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.

     

Nonlocal divergence and flutter instability analysis of embedded fluid-conveying carbon nanotube under magnetic field

This paper deals with nonlocal divergence and flutter instability analysis of carbon nanotubes (CNTs) conveying fluid embedded in an elastic foundation under magnetic field. Nonlocal constitutive equations of Eringen and Euler–Bernoulli beam theory are used in the formulations. Also, the foundation is described by the Winkler and Pasternak models. The governing equation of motion and boundary conditions are derived using extended Hamilton’s variational principle. The extended Galerkin’s approach is adopted to reduce the partial differential equation governing the dynamics of the CNTs to a system of coupled ordinary differential equations. In the present study, four different boundary conditions are considered, namely the pined–pined (P–P), clamped–pined (C–P), clamped–clamped (C–C) and clamped–free (C–F). A detailed parametric study is conducted to elucidate the effects of the nonlocal effect, longitudinal magnetic field, elastic Winkler and Pasternak foundations and geometrically boundary conditions on the instability characteristic of CNTs. It was observed that the only instability type for the investigated CNT with clamped–free boundary condition (cantilever) is flutter, while CNT conveying fluid with both ends supported loses its stability by divergence first and then by flutter with increase in fluid velocity. It was also found that the magnetic field and the Winkler and Pasternak foundations increase the stiffness of the system. Therefore, flutter instability region is enlarged significantly due to the existence of springs, shear foundations and magnetic field. Also, results show that the nonlocal parameter has a prominent effect on the stability behavior of CNTs, in which increasing nonlocal parameter results in the decrease in stability region. Furthermore, it was shown that the stability behavior of CNT is strongly affected by different boundary conditions. Finally, the validity of the present analysis is confirmed by comparing the results with those obtained from 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.

     

Diffusion of a chemically reactive species of a power-law fluid past a stretching surface

A numerical solution for the steady magnetohydrodynamic (MHD) non-Newtonian power-law fluid flow over a continuously moving surface with species concentration and chemical reaction has been obtained. The viscous flow is driven solely by the linearly stretching sheet, and the reactive species emitted from this sheet undergoes an isothermal and homogeneous one-stage reaction as it diffuses into the surrounding fluid. Using a similarity transformation, the governing non-linear partial differential equations are transformed into coupled nonlinear ordinary differential equations. The governing equations of the mathematical model show that the flow and mass transfer characteristics depend on six parameters, namely, the power-law index, the magnetic parameter, the local Grashof number with respect to species diffusion, the modified Schmidt number, the reaction rate parameter, and the wall concentration parameter. Numerical solutions for these coupled equations are obtained by the Keller-Box method, and the solutions obtained are presented through graphs and tables. The numerical results obtained reveal that the magnetic field significantly increases the magnitude of the skin friction, but slightly reduces the mass transfer rate. However, the surface mass transfer strongly depends on the modified Schmidt number and the reaction rate parameter; it increases with increasing values of these parameters. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

Dynamic motion analysis of magnetic particles in microfluidic systems under an external gradient magnetic field

To gain an insightful understanding of motion behavior of paramagnetic particles suspended in a nonmagnetic fluid under a gradient magnetic field, a coupled fluid–structure model based on a direct numerical scheme is developed in this work. The governing equations of magnetic field, fluid flow field and particle motion are simultaneously solved using an Arbitrary Lagrangian–Eulerian method, taking into account magnetic and hydrodynamic interactions between particles in a fully coupled manner. The accuracy of the proposed method is validated using the magnetic particulate flows of two particles under a uniform magnetic field as the test problem and is then applied to investigate effects of magnetic and hydrodynamic interactions between particles on the particle motion behavior. Results show that neighboring magnetic particles are easy to form chain-like clusters along field direction due to magnetic interactions between particles and then move together toward the surface of magnetic source under the action of gradient magnetic force. More importantly, it has been found that both magnetic and hydrodynamic interactions between particles are conducive to the acceleration of particles and the chain formation of particles. The present method and results could help in understanding the basic mechanism underlying the low-gradient magnetophoretic separation process and designing magnetic aggregate-based microfluidic devices.     

