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.     

Evaluation of thermocapillary driving forces in the development of striations during the spin coating process

The evolution of a temperature gradient at the free surface of a coating solution during the spin coating process is examined. Solvent evaporation causes localized cooling at the top that can result in thermocapillary instability within the coating solution, and thereby driving convective flows that may result in non-uniform coatings. We examine the evolution of these temperature gradients by using a one dimensional finite difference model that simultaneously describes the thinning behavior (both by flow and by evaporation) and the temperature evolution within the solution. The entire system is initially isothermal but is subject to evaporation-driven cooling at the free surface of the gradually thinning fluid. The model is then used to determine the magnitude of the thermocapillary effects during the spin coating process. As test systems we simulate the spin coating of several pure alcohol solutions having different volatilities and therefore different evaporative-cooling powers. As the fluid thins, we calculate the instantaneous Marangoni (Mn) number, which signifies the magnitude of thermocapillary-driven convection. We compare these Mn values against their relevant threshold values, determined from prior reports in the literature, in order to deduce the magnitude of the instabilities they represent. If the Mn value is super-critical, then the instability that it represents will be sufficient for the onset of convection cells within a stagnant fluid layer of corresponding thickness. Because the radial outflow is fully laminar under normal conditions, super-critical Mn values imply that similar instabilities would arise within a spinning solution. Super-critical Mn values were observed under numerous conditions suggesting that thermocapillary instability may be responsible for striation features that develop in coatings made by spin coating. Trends related to spin-speed, solvent volatility, and initial solution thickness are discussed with the goal of improving the flatness of coatings that are made by this process.     

Generalized Crane flow induced by continuous surfaces stretching with arbitrary velocities

The boundary-layer flow induced by a permeable sheet stretching with general polynomial velocity distribution is considered. This generalizes the work of Kumaran and Ramanaiah (Acta Mech. 116: 229–233, 1996) who were the first to observe that a Crane-type solution exists for wall motion composed of arbitrary linear and quadratic stretching terms, as long as an appropriate lateral transpiration is applied. We solve explicitly the problem to an arbitrarily high degree of the polynomial stretching. This motivates the second part of our study which provides explicit boundary-layer solutions for arbitrary wall stretching, with suitable transpiration. These solutions describe generalized Crane flows whose reciprocal (dimensionless) thicknesses always coincide with the negative of their (dimensionless) entrainment velocities. The associated heat transfer problem is solved explicitly for arbitrary stretching when an appropriate surface temperature distribution is prescribed.     

Stretching of a liquid layer underneath a semi-infinite fluid

Exact similarity solutions of the Navier–Stokes equation are derived describing the flow of a liquid layer coated on a stretching surface underneath another semi-infinite fluid. In the absence of hydrodynamic instability, the interface remains flat as the layer thickness decreases in time. When the physical properties of the fluids are matched, we obtain Crane’s analytical solution for two-dimensional (2D) flow and a corresponding numerical solution for axisymmetric flow. When the rate of stretching of the surface is constant in time, the temporal evolution of the interface between the layer and the overlying fluid is computed by integrating in time a system of coupled partial differential equations for the velocity in each fluid together with an ordinary differential equation expressing kinematic compatibility at the interface, subject to appropriate boundary, interfacial, and far-field conditions. Multiple solutions are found in certain ranges of the density and viscosity ratios. Additional similarity solutions are presented for accelerated 2D and axisymmetric stretching. The numerical prefactors that appear in the analytical expressions for the interface location and wall shear stress are presented for different ratios of the densities and viscosities of the two fluids.     

A model equation for steady surface waves over a bump

The objective of this paper is to study the solutions of a model equation for steady surface waves on an ideal fluid over a semicircular or semielliptical bump. For upstream Froude number F>1, we show that the numerical solution of the equation has two branches and there is a cut-off value of F below which no solution exists. For F<1, the problem is reformulated to overcome the so-called infinite-mass dilemma. A branch of solutions and a cut-off value of F, above which no solution exists, are found. Furthermore, we also obtain a branch of hydraulic-fall solutions which decrease monotonically from upstream to downstream.     

