Three-dimensional flow with heat transfer of a viscoelastic fluid over a stretching surface with a magnetic field

The present research study deals with the steady flow and heat transfer of a viscoelastic fluid over a stretching surface in two lateral directions with a magnetic field applied normal to the surface. The fluid far away from the surface is ambient and the motion in the flow field is caused by stretching surface in two directions. This result is a three-dimensional flow instead of two-dimensional as considered by many authors. Self-similar solutions are obtained numerically. For some particular cases, closed form analytical solutions are also obtained. The numerical calculations show that the skin friction coefficients in x- and y-directions and the heat transfer coefficient decrease with the increasing elastic parameter, but they increase with the stretching parameter. The heat transfer coefficient for the constant heat flux case is higher than that of the constant wall temperature case.     

A note on the flow over a stretching sheet

Summary This study deals with the viscous incompressible flow over a stretching sheet. The velocity of the sheet is a quadratic polynomial of the distance from the slit and the sheet is subjected to a linear mass flux. A closed form solution is obtained under some restrictions on the linear mass flux. Stream line patterns are plotted and the effect of mass flux on the flow is also studied.     

On the effectiveness of porosity on unsteady flow between parallel plates of a viscoelastic fluid under constant pressure gradient with heat transfer

The unsteady flow in a porous medium of an incompressible non-Newtonian viscoelastic fluid between two parallel horizontal non-conducting porous plates is studied with heat transfer. A sudden uniform and constant pressure gradient and uniform suction and injection through the surface of the plates are applied. The two plates are kept at different but constant temperatures, while the Joule and viscous dissipations are taken into consideration. Numerical solutions for the governing momentum and energy equations are obtained using finite difference approximations. The effect of the porosity of the medium, the parameter describing the non-Newtonian behavior, and the velocity of suction and injection on both the velocity and temperature distributions is examined.     

Combined radiation and mixed convection from a vertical wall with suction/injection in a non–Darcy porous medium

Summary. Mixed convection flow of an absorbing fluid up a uniform non–Darcy porous medium supported by a semi-infinite ideally transparent vertical flat plate due to solar radiation is considered. The external flow field is assumed to be uniform, the effect of the radiation parameter in the boundary layer adjacent to the vertical flat plate with fluid suction/injection through it is analyzed in both aiding and opposing flow situations. It is observed that the similarity solution is possible only when the fluid suction/injection velocity profile varies as x^–1/2. The velocity and temperature profiles in the boundary layer and the heat transfer coefficient are presented for selected values of the parameters. It is observed that the Nusselt number increases with the increase in the radiation parameter and also when the value of the surface mass flux parameter moves from the injection to the suction region.     

Unsteady slot Suction from a High-Reynolds-Number Cross-Flow

The problem of unsteady suction from a high-Reynolds-number cross-flow into a slot is considered in the case where the suction is driven by a time-dependent slot pressure. The model uses linearised asymptotics based on a small parameter that defines the suction strength. An integro differential equation is derived for the mass flow into the slot and this is solved for various time-dependent slot pressures of practical interest. Closed-form expressions are also found for the shape of the shear layer dividing the external flow from the fluid in the slot. For a step function change in the slot pressure, a non-monotonic decay to the steady solution is observed, and for an oscillatory slot pressure there is a phase lag between the slot pressure and the mass flow. For rapidly changing slot pressures it is shown that slot injection can occur, even when the slot pressure remains below the free-stream pressure.     

Unsteady free convection flow over a continuous moving vertical surface

Summary The problem of heat transfer in the unsteady free convection flow over a continuous moving vertical sheet in an ambient fluid has been investigated. Both constant surface temperature and constant surface heat flux conditions have been considered. The nonlinear coupled partial differential equations governing the flow have been solved numerically using the Keller box method and the Nakamura method which both give closely similar solutions. The results indicate that the cooling rate of the sheet can be enhanced by increasing the buoancy parameter or the velocity of the sheet. It is found that a better cooling performance could be achieved by using a liquid as a cooling medium rather than a gas. The overshoot in the velocity occurs near the surface when the buoyancy parameter exceeds a certain critical value.     

