Nonlinear Similarity Equations Research Articles

New Similarity Solutions of Magnetohydrodynamic Flow Over Horizontal Plate by Lie Group with Nonlinear Hydrodynamic and Linear Thermal and Mass Slips

The viscous laminar magnetohydrodynamic convective boundary layer flow with the combined effects of chemical reaction and nonlinear velocity slip and linear thermal and concentration slips have been considered across a flat plate in motion. Using a non-dimensional transformation attained by the single parameter continuous group method, the governing equations are transformed into a system of nonlinear ordinary similarity equations, then, the solutions of the coupled system of equations are constructed for velocity, temperature, and concentration functions by using the numerical methods. Among the parameters that have been looked at are the buoyancy parameter N, the nonlinear slip parameter {mathrm{n}}_{1}, the order of chemical reaction n, the Prandtl number Pr, and the Schmidt number Sc. An investigation was made on the profiles with respect to mixed convection parameter uplambda , order of chemical reaction n, arbitrary index parameter {mathrm{n}}_{1}, velocity slip parameter a, thermal slip parameter b, mass slip parameter c, suction parameter fw, magnetic parameter M. Verification of the results were possible due to comparison of two numerical methods to obtain the solution to the differential equations. The present study indicates that, for a range of values of the magnetic parameter, the wall shear stress decreases with increasing mixed convection. Moreover, for a variety of mixed convection parameter instances, the wall heat transfer decreases with increasing perpendicular magnetic effect.

Dynamics of heat absorbing and radiative hydromagnetic nanofluids through a stretching surface with chemical reaction and viscous dissipation

The hydromagnetic dissipative, heat-absorbing, chemically reactive and optically thick radiative flow of nanofluid over a stretchable surface together with heat and mass transport phenomena is investigated. The prevailing partial differential equations (PDE) of the mathematical model are changed by incorporating apt similarity variables in the form of nonlinear similarity equations. Further, these similarity equations are solved numerically utilizing the Runge-Kutta Fehlberg technique in conjunction with the shooting method. For the numerical explorations, three kinds of nanofluids are prepared by disseminating very fine nanoparticles of titanium oxide (TiO2), aluminium oxide (Al2O3) and copper (Cu) into water. The significance of numerous regulatory flow parameters on the nanofluid velocity, nanofluid temperature, local Nusselt number, species concentration, wall velocity gradient and mass flow rates are examined through different graphical results. Additionally, a quadratic regression approximation analysis is accomplished to analyze the connection between the heat transport rate and regulatory flow parameters. The numerical results reveal that the temperature of Cu-water based nanofluid temperature gets enhanced owing to improvement in the strength of radiation, viscous dissipation and magnetic effects. Further, regression approximation analysis unveils that a slight change in the velocity slip parameter leads in the optimal perturbation in both the shear stress values and heat transfer rate at the stretchable surface. Finally, the validation of numerical results and the developed algorithm of employed computational technique have been done by making a comparison of computed results with the available results under restricted situations.

Radiative bioconvection nanofluid squeezing flow between rotating circular plates: Semi-numerical study with the DTM-Padé approach

