Numerical modelling of second‐grade fluid flow past a stretching sheet

The present numerical study reports the chemically reacting boundary layer flow of a magnetohydrodynamic second‐grade fluid past a stretching sheet under the influence of internal heat generation or absorption with work done due to deformation in the presence of a porous medium. To distinguish the non‐Newtonian behaviour of the second‐grade fluid with those of Newtonian fluids, a very popularly known second‐grade fluid flow model is used. The fourth order momentum equation with four appropriate boundary conditions along with temperature and concentration equations governing the second‐grade fluid flow are coupled and highly nonlinear in nature. Well‐established similarity transformations are efficiently used to reduce the dimensional flow equations into a set of nondimensional ordinary differential equations with the necessary conditions. The standard bvp4c MATLAB solver is effectively used to solve the fluid flow equations to get the numerical solutions in terms of velocity, temperature, and concentration fields. Numerical results are obtained for a different set of physical parameters and their behaviour is described through graphs and tables. The viscoelastic parameter enhances the velocity field whereas the magnetic and porous parameters suppress the velocity field in the flow region. The temperature field is magnified for increasing values of the heat source/sink parameter. However, from the present numerical study, it is noticed that the flow of heat occurs from sheet to the surrounding ambient fluid. Before concluding the considered problem, our results are validated with previous results and are found to be in good agreement.     

Study of an unsteady fluctuative MHD flow of elasticoviscous liquid past a vertical porous plate

Most flows which occur in nature/practical applications are fluctuating. The fluctuating motions superimposed on the main motion are complex. Further, the unsteadiness of the flow is an added reality to applications in various fields. The free convection flow of an electrically conducting fluid past different types of vertical bodies subjected to a magnetic field is studied because of its wide range of applications in astrophysics, geophysics, aerodynamics, electromagnetic pumps, the flow of liquid metals, and so forth. In the present analysis, an attempt has been made to study the thermal radiation effect on the unsteady magnetohydrodynamic flow of an incompressible elasticoviscous liquid (Walters-B' fluid model) along an infinite hot vertical permeable surface embedded in a porous medium with heat source and chemical reaction. The governing equations of motion, energy, and concentration are solved by an approximate analytical method, that is, the successive perturbation technique and numerical method (Runge–Kutta with shooting). The solution procedure rests upon the basic assumption that the unsteady boundary layer involves a steady basic flow superimposed on an unsteady flow. The most striking outcome is that the combined effect of oscillation outflow, the elasticity of the fluid, and thermal as well as mass buoyancy overrides the resistive electromagnetic force and suction at the plate to enhance the velocity so that high values of magnetic strength are not desired. Further, a higher value of the heat source parameter accelerates the momentum diffusion resulting in the escalation of the velocity field. Fall of concentration is relatively faster in cases of heavier species as well as destructive reactions. The heat transfer coefficient assumes positive values indicating the heat flows from the plate to the fluid (cooling of the bounding surface and heating of the fluid). These observations may have industrial (design of heat exchanges) and therapeutic bearings.     

A non‐Fourier heat flux model for magnetohydrodynamic micropolar liquid flow across a coagulated sheet

In this analysis, a heat transfer extrusion system was made by using a modified heat flux model, namely, the Cattaneo‐Christov heat flux. In the present study, we examined the effect of Arrhenius activation energy on magnetohydrodynamic mixed convection stagnation point flow of a micropolar fluid over a variable thickened surface in the attendance of Brownian motion. The fluid motion is assumed to be steady and laminar. The combined influence of heat and mass transfer aspects are scrutinized. First, suitable transformations are considered to modify the governing partial differential equations as ordinary differential equations and revealed by the consecutive application of numerical procedures like shooting and Runge‐Kutta‐Fehlberg. Graphs are delineated to scrutinize the effects of sundry dimensionless parameters on the flow fields. We found that, the present results made a good agreement with the existing results. We observed that there is an enhancement in the fields of concentration with thermophoresis and activation energy parameters but an opposite trend is noticed for the Brownian motion parameter. Also, it is interesting to note that the buoyancy and the primary slip parameters are increasing functions of velocity fields.     