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.

     

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.

     

A numerical model on pulsatile flow of magnetic nanoparticles as drug carrier suspended in Herschel–Bulkley fluid through an arterial stenosis under external magnetic field and body force

ABSTRACT

A theoretical model for blood flow through an artery with stenosis carrying magnetic particles in the presence of magnetic field and periodic body acceleration is analysed. In the present study, blood is assumed to be Herschel–Bulkley fluid carrying iron oxide nanoparticles. The governing equations are highly non-linear and were solved numerically. The effects of model parameters are investigated and the results are represented graphically. The shear stress at the arterial wall and resistive impedance increases with enhancing values of stenotic height, yield stress, flow behaviour index, consistency index, pulsatile Reynolds number, amplitude of body acceleration, particle concentration and particle mass parameters. In order to treat the circulation disorders, control of the parameters involved in blood flow is necessary. The present model is useful in normalizing the parameter values and hence it can be applied in the field of medicine. The study has significant applications in drug delivery for treating cancer.     

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.

     

Behavior of micro-polar flow due to linear stretching of porous sheet with injection and suction

The boundary layer flow of a micro-polar fluid due to a linearly stretching sheet is investigated. The influence of various flow parameters like ‘suction and injection velocity through the porous surface’, ‘viscosity parameter causing the coupling of the micro-rotation field and the velocity field’ and ‘vortex viscosity parameter’ on ‘shear stress at the surface’, ‘fluid velocity’ and ‘micro-rotation’ are studied. The governing equations of the transformed boundary layer are solved analytically using homotopy analysis method (HAM). The convergence of the obtained series solutions is explicitly studied and a proper discussion is given for the obtained results. Comparison between the HAM and numerical solutions showed excellent agreement.     

Linear stability of Hagen–Poiseuille flow in a chemically reacting fluid

In this short paper we study the linearized stability of the flow of a chemically reacting fluid in a cylindrical pipe, under the assumption that the length of the pipe is far greater than its diameter. The fluid models that are considered have relevance to the flow of both polymeric liquids that are capable of undergoing chemical reactions and biological fluids such as the synovial fluid whose viscosity changes due to the concentration of the hyaluronan. The viscosity of the class of fluids that we consider can increase or decrease due to the concentration of the chemical that is being carried by the fluid and it can also shear thin or shear thicken. We non-dimensionalize the equations governing the motion of the fluid and then carry out an approximation wherein we retain terms that are of order unity in the Reynolds number and Péclet number. We further simplify the problem by seeking a special semi-inverse solution, in the same spirit as that which is used in the study of classical Hagen–Poiseuille flow, and look for solutions for the velocity field and the concentration that vary only with the radial coordinate. Under the above mentioned approximation, one can obtain an exact solution for the basic flow which then allows us to analytically consider the stability of the base flow to sufficiently small disturbances. On the basis of earlier studies of such fluids in the modeling of biological fluids, especially the synovial fluid, we consider two types of variation of the viscosity with the concentration. We find that flows in the cylindrical pipe, within the context of our approximation, are stable to sufficiently small disturbances, for both variations of the viscosity that are considered.     

Nanofluid flow over a shrinking sheet with surface slip

In the present study, the effects of partial slip on mixed convection stagnation point flow and heat transfer of nanofluid impinging normally toward a shrinking sheet are investigated numerically. In particular, focus is on Cu–water and Al₂O₃–water nanofluids. Similarity transformation technique is adopted to obtain the self-similar ordinary differential equations and then solved numerically using Runge–Kutta–Fehlberg method with shooting technique. A parametric study is performed to explore the effects of various governing parameters on the fluid flow and heat transfer characteristics. Both the cases of assisting and opposing flows are considered. The physical aspects of the problem are highlighted and discussed.     

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.