Modern biomedical and tribological systems are increasingly deploying combinations of nanofluids and bioconvecting microorganisms which enable improved control of thermal management. Motivated by these developments, in this study, a new mathematical model is developed for the combined nanofluid bioconvection axisymmetric squeezing flow between rotating circular plates (an important configuration in, for example, rotating bioreactors and lubrication systems). The Buongiorno two-component nanoscale model is deployed, and swimming gyrotactic microorganisms are considered which do not interact with the nanoparticles. Thermal radiation is also included, and a Rosseland diffusion flux approximation is utilized. Appropriate similarity transformations are implemented to transform the nonlinear, coupled partial differential conservation equations for mass, momentum, energy, nanoparticle species and motile microorganism species under suitable boundary conditions from a cylindrical coordinate system into a dimensionless nonlinear ordinary differential boundary value problem. An efficient scheme known as differential transform method (DTM) combined with Padé-approximations is then applied to solve the emerging nonlinear similarity equations. The impact of different non-dimensional parameters i.e. squeezing Reynolds number, rotational Reynolds number, Prandtl number, thermophoresis parameter, Brownian dynamics parameter, thermal radiation parameter, Schmidt number, bioconvection number and Péclet number on velocity, temperature, nanoparticle concentration and motile gyrotactic microorganism density number distributions is computed and visualized graphically. The torque effects on both plates, i.e. the lower and the upper plate, are also determined. From the graphical results, it is seen that momentum in the squeezing regime is suppressed clearly as the upper disk approaches the lower disk. This inhibits the axial flow and produces axial flow retardation. Similarly, by enhancing the value of squeezing Reynolds number, the tangential velocity distribution also decreases. More rigorous squeezing clearly therefore also inhibits tangential momentum development in the regime and leads to tangential flow deceleration. Tables are also provided for multiple values of flow parameters. The numerical values obtained by DTM-Padé computation show very good agreement with shooting quadrature. DTM-Padé is shown to be a precise and stable semi-numerical methodology for studying rotating multi-physical flow problems. Radiative heat transfer has an important influence on the transport characteristics. When radiation is neglected, different results are obtained. It is important therefore to include radiative flux in models of rotating bioreactors and squeezing lubrication dual disk damper technologies since high temperatures associated with radiative flux can impact significantly on combined nanofluid bioconvection which enables more accurate prediction of actual thermofluidic characteristics. Corrosion and surface degradation effects may therefore be mitigated in designs.

Heat Transfer From a Concentrated Tip Source in Falkner–Skan Flow

Abstract The Falkner–Skan flow over a wedge is classic in boundary layer theory. We consider the heat or mass transfer from a source at the vertex of the wedge. The interactions of the thermal boundary layer and momentum boundary layer lead to nonlinear similarity equations which are integrated numerically. There exists a mixing index that depends on the Prandtl number and the wedge opening angle. Attention is paid to special cases such as forced convection in Blasius flow past a semi-infinite plate and the Hiemenz stagnation flow normal to a plate.

Controlling of the melting through porous medium and magnetic field

The melting procedure with a direct contact of phase change material is taken into account to consider the porous medium in the presence of a uniform and transverse magnetic field. A permeable rotating disk is taken as a heater in the melting progression of solid phase change material. The three-dimensional melting layer takes place due to the accruing of the temperature difference among the porous disk and solid material. Movement is subject to the effect of pressure loading (counting the weight of solid), direct relation with solid and rotation due to centrifugal force. The removal of melting is controlled due to the joint exertions of the porous media, wall permeability and resistive force generated due to the applied magnetic field. The motion of the melting layer is assumed unsteady and governed the nonlinear similarity equations. Furthermore, magnetic field, porosity, external load and wall suction enhance melting and heat transfer rates at the thin melts film thickness. The melting rate, momentum and thermal boundary layers are estimated under the impact of Stefan number, magnetic field, porosity parameter and unsteadiness parameter. The Eckert number enhances the thermal boundary layer, and consequently the larger amount of melting received. The governing PDEs is highly nonlinear; thus for the solution we use analytical method of HAM and BVPh 2.0 package. The important outputs of the thickness of the thin layer during melting process in the presence and absence of wall suction are mainly focused.

Mixed Convection Flow Of Viscoelastic Nanofluid Past A Horizontal Circular Cylinder In Presence Of Heat Generation

The steady two-dimensional mixed convection boundary layer flow of viscoelastic nanofluid past a horizontal circular cylinder with convective boundary condition in presence of heat generation has been studied numerically. Carboxymethyl cellulose solution (CMC) is chosen as the base fluid and copper as a nanoparticle with the Prandtl number Pr = 6.2. The Tiwari and Das model has been considered in this study. The governing partial differential equations are reduced to a system of ordinary differential equations by introducing similarity transformations. The nonlinear similarity equations are solved numerically by applying the Keller-box method. The numerical results are presented graphically for different values of the parameters including the heat generation parameter, nanoparticles volume fraction, and Biot number. A systematic study is discussed to analyze the effect of these parameters on the velocity and temperature profiles as well as the skin friction and heat transfer coefficient. The thermal boundary layer shows the changes in variation behavior when the nanoparticles volume fraction, heat generation and Biot number are increased. Heat transfer coefficient is increasing function of heat generation parameter. Nanoparticles volume fraction on heat transfer coefficient have opposite effect when compared with heat generation parameter.