Flows induced by the surface motions of a cylindrical sheet

Fluid flows induced by the surface stretching or shearing motion of cylindrical sheets are investigated. Steady and unsteady exact solutions of the Navier–Stokes equations are found for periodic axial shearing of an impermeable sheet and for periodic azimuthal stretching of a permeable sheet. Steady Stokes-flow solutions induced by the periodic axial stretching and the periodic azimuthal stretching of impermeable cylindrical sheets are also reported. In each case flows interior and exterior to a cylinder are considered, as well as the flow in the annulus between concentric cylinders.     

Exact-solution of entropy generation for MHD nanofluid flow induced by a stretching/shrinking sheet with transpiration: Dual solution

In this article, attempts are made to present an exact solution for the fluid flow and heat transfer and also entropy generation analysis of the steady laminar magneto-hydrodynamics (MHD) nanofluid flow induced by a stretching/shrinking sheet with transpiration. This paper is the first contribution to the study of entropy generation for the nanofluid flow via exact solution approach. The governing partial differential equations are transformed into nonlinear coupled ordinary differential equations via appropriate similarity transformations. The current exact solution illustrates very good correlation with those of the previously published studies in the especial cases. The entropy generation equation is derived as a function of the velocity and the temperature gradients. The influences of the different flow physical parameters including the nanoparticle volume fraction parameter, the magnetic parameter, the mass suction/injection parameter, the stretching/shrinking parameter, and the nanoparticle types on the fluid velocity component, the temperature distribution, the skin friction coefficient, the Nusselt number and also the averaged entropy generation number are discussed in details. This study specifies that nanoparticles in the base fluid offer a potential in increasing the convective heat transfer performance of the various liquids. The results show that the copper and the aluminum oxide nanoparticles have the largest and the lowest averaged entropy generation number, respectively, among all the nanoparticles considered.     

Magnetohydrodynamic heat transfer over a non-isothermal stretching sheet

Summary This paper presents solutions of the energy equation for the boundary layer flow of an electrically conducting fluid under the influence of a constant transverse magnetic field over a linearly stretching non-isothermal flat sheet. Effects due to dissipation, stress work and heat generation are considered. Analytical solutions of the resulting linear nonhomogeneous boundary value problems, expressed in terms of Kummer's functions, are presented for the case of prescribed surface temperature as well as the case of prescribed wall heat flux, both of which are assumed to be quadratic functions of distance. The boundary value problems are also solved by direct numerical integration yielding results in excellent agreement with the analytical solutions.     

Unsteady MHD forced flow due to a point sink

Summary An analysis is performed to study the unsteady laminar incompressible boundary-layer flow of an electrically conducting fluid in a cone due to a point sink with an applied magnetic field. The unsteadiness in the flow is considered for two types of motion, viz. the motion arising due to the free stream velocity varying continuously with time and the transient motion occurring due to an impulsive change either in the strength of the point sink or in the wall temperature. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The magnetic field increases the skin friction but reduces heat transfer. The heat transfer and temperature field are strongly influenced by the viscous dissipation and Prandtl number. The velocity field is more affected at the early stage of the transient motion, caused by an impulsive change in the strength of the point sink, as compared to the temperature field. When the transient motion is caused by a sudden change in the wall temperature, both skin friction and heat transfer take more time to reach a new steady state. The transient nature of the flow and heat transfer is active for a short time in the case of suction and for a long time in the case of injection. The viscous dissipation prolongs the transient behavior of the flow.     