Numerical study for micropolar flow over a stretching sheet

The paper presents a numerical study of the steady flow of a micropolar fluid flow from a stretching sheet. Approximate analytical solution of high nonlinear momentum, angular momentum and confluent hypergeometric similarity solution of the heat transfer equation are obtained for a particular case when the vortex viscosity is neglected. Accuracy of the analytical solution is verified by numerical solutions obtained by employing finite element and Chebyshev finite difference methods. The good agreement between the numerical results of both methods, together with an excellent agreement with the analytical solutions for the special case, ensures the reliability of the obtained results. The velocity, microrotation and temperature functions are shown graphically and the effect of the permeability parameter is studied.     

Heat transfer in a viscoelastic boundary layer flow through a porous medium

This paper examines the viscoelastic fluid flow and heat transfer characteristics in a saturated porous medium over an impermeable stretching surface with frictional heating and internal heat generation or absorption. The heat transfer analysis has been carried out for two different heating processes, namely (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). The governing equations for the boundary layer flow problem result similar solutions. For the specified five boundary conditions, it is not possible to solve directly the resulting sixth-order nonlinear ordinary differential equation. For the present incompressible boundary layer flow problem with constant physical parameters, the momentum equation is decoupled from the energy equation. Two closed–form solutions for the momentum equation are obtained and identified the realistic solution of the physical problem. Exact solution for the velocity field and the skin-friction are obtained. Also, the solution for the temperature and the heat transfer characteristics are obtained in terms of Kummers function. Asymptotic results for the temperature function for large Prandtl numbers are presented. The work due to deformation in the energy equation, which is essential and escaped from the attention of researchers while formulating the visco-elastic boundary layer flow problems, is considered. Drastic variation in the values of heat transfer coefficient is observed when the work due to deformation is ignored.The authors would like to thank the reviewers for their valuable comments/ suggestions to improve the clarity of the paper.     

An inverse solution for reconstruction of the heat transfer coefficient from the knowledge of two temperature values in a solid substrate

Reconstruction of the heat transfer coefficient from the knowledge of temperature distribution is an inverse problem. The main focus of this study was to develop an inverse model that could be used to determine the heat transfer coefficient associated with a fluid in contact with a solid surface from the knowledge of two measured temperature values (T₁ and T_M) in the solid substrate. The temperature distribution for the inverse problem was numerically generated, for a situation with a known heat transfer coefficient, using an implicit finite-differencing scheme. The solution domain was first discretized in to finite number of small regions and nodes. Conservation of energy was then applied to each of the control volume about the nodal regions. This approach resulted in a set of linear equations that was solved simultaneously. Two nodal temperatures in the substrate, from the direct solution, were then used in the inverse problem to reconstruct the heat transfer coefficient. To solve the inverse problem, the solution domain was divided into two distinct regions (Region I and Region II). Region I contained the solution domain between the two known temperatures (T₁ and T_M), and Region II included the region between T_M and the surface with the convective boundary condition. Again, a finite-differencing scheme was employed to generate a set of linear equations in each region. First, the set of linear equations in Region I was solved simultaneously and the results were then used to reconstruct the nodal temperatures in Region II. The convective surface temperature was then utilized to determine the heat transfer coefficient. A series of numerical experiments were conducted to test the validity of the inverse model. Comparison of the inverse solutions with the direct solutions confirms that the heat transfer coefficient can be reconstructed, with good accuracy, from the knowledge of two temperature points in the solid substrate.     

Flow of a second-order fluid over a stretching surface having power-law temperature

Summary The important physical quantities such as the coefficients of skin-friction and heat transfer are obtained from the closed-form solutions for the boundary layer equations of the flow of a second-order fluid over a stretching surface having power-law temperature.     

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.     

Hydromagnetic convection at a continuous moving surface

Summary Exact solution for hydromagnetic convection at a continous moving surface with uniform suction is obtained. Flow of this type represents a new class of boundary-layer problems, with solutions substantially different from those for boundary-layer flow at a surface of finite length. Moreover, this is an exact solution of the complete Navier-Stokes equations (including, buoyancy force-term). The solutions for the velocity and skin friction are evaluated numerically for several sets of values of the parameters;G, the Grashof number,P, the Prandtl number, andM, the Hartmann number. It is observed that there is a reverse flow in the boundary-layer due to heating of the fluid (close to the moving surface). That is, whenT _w>T , the fluid in the boundary-layer will be heated up and thus the free convection currents will set in. Also, it is observed that, there is an increase in the skin friction (absolute) with increasingG, P andM.With 7 Figures     