Buoyancy forces and activation energy on the MHD radiative flow over an exponentially stretching sheet with second‐order slip

The current study explores the effects of second‐order slip and activation energy (AE) on the magnetohydrodynamic and radiative fluid flow caused by a surface with exponential stretching. The binary chemical reaction with mixed convection is considered in this physical model to discover the heat transfer phenomenon. The governing system of equations leads to a set of nonlinear ordinary differential equations by using scaling analysis. The transformed system is calculated computationally by using the most powerful Shooting procedure with the support of MATLAB software. The characteristics of various flow parameters on the governing flow field are exhibited pictorially and deliberated. The results revealed that the coefficient of heat and mass transfer upsurge with growing values of the second‐order slip parameter and skin friction coefficient has a reverse effect on the first‐order slip parameter. The thermal measure of the fluid in the presence of suction and slip conditions is seen to be lesser than that with the nonslip and nonsuction conditions. The heat measure of the fluid augments with the rising buoyancy parameter. The influence of slips coupled with AE is significant in the fluid flow and heat transfer characteristics. The outputs of the current investigation are validated by comparing the Nusselt number with the available results and are found to closely agree as a limiting case.     

Effect of Joule heating on MHD non‐Newtonian fluid flow past an exponentially stretching curved surface

The current endeavor examines the convective heat transfer characteristics on magnetohydrodynamic stagnation point flow of micropolar fluid past an exponential curved surface. The flow is supposed to be laminar and time‐independent. The influence of radiation, irregular heat source/sink, Joule heating, and variable thermal conductivity are supposed. Suitable similarity renovations are considered to transform the original partial differential equations as ordinary ones and then resolved by shooting and fourth‐order Runge–Kutta methods. Graphs are drawn to inspect the impacts of sundry nondimensional parameters on the distributions of velocity, microrotation, and temperature. We detect that there is an escalation in temperature with Eckert number and variable heat source/sink parameters. Also, it is motivating to comment that Biot number is an increasing function of local Nusselt number.     

Flow and heat transfer in a porous medium saturated with a Sisko nanofluid over a nonlinearly stretching sheet with heat generation/absorption

This work is focused on steady flow and heat transfer in a porous medium saturated with a Sisko nanofluid (non‐Newtonian power‐law) over a nonlinearly stretching sheet in the presence of heat generation/absorption. Nonlinear PDEs are transformed into a system of coupled nonlinear ODEs with related boundary conditions using similarity transformation. The reduced equations are then solved numerically using the Runge–Kutta–Fehlberg fourth–fifth order method (RKF45) with Maple 14.0 software. The solutions depend on the power‐law index n and the effect of pertinent parameter such as the Brownian motion parameter, thermophoresis parameter, Lewis number, the permeability, and the heat generation/absorption on the dimensionless velocity, temperature, and nanoparticles volume fraction and also on the skin friction, local Nusselt, and Sherwood numbers are produced for values of the influence parameter. A rapprochement of the numerical results of the actual study with formerly published data detected an excellent agreement.     

Influence of relative motion of magnetic field on time‐dependent MHD natural convection flow

The effects of relative motion of magnetic field on unsteady magnetohydrodynamic free convection flow with ramped motion and temperature‐dependent heat source/sink have been analyzed. The motion of the inner cylinder is ramped while the motion of the outer cylinder is fixed. The momentum and energy equations are solved using the well‐known Laplace transform. The time‐domain solution is obtained using the Riemann‐sum approximation method. The influence of the governing parameters on fluid velocity, fluid temperature, volume flow rate, and rate of heat transfer are discussed with the help of line graphs. It is found that Hartmann number has a retarding effect on fluid velocity, skin friction at the outer surface of the inner cylinder, and mass flow rate when the magnetic field is fixed with the fluid and when the velocity of the magnetic field is less than the velocity of the moving cylinder. Whereas, the reverse effect is noticed when the magnetic field is fixed with the moving cylinder.     