Analysis of transient flow of MHD nanofluid past a non-linear stretching sheet considering Navier’s slip boundary condition

Induced boundary layer flow in the nanofluid caused by impulsively started nonlinear stretching sheet is analyzed in the presence of thermal radiation and magnetic field considering Navier’s velocity sip boundary condition. The similarity solution is achieved numerically using Galerkin finite element technique. The variation of flow characteristics viz. nanofluid velocity, temperature and nanoparticle concentration is examined corresponding to different flow parameters considered in the problem. The findings of the investigation reveal that nonlinearity in the stretching sheet leads to a deceleration in the nanofluid velocity, temperature and nanoparticle concentration while the unsteadiness in the stretching tends to decelerate the velocity whereas temperature and nanoparticle concentration are found to be increasing with increase in unsteadiness. The problem investigated in this article is basically an extension for unsteady case of the previously done works on nanofluid flow over a sheet stretching with a power-law velocity.     

Deceleration of a porous rotating disk in a viscous fluid

The flow due to a rotating disk decelerating with an angular velocity inversely proportional to time with either surface suction (or injection) which again varies with time is investigated. The unsteady Navier-Stokes equations are transformed to non-linear ordinary differential equations using similarity transformations. The resulting equations are solved numerically using a globally convergent homotopy method. The flow depends on two non-dimensional parameters, namely an unsteadiness parameter $\text{S}$ and a suction (or injection) parameter $\text{A}$. Some interesting numerical results are presented graphically and discussed.     

Unsteady MHD rotating flow over a rotating sphere near the equator

The unsteady rotating flow of a laminar incompressible viscous electrically conducting fluid over a rotating sphere in the vicinity of the equator has been studied. The fluid and the body rotate either in the same direction or in opposite directions. The effects of surface suction and magnetic field have been included in the analysis. There is an initial steady state that is perturbed by a sudden change in the rotational velocity of the sphere, and this causes unsteadiness in the flow field. The nonlinear coupled parabolic partial differential equations governing the boundary-layer flow have been solved numerically by using an implicit finite-difference scheme. For large suction or magnetic field, analytical solutions have also been obtained. The magnitude of the radial, meridional and rotational velocity components is found to be higher when the fluid and the body rotate in opposite directions than when they rotate in the same direction. The surface shear stresses in the meridional and rotational directions change sign when the ratio of the angular velocities of the sphere and the fluid ₀. The final (new) steady state is reached rather quickly which implies that the spin-up time is small. The magnetic field and surface suction reduce the meridional shear stress, but increase the surface shear stress in the rotational direction.     

Thermal boundary layers over a shrinking sheet: an analytical solution

In this paper, the heat transfer over a shrinking sheet with mass transfer is studied. The flow is induced by a sheet shrinking with a linear velocity distribution from the slot. The fluid flow solution given by previous researchers is an exact solution of the whole Navier–Stokes equations. By ignoring the viscous dissipation terms, exact analytical solutions of the boundary layer energy equation were obtained for two cases including a prescribed power-law wall temperature case and a prescribed power-law wall heat flux case. The solutions were expressed by Kummer’s function. Closed-form solutions were found and presented for some special parameters. The effects of the Prandtl number, the wall mass transfer parameter, the power index on the wall heat flux, the wall temperature, and the temperature distribution in the fluids were investigated. The heat transfer problem for the algebraically decaying flow over a shrinking sheet was also studied and compared with the exponentially decaying flow profiles. It was found that the heat transfer over a shrinking sheet was significantly different from that of a stretching surface. Interesting and complicated heat transfer characteristics were observed for a positive power index value for both power-law wall temperature and power-law wall heat flux cases. Some solutions involving negative temperature values were observed and these solutions may not physically exist in a real word.     

Oscillating cascade unsteady aerodynamics including separated flow effects

A mathematical model is developed to predict the effect of flow separation on the unsteady aerodynamic lift and moment acting on a two-dimensional flat plate cascade which is harmonically oscillating in a subsonic flow field. The unsteady flow is considered to be a small perturbation to the uniform steady flow, with the steady flow assumed to separate at a specified fixed position on the airfoil suction surface. This formulation does not require the difference in the upwash velocity across the airfoil in the separated flow region to be determined before calculating the unsteady pressure difference across the chordline of the airfoils, thereby eliminating the assumption that the upwash difference is zero at the trailing edge when the steady flow is separated. Results obtained demonstrate that although flow separation decreases bending mode stability, it does not result in bending mode flutter. However, flow separation can result in torsion mode flutter, with this instability being a function of the location of both the separation point and the elastic axis.     