Direct contact melting due to a permeable rotating disk

The melting process of a solid phase change material sitting above a heater of rotating as well as permeable disk is revisited. The melting starts due to the temperature difference between the solid and the disk, and a melt layer of thin thickness is formed subject to a three dimensional motion. The motion is subject to the effects of pressure load (including the weight of the solid) directly acting on the solid and a centrifugal force created by the rotation. The amount of melt is then controlled by the action of wall permeability. The removal of melt is assumed to be steady, governed by the nonlinear similarity equations. Numerical simulations are performed to seize the behaviors of fluid and temperature fields of the melt. Moreover, the physical features of melting rate, heat transfer at the solid liquid interface and at the disk surface, and radial and tangential stresses are analyzed in detail. These physical properties are shown to be in excellent agreement with those determined from the thin melting film approximations. External load and suction are found to enhance the melting and heat transfer rates at sufficiently small melt film thicknesses.

Effects of the magnetic field on direct contact melting transport processes during rotation

Contact melting heat transfer occurs via relative motion between the heating source and a phase change material (PCM) during melting in various applications. In this study, we investigated the physics of the close contact melting process generated by rotation and when subjected to an applied magnetic field. We transformed the physical model comprising the three-dimensional mass, momentum, and energy equations of the liquid melt layer in the cylindrical coordinate system, including the effects of the Lorentz forces and coupled with an interfacial energy jump condition, into a set of nonlinear similarity equations. Various characteristic dimensionless variables were identified, including an external force parameter σ, which defines the relationship between the external load on the PCM and the centrifugal force due to rotation, and a magnetic field parameter M. Numerical results were obtained and we systematically studied and interpreted the effects of various dimensionless variables on the contact melting and heat transfer processes during rotation, including the structures of the flow and thermal fields, melt layer thickness, and the melting and heat transfer rates. In particular, our results demonstrate that the melting and heat transfer rates increase while the liquid melt film becomes thinner as the external force parameter σ increases. By contrast, an increase in the magnetic field parameter M decreases the melting and heat transfer rates, while yielding relatively thicker melt layers.

MHD slip flow and heat transfer of Al2O3-water nanofluid over a horizontal shrinking cylinder using Buongiorno’s model: Effect of nanolayer and nanoparticle diameter

This paper considers the study of electrically conducting alumina-water nanofluid flow induced by horizontal shrinking cylinder under the effect of velocity and thermal slip at the surface. The Buongiorno’s non-homogenous model is implemented which accounted with the nanolayer based effective thermal conductivity and Nielsen’s effective viscosity model. The governing transport model for nanofluid has been transformed into a set of nonlinear similarity equations and solved with a shooting technique using MATLAB ode solver and ‘fsolve’ command. The behavior of skin friction, velocity, Nusselt number and temperature are investigated numerically. The increasing trends of Nusselt number as well as skin friction are noticed as nanoparticle concentration increases at the outer surface of the boundary layer. Nusselt number is found to be decreasing nonlinearly with nanoparticle diameter ranging from 5nm to 60nm. Two branches of solution are also captured in some fixed range of involving parameter, namely mass transfer (suction), curvature and shrinking parameters. The physical existence of these branches is also discussed.

A novel investigation of a micropolar fluid characterized by nonlinear constitutive diffusion model in boundary layer flow and heat transfer.

The rheological and heat-conduction constitutive models of micropolar fluids (MFs), which are important non-Newtonian fluids, have been, until now, characterized by simple linear expressions, and as a consequence, the non-Newtonian performance of such fluids could not be effectively captured. Here, we establish the novel nonlinear constitutive models of a micropolar fluid and apply them to boundary layer flow and heat transfer problems. The nonlinear power law function of angular velocity is represented in the new models by employing generalized "n-diffusion theory," which has successfully described the characteristics of non-Newtonian fluids, such as shear-thinning and shear-thickening fluids. These novel models may offer a new approach to the theoretical understanding of shear-thinning behavior and anomalous heat transfer caused by the collective micro-rotation effects in a MF with shear flow according to recent experiments. The nonlinear similarity equations with a power law form are derived and the approximate analytical solutions are obtained by the homotopy analysis method, which is in good agreement with the numerical solutions. The results indicate that non-Newtonian behaviors involving a MF depend substantially on the power exponent n and the modified material parameter [Formula: see text] introduced by us. Furthermore, the relations of the engineering interest parameters, including local boundary layer thickness, local skin friction, and Nusselt number are found to be fitted by a quadratic polynomial to n with high precision, which enables the extraction of the rapid predictions from a complex nonlinear boundary-layer transport system.