Thermocapillary thin film flow upon a porous heated stretchable cylinder

Earlier research work related to the unsteady flow of thin liquid film on a heated stretching cylinder has considered the boundary-layer approximation under the assumption that the film thickness is thick. In this article, we have considered the full set of Navier–Stokes equations to study the unsteady film development over a porous heated stretching cylinder with the assumption that film height either coincides or lies within the boundary-layer thickness. The effects of magnetic field, suction/injection, and cooling or heating of the cylinder have been considered for investigation. The governing set of coupled nonlinear partial differential equations is solved numerically by the implicit finite difference scheme (Crank–Nicolson). It was found that the film thinning rate diminishes with rising values of the porosity parameter and Hartmann number. It is also noted that the film thickness is enhanced for injection whereas opposite results are observed for suction through the porous cylinder. The thermocapillary effect showed that the thinning rate of the liquid film decreases or increases according to the surface of the cylinder being heated or cooled. It is observed that continuously increasing the stretching speed of the cylinder produces faster thinning of the liquid film. The temperature of the liquid film is the maximum at the surface of the cylinder and it decreases toward the free surface when the cylinder is heated. But the opposite phenomenon occurs when the cylinder is cooled.     

Characteristics of MHD three-dimensional flow of nanofluid over a permeable stretching porous sheet

The aim of the present work is to focus on heat and mass transfer characteristics of the magnetohydrodynamic three-dimensional flow of nanofluid over a permeable stretching porous sheet. The significance of this study is the consideration of copper-based and aluminum oxide-based nanofluids. The physical parameters like a chemical reaction, Soret effect, radiation, and heat generation, and radiation absorption being involved in this examination are novel. The nonlinear partial differential equations are transformed into ordinary differential equations by adopting suitable similarity transformations. The numerical solutions are obtained by applying the Runge–Kutta method of fourth-order with the Shooting technique using MATLAB. The results obtained are presented through graphs and tables for various parameters. A comparison with published results has been done to validate the methodology and found good coincidence. It is claimed that the increase in heat generation parameters results in increasing the temperature. With an increase in the Soret effect, the skin friction coefficient along x-axis increases and skin friction coefficient along the y-axis, Nusselt number and Sherwood number decrease.     

MHD boundary‐layer flow over a stretching surface with internal heat generation or absorption

This article is concerned with the steady laminar magnetohydrodynamic boundary‐layer flow past a stretching surface with uniform free stream and internal heat generation or absorption in an electrically conducting fluid. A constant magnetic field is applied in the transverse direction. A uniform free stream of constant velocity and temperature is passed over the sheet. The effects of free convection and internal heat generation or absorption are also considered. The governing boundary layer and temperature equations for this problem are first transformed into a system of ordinary differential equations using similarity variables, and then solved by a new analytical method and numerical method, by using a fourth‐order Runge–Kutta and shooting method. Velocity and temperature profiles are shown graphically. It is shown that the differential transform method solutions are only valid for small values of independent variables but the results obtained by the DTM‐Padé are valid for the entire solution domain with high accuracy. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21054     

A novel analytical solution of heat transfer of a micropolar fluid through a porous medium with radiation by DTM‐Padé

In this paper, the differential transform method (DTM) was applied to heat transfer of a micropolar fluid through a porous medium with radiation. The governing equations can be written as a system of nonlinear ordinary differential equations. The approximate solutions of these equations were obtained in the form of series with easily computable terms. Then, Padé approximant was applied to increase the convergence rate of the series. The results obtained in this study were compared with the numerical results (fourth‐order Runge–Kutta method). © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res, 39(8), 575–589, 2010; Published online 26 July 2010 in Wiley Online Library ( wileyonlinelibrary.com ). DOI 10.1002/htj.20317     

Radiation and Hall effects on a 3D flow of MHD Williamson fluid over a stretchable surface

Magnetohydrodynamics (MHD) three-dimensional flow of an unsteady Williamson fluid on an enlarging surface with Hall current, radiation, heat source/sink, and chemical reaction is investigated in this article. The basic governing equations are transformed into a system of ordinary differential equations by using an appropriate similarity transformation. The system is deciphered using the shooting method. The properties of influential parameters such as parameters of magnetic field, Hall current, radiation, and so forth, on the flow are discussed with the help of graphs and tables. We noticed that the increase in the magnetic field reduces the velocity in x-direction and the rate of heat and mass transfer. We also acknowledged that the growing values of Hall current parameter boost the velocity in z-direction but it reduce the temperature and concentration distributions, respectively. The results of this study represent many applications in biomedical engineering and these results are helpful for further study of non-Newtonian fluids in various circumstances.     

