Group method analysis of unsteady free-convective laminar boundary-layer flow on a nonisothermal vertical flat plate

The transformation group theoretic approach is applied to present an analysis of the problem of unsteady laminar free convection from a non-isothermal vertical flat plate. The application of two-parameter groups reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature variations with position and time are derived. The ordinary differential equations are solved numerically using a fourth-order Runge-Kutta scheme and the gradient method. The heat-transfer characteristics for finite values of the Prandtl number Pr are presented, as temperature and velocity distributions.     

Group method analysis of mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder

Summary The transformation group theoretic approach is applied to the system of equations governing the unsteady mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder. The application of a two-parameter group reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature T_w, potential velocity U and sin with position and time are derived in steady and unsteady cases. New formulae of dimensionless temperature are presented using the group method analysis. Hiemenz and Falkner-Skan equations are obtained as special cases. The new similarity representations and similarity transformations in steady/unsteady states are obtained. The family of ordinary differential equations has been solved numerically using a fourth-order Runge-Kutta algorithm with the shooting technique. The effect of varying parameters governing the problem is studied.     

Similarity analysis in magnetohydrodynamics: Hall effects on forced convective heat and mass transfer of non–Newtonian power law fluids past a semi-infinite vertical flat plate

Summary The forced convective heat and mass transfer along a semi-infinite vertical flat plate is investigated for non–Newtonian power law fluids in the presence of a strong nonuniform magnetic field, and the Hall currents are taken into account. The similarity solutions are obtained using transformations group theory. These are the only symmetry transformations admitted by the field equations. The application of one-parameter groups reduces the number of independent variables by one, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with the appropriate boundary conditions. Furthermore the similarity equations are solved numerically by using a fourth-order Runge-Kutta scheme with the shooting method. Numerical results for the velocity profiles, the temperature profiles and the concentration profiles are presented graphically for various values of the power-law viscosity index n, generalized Schmidt number Sc, generalized Prandtl number Pr, the magnetic parameter M and the Hall parameter m.     

Numerical solution of linear two-point boundary problems via the fundamental-matrix method

A numerical method is presented for the-solution of linear systems of differential equations with initial-value or two-point boundary conditions. For y ′(x) = A (x) y (x) + f (x) the domain of interest [a,b] is divided into an appropriate number L of subintervals. The coefficient matrix A (x) is replaced by its value A_k at a point x_k within the Kth subinterval, thus replacing the original system by the L discretized systems y k(x) = A _k y _k(x) + f _k(x), k = 1,2,…, L. The fundamental matrix solution Φ_k(x, x_k) over each subinterval is found by computing the eigenvalues and eigenvectors of each A _k. By matching the solutions y _k(x) at the L – 1 equispaced grid points defining the limits of the subintervals and the boundary conditions, the two-point problem is reduced to solving a system of linear algebraic equations for the matching constants characterizing the different y _k(x). The values of y ₁(a) and y _L(b) are used to calculate the missing boundary conditions. For initial-value problems this method is equivalent to a one-step method for generating approximate solutions. By means of a coordinate transformation, as in the multiple shooting method,¹ the method becomes particularly suitable for stiff systems of linear ordinary differential equations. Five examples are discussed to illustrate the viability of the method.     

Injection of a non-Newtonian fluid through one side of a long vertical channel

Summary The problem considered here is the injection of a non-Newtonian fluid with elastic properties through one side of a long vertical channel. Using the transformations proposed by Wang and Skalak [14] for the velocity components, the basic equations governing the flow and heat transfer are reduced to a set of ordinary differential equations. These equations have been solved approximately subject to the relevant boundary conditions. The effect of the non-Newtonian parameter,S, on the velocity field and heat transfer on the walls is examined carefully.     

Estimation of magnetohydrodynamic radiative nanofluid flow over a porous non‐linear stretching surface: application in biomedical research

The present study investigates the effects of thermal radiation and chemical reaction on magnetohydrodynamic flow, heat, and mass transfer characteristics of nanofluids such as Cu–water and Ag–water over a non‐linear porous stretching surface in the presence of viscous dissipation and heat generation. Using similarity transformation, the governing boundary layer equations of the problem are transformed into non‐linear ordinary differential equations and solved numerically by the shooting method along with the Runge–Kutta–Fehlberg fourth–fifth‐order integration scheme. The influences of various parameters on velocity, temperature, and concentration profiles of the flow field are analysed and the results are plotted graphically. A backpropagation neural network is applied to predict the skin friction coefficient, Nusselt number, and Sherwood number and these results are presented through graphs. The present numerical results are compared with the existing results and are found to be in good agreement. The results of artificial neural network and the obtained numerical values agree well with an error <5%.Inspec keywords: silver, copper, transforms, nanofluidics, friction, backpropagation, heat radiation, water, external flows, partial differential equations, nonlinear differential equations, boundary layers, Runge‐Kutta methods, mass transfer, flow through porous media, magnetohydrodynamicsOther keywords: magnetohydrodynamic radiative nanofluid flow, nonlinear stretching surface, biomedical research, thermal radiation, chemical reaction, magnetohydrodynamic flow, nonlinear porous stretching surface, viscous dissipation, similarity transformation, governing boundary layer equations, nonlinear ordinary differential equations, shooting method, Runge–Kutta–Fehlberg fourth–fifth‐order integration scheme, flow field, backpropagation neural network, Cu–water nanofluid, Ag–water nanofluid, skin friction coefficient, Nusselt number, Sherwood number, artificial neural network, Ag‐H₂ O, Cu‐H₂ O     