     

Space approach to the hydro-magnetic flow of a dusty fluid through a porous medium

The technique of the state space approach and the inversion of the Laplace transformation method are applied to the non-dimensional equations of an unsteady laminar free convection flow of an incompressible, viscous, electrically conducting dusty fluid through a porous medium, which is bounded by an infinite vertical plane surface of constant temperature, in the presence of a constant magnetic field. The technique is applied to the thermal shock problem. The inversion of the Laplace transforms is carried out using a numerical approach. The numerical results of the dimensionless temperature, velocity, and induced magnetic and electric field distributions are given and illustrated graphically. The effects of the material’s parameters such as the Grashof number, the Prandtl number, the permeability parameter, the mass concentration of the particle phase, the Alfven velocity, the thermal relaxation time and the relaxation time of the particle phase on the temperature, velocity and the induced magnetic and electric fields are discussed.     

Natural convection boundary layer of a non-Newtonian fluid about a permeable vertical cone embedded in a porous medium saturated with a nanofluid

An analysis was performed to study the effect of uniform transpiration velocity on free convection boundary-layer flow of a non-Newtonian fluid over a permeable vertical cone embedded in a porous medium saturated with a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing partial differential equations are transformed into a set of non-similar equations and solved numerically by an efficient implicit, iterative, finite-difference method. Comparisons with previously published work are performed and excellent agreement is obtained. A parametric study of the physical parameters is conducted and a representative set of numerical results for the velocity, temperature, and volume fraction profiles as well as the local Nusselt and Sherwood numbers is illustrated graphically to show interesting features of the solutions.     

Pulsatile vibrations of viscoelastic microtubes conveying fluid

In this paper, the pulsatile coupled vibrations of a viscoelastic microtube conveying pulsatile fluid is examined for the first time. The problem is grouped into the class of parametrically excited, internally damped, gyroscopic where both Coriolis and parametric forces are present in the presence of viscosity. The Kelvin–Voigt approach of the viscosity, the Euler–Bernoulli for the deformation, the modified couple stress theory for the small size, and Hamilton’s principle for deriving differential equations are used. Parametric frequency–response curves are obtained in the vicinity of the parametric resonance near the critical speed for both subcritical and supercritical regimes. The effect of the flow pulsation on the oscillations is investigated.

     

Buckling and post-buckling of magneto-electro-thermo-elastic functionally graded porous nanobeams

The problem of the nonlinear thermal buckling and post-buckling of magneto-electro-thermo-elastic functionally graded porous nanobeams is analyzed based on Eringen’s nonlocal elasticity theory and by using a refined beam model. The beams with immovable clamped ends are exposed to the external electric voltages, magnetic potentials, a uniform transverse load and uniform temperature change. For the first time, the four types of porosity distribution in the nanobeam are considered and compared in complex electric–magnetic fields. Besides, the new formula of the effective material properties is proposed in this paper to simultaneously estimate the material distribution and porosity distribution in the thickness direction. The generalized variation principle is used to induce the governing equations, then the approximate analytical solution of the METE-FG nanobeams based on physical neutral surface is obtained by using a two-step perturbation technique. Finally, detailed parametric analyses are performed to get an insight into the effects of different physical parameters, including the slenderness ratio, small-scale parameter, volume fraction index, external electric voltages, magnetic potentials, porosity coefficient and different porosity distributions, for providing an effective way to improve post-buckling strength of porous beams.

     

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.

     

Numerical study of unsteady hydromagnetic Generalized Couette flow of a reactive third-grade fluid with asymmetric convective cooling

Unsteady hydromagnetic Generalized Couette flow and heat transfer characteristics of a reactive variable viscosity incompressible electrically conducting third grade fluid in a channel with asymmetric convective cooling at the walls in the presence of uniform transverse magnetic field is studied. It is assumed that the chemical kinetics in the flow system is exothermic and the convective heat transfer at the channel surface with the surrounding environment follow the Newton’s law of cooling. The coupled nonlinear partial differential equations governing the problem are derived and solved numerically using an unconditionally stable and convergent semi-implicit finite difference scheme. Both numerical and graphical results are presented and physical aspects of the problem are discussed with respect to various parameters embedded in the system.