Research on a new heat transfer model for liquid film convective evaporation in annular flow

Based on the phenomenon of turbulence restraint in liquid‐vapor interface, an analytical model is proposed for annular flow with a velocity distribution. The liquid‐vapor interface affecting district mixing length model was amended, and a new liquid film convective evaporation heat transfer model at the annular flow was developed. Compared with the experimental data, the results show that the new model is better than the model based on full tube flow velocity distribution. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 31(7): 524–530, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10051     

Homogeneous‐heterogeneous chemical action and non‐Fourier flux theory effects in a flow with carbon nanotubes

A theoretical outline is sketched to figure out the stream of polyvinyl alcohol (PVA)–based CNTs along a moving surface with multiple slip effects on homogeneous and heterogeneous chemical reactions. The nonlinear radiant, heat, and nonuniform heat source/sink is incorporated in the energy equation to discuss the heat transport phenomenon. The novel theory of non‐Fourier flux model of heat diffusion is also merged in the energy equation. Two kinds of nanoparticles are considered, namely, single wall carbon nanotubes and multiple wall carbon nanotubes, and are suspended in the working PVA to illustrate the flow behavior. Due to a large variety of applications, PVA is used in papermaking, textiles, and a variety of coatings. The principal structure of mathematical model is based upon the system of differential equations, which has been tackled through Runge–Kutta–Fehlberg method in MATLAB software. Graphical images are made vs different physical parameters which emerged in the model. The heat transfer of the fluid enhances with the increase of nonuniform thermal radiation. The homogeneous and heterogeneous chemical reactions slow down the concentration rate.     

Multiple slip impacts on unsteady MHD micropolar fluid past a stretching sheet with non-Darcy porous medium and uneven heat source/sink

The major scope of this research is to scrutinize the effects of multiple slips on unstable magnetohydrodynamic micropolar fluid past a stretched sheet with a non-Darcy porous medium. In the momentum equation, the non-Darcy porous medium effect is also taken into consideration. The effects of uneven heat source/sink and thermal radiation in the energy equation are also analyzed. By implementing the similarity transmission, the mathematical modeling of the set of managing partial differential equations is reframed into nonlinear ordinary differential equations. These equations are numerically solved by applying Matlab built-in solver bvp5c. The implications of foremost parameters such as micropolar parameter, magnetic parameter, permeability parameter, Prandtl, Eckert, and Schmidt numbers, Chemical reaction, slip parameters on velocity, microrotation, temperature as well as concentration profiles are displayed pictorially and explained. It is worthwhile to mention that the improving values of micropolar parameter $K$ escalate the velocity as well as microrotation profiles. However, the upsurge in non-Darcy porous medium $F_s$ will cause a declining nature in the velocity profile. Also, an enhancement in the unsteadiness parameter $A$ brings about a lessening in all the profiles. Increment in all the three usual slip parameters will bring a declining nature in the respective profiles. An increase in Schmidt number will give a deduction nature in velocity as well as concentration profile. Moreover, the physical quantities are defined and Nusselt numbers are formulated in the table, and it enlarges while boosting up $Pr$ and $R$, whilst a reverse nature is noticed for others. This present study compared with the earlier studies in special cases holds a better agreement.     

Free surface dynamics of MHD third-grade fluid model over a heated stretching sheet with variable fluid properties

A thorough investigation of MHD third-grade differential-type fluid flow over a heated stretching sheet is performed in this work. In particular, we analyze the film thinning process, when the thermal sensitive fluid parameters vary due to the effect of heat supplied to the stretching sheet. Starting with a two-dimensional (2D) free surface boundary value problem of non-Newtonian third-grade fluid, we present a systematic derivation of a 1D transient thin-film height equation using longwave analysis with respect to the small aspect ratio of the fluid domain. The derived model is used to study the impact of Newtonian and non-Newtonian parameters with variable fluid properties on the thin film height. The model is discretized using an upwind discretization in space and implicit time integration to guarantee first-order convergence. The model is analyzed thoroughly with the help of numeric computing software MATLAB. The existing findings for a Newtonian fluid are in excellent agreement with derived evidence. In comparison to Newtonian fluid, the study finds that the third-grade parameter causes thinning under different parametric restrictions. Simulations on the coupling effect explain that, the film thickness can be reduced with a high Marangoni number for highly viscous fluids. Also, since the effect of the conductivity parameter can be reduced at a low Prandtl number, the fluid shows a thinning effect. The film thinning rate, on the other hand, is reduced by the magnetic field.     