MHD Free-Convective and Mass Transfer Flow Past a Continuously Moving Semi-Infinite Vertical Porous Plate with Thermal Diffusion

A steady two dimensional MHD free convective and mass transfer flow of an incompressible fluid past a continuously moving vertical infinite porous plate in the presence of thermal diffusion has been studied. The governing partial differential equations of the MHD free convective boundary layer flow are reduced to nonlinear ordinary differential equations by introducing suitable similarity transformations. The nonlinear similarity equations are then solved numerically by Nachtshiem-Swigert iteration technique. The results of the numerical solution are then presented graphically in the form of velocity, temperature and concentration profiles. The corresponding skin friction coefficient, the Nusselt number and the Sherwood number are also calculated and displayed in tables showing the effects of various parameters on them.

Unsteady MHD Non-Newtonian Casson Fluid Flow due to a Porous Rotating Disk with Uniform Electric Field

The unsteady MHD Non-Newtonain incompressible casson fluid flow due to a porous rotating disk with a uniform angular velocity in the presence of an axial uniform magnetic field and a uniform electric field is examined. The Hall current is not considered in this paper. The governing equations of the problem are then reduced to non-linear ordinary differential equations by introducing suitable similarity parameters. The similarity equations are derived in such a way that one can see the effects of the angular velocity explicitly unlike the usual similarity transformation of rotating disk problems. Numerical solutions to the reduced non-linear similarity equations are then obtained by adopting shooting method using the Nachtsheim-Swigert iteration technique.

The exterior unsteady viscous flow and heat transfer due to a porous expanding or contracting cylinder.

Since the vessels in the biological tissues are characterized by low seepage Reynolds numbers and contracting or expanding walls, more attention is paid on the viscous flow outside the porous pipe with small expansion or contraction. This paper presents a numerical solution of the flow and heat transfer outside an expanding or contracting porous cylinder. The coupled nonlinear similarity equations are solved by Bvp4c, which is a collocation method with MATLAB. The effects of the different physical parameters, namely the permeability Reynolds number,the expansion ratio and the Prandtl number, on the velocity and temperature distribution are obtained and the results are shown graphically.

A BINARY CHEMICAL REACTION ON UNSTEADY FREE CONVECTIVE BOUNDARY LAYER HEAT AND MASS TRANSFER FLOW WITH ARRHENIUS ACTIVATION ENERGY AND HEAT GENERATION/ABSORPTION

We investigate a local similarity solution of an unsteady natural convection heat and mass transfer boundary layer incompressible fluid flow past a moving vertical porous plate in the presence of the heat absorption and generation. The effects of chemical reaction rate which is function of temperature and Arrhenius activation energy on the velocity, temperature and concentration are also studied in this paper. The governing partial differential equations are reduced to ordinary differential equations by introducing local similarity transformation (Maleque, 2010a). Numerical solutions to the reduced non-linear similarity equations are then obtained by adopting Runge-Kutta and shooting methods using the Nachtsheim- Swigert iteration technique. The results of the numerical solution are then presented graphically in the form of velocity, temperature and concentration profiles. The corresponding skin friction coefficient, the Nusselt number and the Sherwood number are also calculated and displayed in table showing the effects of various parameters on them.

Effects of Exothermic/Endothermic Chemical Reactions with Arrhenius Activation Energy on MHD Free Convection and Mass Transfer Flow in Presence of Thermal Radiation

A local similarity solution of unsteady MHD natural convection heat and mass transfer boundary layer flow past a flat porous plate within the presence of thermal radiation is investigated. The effects of exothermic and endothermic chemical reactions with Arrhenius activation energy on the velocity, temperature, and concentration are also studied in this paper. The governing partial differential equations are reduced to ordinary differential equations by introducing locally similarity transformation (Maleque (2010)). Numerical solutions to the reduced nonlinear similarity equations are then obtained by adopting Runge-Kutta and shooting methods using the Nachtsheim-Swigert iteration technique. The results of the numerical solution are obtained for both steady and unsteady cases then presented graphically in the form of velocity, temperature, and concentration profiles. Comparison has been made for steady flow () and shows excellent agreement with Bestman (1990), hence encouragement for the use of the present computations.