On extra boundary condition in the stagnation point flow of a second grade fluid

The two dimensional stagnation point flow of a second grade fluid is considered. The flow is governed by a boundary value problem in which the order of differential equations is one more than the number of available boundary conditions. It is shown that without augmenting the boundary conditions at infinity it is possible to obtain a numerical solution of the problem for all values of K, where K is the dimensionless viscoelastic fluid parameter. The numerical results using the algorithm foreshadow an asymptotic behavior for large K. The asymptotic solution is derived up to terms of O(K⁻¹). Perturbation solutions are also obtained up to the terms of O(K²). Finally an approximate solution is developed, based on stretching of the independent variable and minimizing the residual of the differential equation in the least square sense. All these solutions are compared with the exact numerical solution and the appropriate conclusions are drawn.     

Flow of a micropolar fluid over a curved stretching surface

The paper is devoted to the numerical solution of the problem of flow of a micropolar fluid over a curved stretching surface. A similarity transformation is applied to reduce a system of partial differential equations to a system of ordinary differential equations. The numerical solution of these coupled equations is carried out by the shooting method using the Runge–Kutta algorithm. The physical quantities of interest, like the fluid velocity, microrotation velocity, and pressure, are obtained and discussed as functions of the nondimensional curvature radius. It is evident from the results that the pressure inside the boundary layer cannot be neglected for a curved stretching sheet, as distinct from a flat stretching sheet.     

Propagation of a penny-shaped hydraulic fracture parallel to the free-surface of an elastic half-space

In this paper we analyze the problem of a penny-shaped hydraulic fracture propagating parallel to the free-surface of an elastic half-space. The fracture is driven by an incompressible Newtonian fluid injected at a constant rate at the center of the fracture. The flow of viscous fluid in the fracture is governed by the lubrication equation, while the crack opening and the fluid pressure are related by singular integral equations. We construct two asymptotic solutions based on the assumption that either the solid has no toughness or that the fluid has no viscosity. These asymptotic solutions must be understood as corresponding to limiting cases when the energy expended in the creation of new fracture surfaces is either small or large compared to the energy dissipated in viscous flow. It is shown that the asymptotic solutions, when properly scaled, depend only on the dimensionless parameter cal R cal, the ratio of the fracture radius over the distance from the fracture to the free-surface. The scaled solutions can thus be tabulated once and for all and the dependence of the solution on time can be retrieved for specific parameters, through simple scaling and by solving an implicit equation.     

Analytical series solutions for three-dimensional supercritical flow over topography

An analytical series solution method for three-dimensional, supercritical flow over topography is presented. Steady, nonlinear solutions are calculated for a single layer of inviscid, constant-density fluid that flows irrotationally over an obstacle that varies significantly in the x-, y- and z-directions. Accurate series solutions for the free surface and a series of stream tubes throughout the flow region are calculated to demonstrate the three-dimensional properties of the problem. These solutions provide valuable insight into the three-dimensional interactions between the fluid and obstacle which is impossible to gain from any two-dimensional model. The model is described by a Laplacian free-boundary problem with fully nonlinear boundary conditions. The solution method consists of iteratively updating the location of the free surface (on top of the fluid) using a cost function which is derived from the Bernoulli equation. Root-mean-square errors in the boundary conditions are used as convergence criteria and a measure of the accuracy of the solution. This method has been used to solve the two-dimensional version of this problem in the past. Here, we detail the extensions required for three-dimensional flow.     

Forced-convection heat transfer over a circular cylinder with Newtonian heating

A mathematical model for the forced convection boundary-layer flow over a circular cylinder is considered when there is Newtonian heating on the surface of the cylinder through which the heat transfer is proportional to the local surface temperature. The dimensionless version of the boundary-layer equations involve two parameters, the Prandtl number σ and γ measuring the strength of the surface heating. The solution near the stagnation point is considered first and this reveals that, to get a physically acceptable solution, γ must be less than some critical value γ _c , dependent on σ. Numerical solutions to the full boundary-layer problem are obtained which show that the surface temperature increases as the flow develops from the stagnation point.     

