristic of materials and boundary conditions on the dynamic instability of system are deliberated. The results indicate that increase in the pulsation frequency is caused by the decrease of nonlocal parameter and the increase of strain gradient parameter. Besides, it is revealed that by increasing Knudsen number the pulsation frequency decreases. Furthermore, the dynamic instability region and pulsation frequency of CNT can be enhanced due to the magnetic field effects.

     

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.

     

Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study

A numerical simulation for mixed convective three-dimensional slip flow of water-based nanofluids with temperature jump boundary condition is presented. The flow is caused by nonlinear stretching surface. Conservation of energy equation involves the radiation heat flux term. Applied transverse magnetic effect of variable kind is also incorporated. Suitable nonlinear similarity transformations are used to reduce the governing equations into a set of self-similar equations. The subsequent equations are solved numerically by using shooting method. The solutions for the velocity and temperature distributions are computed for several values of flow pertinent parameters. Further, the numerical values for skin-friction coefficients and Nusselt number in respect of different nanoparticles are tabulated. A comparison between our numerical and already existing results has also been made. It is found that the velocity and thermal slip boundary condition showed a significant effect on momentum and thermal boundary layer thickness at the wall. The presence of nanoparticles stabilizes the thermal boundary layer growth.

     

Effect of complex turbulator on heat transfer of nanomaterial considering turbulent flow

To exhibit the nanomaterial hydrothermal behavior within a duct, FVM has been utilized and to augment the performance, helical complex device was incorporated. To achieve the formulation of problem, K–ɛ model was applied with considering CuO–water nanomaterial. Outputs in forms of velocity and temperature contours have been extracted. Stronger tangential contact of carrier fluid with outer wall guarantees the thinner boundary layer with rise of inlet velocity and friction loss augments. The increase of turbulator width results in augment of Nu owing to greater tangential flow and more fluctuations.

     

Discrete radiation boundary conditions for a semi-infinite layer of fluid

The paper deals with the discrete formulation of radiation boundary conditions for a layer of fluid. The problem is examined with the help of the finite difference method. The proposed radiation boundary enables us to replace an infinite layer by a finite domain. The conditions ensure near equivalence between the infinite layer and the proposed finite model. The method is consistent itself and operates on a finite number of points. The results of numerical solutions are in good agreement with the results of analytical solutions of the problem.     

Numerical study of radiative Maxwell viscoelastic magnetized flow from a stretching permeable sheet with the Cattaneo–Christov heat flux model

In this article, the Cattaneo–Christov heat flux model is implemented to study non-Fourier heat and mass transfer in the magnetohydrodynamic flow of an upper-convected Maxwell fluid over a permeable stretching sheet under a transverse constant magnetic field. Thermal radiation and chemical reaction effects are also considered. The nonlinear partial differential conservation equations for mass, momentum, energy and species conservation are transformed with appropriate similarity variables into a system of coupled, highly nonlinear ordinary differential equations with appropriate boundary conditions. Numerical solutions have been presented for the influence of elasticity parameter (α), magnetic parameter (M ²), suction/injection parameter \((\lambda ),\) Prandtl number (Pr), conduction–radiation parameter (R _d ), sheet stretching parameter (A), Schmidt number (Sc), chemical reaction parameter \(\left( {\gamma_{c} } \right)\), modified Deborah number with respect to relaxation time of heat flux (i.e., non-Fourier Deborah number) on velocity components, temperature and concentration profiles using the successive Taylor series linearization method (STSLM) utilizing Chebyshev interpolating polynomials and Gauss–Lobatto collocation. The effects of selected parameters on skin friction coefficient, Nusselt number and Sherwood number are also presented with the help of tables. Verification of the STSLM solutions is achieved with existing published results demonstrating close agreement. Further validation of skin friction coefficient, Nusselt number and Sherwood number values computed with STSLM is included using Mathematica software shooting quadrature.