Unsteady Natural Convection Boundary Layer Flow with Mass Transfer and a Binary Chemical Reaction

We investigate a locally similarity solution of unsteady natural convection heat and mass transfer boundary layer flow past a flat porous plate. The effects of chemical reaction rate and Arrhenius activation energy on the velocity, temperature and concentration are also studied in this paper. The governing partial differential equations are reduce d to ordinary differential equations by introducing locally similarity transformation Maleque [14]. Numerical solutions to the reduced nonlinear similarity equations are then obtained by adopting Runge-Kutta and shooting method using the Nachtsheim-Swigert iteration technique. The results of the numerical solution are then presented graphically in the form of velocity, temperature and concentration profiles. The corresponding skin friction coefficient, the Nusselt number and the Sherwood number are also ca lculated and displayed inTableshowing the effects of various parameters on them.

A porous slider with velocity slip

A porous slider, i.e. a solid levitated by fluid injection from its porous bottom to reduce the sliding drag, is studied theoretically in this paper. Slip is present on both the slider and the ground. This three-dimensional problem is transformed to a set of high-order, nonlinear similarity equations. Asymptotic solutions for high or low cross-flow Reynolds numbers and numerical integration are used for both the strip slider and the circular slider. We find that the Reynolds number and the surface slip have a significant effect on the lift and drag of the slider.

An integral treatment for combined heat and mass transfer by mixed convection along vertical surface in a saturated porous medium

Combined free and forced convection heat and mass transfer over an impermeable vertical surface embedded in a saturated porous medium is considered. A similarity transformation has been used to study the mixed convection boundary-layer flow over a semi-infinite vertical flat plate. Integral solutions are derived for the coupled nonlinear similarity equations of coupled heat and mass transfer in porous media for the case where the free stream velocity and temperature and concentration near the wall are kept constant. The governing parameters for the problem under consideration are the Lewis number Le, the buoyancy ratio N and mixed convection parameter (). The results for the heat and mass transfer coefficients in terms of Nusselt and Sherwood numbers are represented graphically for the various values of governing parameters of the problem. The heat and mass transfer in the boundary layer region has been analyzed for aiding and opposing flows. Key words: Convection in porous media, heat and mass transfer, integral method, boundary layer.

Mass Transfer and Heat Generation Effects on MHD Free Convection Flow past an Inclined Vertical Surface in a Porous Medium

A steady two-dimensional MHD free convection and mass transfer flow past an inclined semi-infinite vertical surface in the presence of heat generation and a porous medium has been studied numerically. The governing partial differential equations are reduced to a system of ordinary differential equations by introducing similarity transformations. The non-linear similarity equations are solved numerically by applying the Runge-Kutta method of fourth order with shooting technique. The numerical results are presented graphically for different values of the parameters. Finally, the numerical values of the local skin-friction coefficient, local Nusselt number and Sherwood number are shown in Table 1.

Interaction of Kelvin force and transport across a melting substrate in a microgravity environment

Gradients in magnetic field applied to electrically insulating fluids result in an effective body force-the Kelvin force, which can potentially be used to control flow and thermal processes, particularly in microgravity environments. We study the influence of the Kelvin force on the mass, momentum, and energy transport of fluid arising from the melting of a semi-infinite solid substrate subjected to temperature and magnetic-field gradients. The governing equations of the magnetothermal free convection of the melt under the boundary layer approximation with a suitable transformation lead to a similarity solution. Closed-form analytical solutions for the limiting case of smaller Prandtl numbers are presented. The nonlinear similarity equations are solved numerically using an iterative boundary-value technique based on a finite-difference approach. Based on the numerical results, the effect of various characteristic nondimensional parameters on the structure of the melt boundary layer and transport rates of heat and mass across the substrate are elucidated and discussed.