Comparison of Solutions from Parabolic and Hyperbolic Models for Transient Heat Conduction in Semi-Infinite Medium

The expression for the transient temperature during damped wave conduction and relaxation developed by Baumeister and Hamill by the method of Laplace transforms was further integrated. A Chebyshev polynomial approximation was used for the integrand with a modified Bessel composite function in space and time. A telescoping power series leads to a more useful expression for the transient temperature. By the method of relativistic transformation, the transient temperature during damped wave conduction and relaxation was developed. There are four regimes to the solution. These include: (i) a regime comprising a Bessel composite function in space and time, (ii) another regime comprising a modified Bessel composite function in space and time, (iii) the temperature solution at the wave front was also developed separately, and (iv) the fourth regime at a given location X in the medium is at times less than the inertial thermal lag time. In this regime, the temperature was found to be unchanged at the initial condition. The solution for the transient temperature from the method of relativistic transformation is compared side by side with the solution for the transient temperature from the method of Chebyshev economization. Both solutions are within 12% of each other. For conditions close to the wave front, the solution from the Chebyshev economization is expected to be close to the exact solution and was found to be within 2% of the solution from the method of relativistic transformation. Far from the wave front, i.e., close to the surface, the numerical error from the method of Chebyshev economization is expected to be significant and verified by a specific example. The solution for transient surface heat flux from the parabolic Fourier heart conduction model and the hyperbolic damped wave conduction and relaxation models are compared with each other. For τ > 1/2 the parabolic and hyperbolic solutions are within 10% of each other. The parabolic model has a “blow-up” as τ → 0, and the hyperbolic model is devoid of singularities. The transient temperature from the Chebyshev economization is within an average of 25% of the error function solution for the parabolic Fourier heat conduction model. A penetration distance beyond which there is no effect of the step change in the boundary is predicted using the relativistic transformation model.     

Numerical modeling of Kelvin–Helmholtz instability using smoothed particle hydrodynamics

This paper presents a Smoothed Particle Hydrodynamics (SPH) solution for the Kelvin–Helmholtz Instability (KHI) problem of an incompressible two‐phase immiscible fluid in a stratified inviscid shear flow with interfacial tension. The time‐dependent evolution of the two‐fluid interface over a wide range of Richardson number (Ri) and for three different density ratios is numerically investigated. The simulation results are compared with analytical solutions in the linear regime. Having captured the physics behind KHI, the effects of gravity and surface tension on a two‐dimensional shear layer are examined independently and together. It is shown that the growth rate of the KHI is mainly controlled by the value of the Ri number, not by the nature of the stabilizing forces. It was observed that the SPH method requires a Richardson number lower than unity (i.e. Ri?0.8) for the onset of KHI, and that the artificial viscosity plays a significant role in obtaining physically correct simulation results that are in agreement with analytical solutions. The numerical algorithm presented in this work can easily handle two‐phase fluid flow with various density ratios. Copyright © 2011 John Wiley & Sons, Ltd.     

Coupled heat and mass transfer in mixed convection over a VHF/VMF wedge in porous media: The entire regime

Summary The development of flow and heat transfer of a viscous electrically conducting fluid in the stagnation point region of a three-dimensional body with an applied magnetic field is studied when the external stream is set into an impulsive motion from rest and at the same time the surface temperature is suddenly raised from that of the surrounding fluid. This analysis includes both short time solution (Rayleigh-type of solution) and the steady-state solution as time tends to infinity (Falkner-Skan type of solution). The unsteady three-dimensional boundary layer equations represented by a system of parabolic partial differential equations are solved numerically using an implicit finite-difference scheme. For certain particular cases analytical solutions are obtained. In the absence of the magnetic field, the reverse flow occurs in the transverse component of the velocity in a certain portion of the saddle-point region (–1c<–0.4, wherec=b/a is the ratio of the velocity gradients in they- andx-directions at the edge of the boundary layer). The magnetic field delays or prevents the reverse flow. The surface shear stresses in the principal and transverse directions and the surface heat transfer increase with the magnetic field both in nodal point (0c1) and saddle point (–1c<0) regions. For a fixed magnetic field, the surface shear stress inx-direction and the surface heat transfer increase with time in nodal and saddle point regions, but the surface shear stress in the transverse direction increases with time for 0<c1 and decreases with increasing time for –1c<0.     

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.