Viscous flow due to an exponentially shrinking permeable sheet in nanofluid in presence of slip

Classical problem of steady boundary layer flow of nanofluid over an exponentially permeable shrinking sheet in presence of slip is investigated. The model used for nanofluid includes Brownian motion and thermophoresis effects. The governing equations for momentum, energy, and nanofluid solid volume fraction are transformed to ordinary differential equations with the help of similarity transformations and then solved numerically using fourth order Runge–Kutta method with shooting technique. It is found that the governing parameters, viz. the suction/blowing parameter, velocity slip, thermal and mass slip parameters, Brownian motion parameter, thermophoresis parameter, Prandtl number, and Lewis number significantly affect the flow field, heat, and mass transfer. The results obtained indicate that the dual solutions exist for certain values of the mass suction parameter. Velocity increases whereas the temperature and nanoparticle volume fraction decrease due to suction through the porous sheet. It is noted that with the increase in velocity slip fluid velocity increases whereas temperature and concentration decrease. Due to increase in thermal slip and mass slip both temperature and concentration decrease.     

Nonlinear streaming due to the oscillatory stretching of a sheet in a viscous fluid

Summary An elastic sheet is stretched back and forth in a viscous fluid. The problem is governed by a nondimensional parameterS which represents the relative magnitude of frequency to stretching rate. The Navier-Stokes equations are solved by matched asymptotic expansions for largeS. Due to nonlinearity there exists boundary layers ofO(S ^–1/2). The unsteady oscillatory flow contains both basic and higher harmonic oscillations. The induced steady streamlines show a saddle like flow which is different from that of acoustic streaming.With 5 Figures     

MHD flow of a viscoelastic fluid past a stretching sheet with suction

Summary The laminar flow of a viscoelastic fluid past a stretching sheet in the presence of a magnetic field, when the fluid is extracted from the sheet at a uniform rate, is considered. An exact analytical solution exists for the problem. It is shown that when there is a suction of the fluid, the solutions are possible only upto a critical value of the viscoelastic parameter. Also, for values less than this critical value, dual solutions exist.     

Thermocapillary migration of a planar droplet at moderate and large Marangoni numbers

Thermocapillary migration of a planar non-deformable droplet in flow fields with two uniform temperature gradients at moderate and large Marangoni numbers is studied numerically by using the front-tracking method. It is observed that the thermocapillary motion of planar droplets in the uniform temperature gradients is steady at moderate Marangoni numbers, but unsteady at large Marangoni numbers. The instantaneous migration velocity at a fixed migration distance decreases with increasing Marangoni numbers. The simulation results of the thermocapillary droplet migration at large Marangoni numbers are found in qualitative agreement with those of experimental investigations. Moreover, the results concerned with steady and unsteady migration processes are further confirmed by comparing the variations of temperature fields inside and outside the droplet. It is evident that at large Marangoni numbers the weak transport of thermal energy from outside of the droplet into inside cannot satisfy the condition of a steady migration process, which implies that the advection around the droplet is a more significant mechanism for heat transfer across/around the droplet at large Ma numbers. Furthermore, from the condition of overall steady-state energy balance in the flow domain, the thermal flux across its surface is studied for a steady thermocapillary droplet migration in a flow field with uniform temperature gradient. By using the asymptotic expansion method, a non-conservative integral thermal flux across the surface is identified in the steady thermocapillary droplet migration at large Marangoni numbers. This non-conservative flux may well result from the invalid assumption of a quasi-steady state, which indicates that the thermocapillary droplet migration at large Marangoni numbers cannot reach a steady state and is thus an unsteady process.