Impact of homogeneous‐heterogeneous reactions in a hybrid nanoliquid flow due to porous medium

Chemical responses are investigated as a piece of the modern applications like hydrometallurgical industry, food processing, and polymer production. Numerous chemical responding structures consolidate homogeneous and heterogeneous response and it is especially eccentric. Hence, this paper explains how the hybrid nanoliquid flow is handy in accelerate the thermochemical possessions of the base fluid in existence of homogeneous‐heterogeneous reactions. Here, three different types of hybrid nanoliquid used are copper‐Al₂O ₃/water, silver‐Al ₂O ₃/water, and gold‐Al ₂O ₃/water. Outcoming differential systems are resolved numerically by adopting fourth‐order and fifth‐order Runge‐Kutta‐Fehlberg method. To get better view of the topic, the flow field, temperature behavior, and concentration curves are investigated for particular estimations of critical elements. The results predict that gold‐Al ₂O ₃/water nanoliquid has good impact of heat rate coefficient and further porosity parameter decelerates the velocity and accelerates the temperature of the hybrid nanofluid.     

Numerical investigation of ferromagnetic liquid film flow over an unsteady stretching surface in the presence of radiation and aligned magnetic field

An investigation of the two-dimensional unsteady flow of a thin layer of ferromagnetic liquid past a stretching sheet is performed. The flow is exposed to an external magnetic field in the axial direction along with the thermal radiation effect. Relevant Maxwell's equations are considered together with the conservation laws of fluid dynamics to model the problem. The mathematical model is constructed using a system of partial differential equations with relevant boundary conditions, which are transformed into two-point boundary value problem (BVP) using similarity transformations. The resultant BVP is numerically solved by a shooting technique that involves Runge–Kutta–Fehlberg (RKF45) method to integrate the initial value problem and the Newton–Raphson method to refine the guessed initial values. The influence of the dimensionless parameters on the flow and heat exchange characteristics is graphically analyzed. It is found that the thickness of the film increases for higher values of the thermal radiation parameter. The thermal profile shows increasing behavior with the radiation parameter and reverse effect with the Prandtl number.     

The flow characteristics of an upward gas‐liquid two‐phase flow in a vertical tube with a wire coil: Part 2. Effect on void fraction and liquid film thickness

An experimental study has been carried out to clarify the characteristics of the void fraction and the liquid film thickness of the air‐water two‐phase flow in vertical tubes of 25‐mm inside diameter with wire coils of varying wire diameter and pitch. The flow pattern in the experiment on the average void fraction and the local void fraction distribution in cross section was a bubble flow, and the liquid film thickness was in the region of semiannular and annular flows. It is clarified from these experiments that the average void fraction in tubes with wire coils is lower than that in a smooth tube and decreases with the wire diameter owing to the centrifugal force of the swirl flow which concentrates bubbles at the center of the tube, that the local liquid film thickness becomes more uniform with a decrease in the pitch of the wire coil, and that the liquid film becomes thicker after the passage through the wire coil with an increase in the wire diameter. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 31(8): 652–664, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10067     

Radiation effect on MHD flow past a porous vertical plate in the presence of heat sink

This study investigates heat and mass transfer in MHD convective flow through a vertical plate via porous media in the presence of radiation and a heat source/sink. It is assumed that a uniform magnetic field of strength is imposed perpendicular to the plate and directed into the fluid area. The governing nondimensional equations are solved using the perturbation technique. We further derived the skin friction, Nusselt number, and Sherwood number. The computation of results is performed with the aid of mathematical software and results are presented in graphical and tabular forms for distinct flow impacting parameters. It is observed that fluid motion is retarded due to the application of the magnetic field. Furthermore, the fluid temperature comprehensively falls under the Prandtl number as well as the thermal radiation effect. It is important to note that the heat sink causes fluid velocity and fluid temperature to fall drastically.