A numerical method for solving two-point boundary value problems for nonlinear ordinary differential equations

A numerical method for solving two-point boundary value problems associated with systems of first-order nonlinear ordinary differential equations is described. It needs four function evaluations at each point and is of order h⁶, where h is the space chop. Results of computational experiments, which include perturbation of the initial conditions, comparing this method with other known methods are given.     

Unsteady three-dimensional combined heat and mass free convective flow over a stretching surface with time-dependent chemical reaction

Summary This paper investigates the combined effects of the free convective heat and mass transfer on the unsteady three-dimensional laminar boundary layer flow over a stretching surface. The stretching rates of the surface are assumed to vary as a reciprocal of a linear function of time. Generation or consumption of the diffusing species due to a homogeneous chemical reaction is considered. The chemical reaction rate is assumed to vary with time according to a power law. With appropriate similarity transformation, the boundary layer equations governing the flow are reduced to ordinary differential equations, which are numerically solved by applying a fifth-order Runge-Kutta-Fehlberg scheme with the shooting technique. The effects of the Prandtl number Pr, Schmidt number Sc, the unsteadiness parameter λ, the chemical reaction parameter γ₀ and the reaction order n are examined on the velocity, temperature and concentration distributions. Numerical data for the skin-friction coefficients, Nusselt and Sherwood numbers have been tabulated for various values of the parameters. A comparison is made between the present work and previous results.     

Heat Transfer of Heat-Releasing Fluid in the Top Portion of a Closed Volume

The method of analytic estimates is used to determine the characteristics of steady-state free-convection heat transfer of a fluid with internal heat sources in the top part of a closed volume with different conditions of heat removal on the top horizontal boundary at the Prandtl number value on the order of unity. It is demonstrated that, in the case of adiabatic condition on the top boundary of the volume, the maximal heat flux q _max attained in the region of intersection of the top horizontal and vertical boundaries depends only on the maximal temperature in the volume T _max and on the thermal characteristics of the fluid. The correction to the bulk temperature (outside of the boundary layers) T _b z ^1/4, which is a function of the vertical coordinate z, significantly prevails over perturbations in the horizontal section. When the turbulent Rayleigh-Benard (RB) convection arises, the heat removal through the top boundary is defined only by the energy release in the RB-layer. Given a fixed power of heat release Q, the RB-layer thickness increases by the linear law h=q/Q with increasing heat flux q through the top horizontal boundary.     

Nonlocal,strain gradient and surface effects on vibration and instability of nanotubes conveying nanoflow

ABSTRACT

In this paper, the size-dependent vibration and instability of nanoflow-conveying nanotubes with surface effects using nonlocal strain gradient theory (NSGT) are examined. Hence, based on Gurtin-Murdoch theory, the nonclassical governing equations are derived by extended Hamilton's principle. To study the small-size effects on the flow field, the Knudsen number is applied. Applying Galerkin's approach, the partial differential equations converted to ordinary differential equations. The effects of the main parameters like nonlocal and strain gradient parameters, length to diameter ratio, thickness, surface effects, Knudsen number and different boundary conditions on the eigenvalue and critical fluid velocity of the nanotube are explained.     

A numerical study of steady plane granular chute flows using the Jenkins–Savage model and its extension

The granular flow model proposed by Jenkins and Savage and extended by us is used here to construct numerical solutions of steady chute flows thought to be typical of granular flow behaviour. We present the governing differential equations and discuss the boundary conditions for two flow cases: (i) a fully fluidized layer of granules moving steadily under rapid shear and (ii) a fluidized bottom-near bed covered by a rigid slab of gravel in steady motion under its own weight. The boundary value problem is transformed into a dimensionless form and the emerging system of non-linear ordinary differential equations is numerically integrated. Singularities at the free surface and (in one case) also at an unknown point inside the solution interval make the problem unusual. Since the non-dimensionalization is performed with the maximum particle concentration and the maximum velocity, which are both unknown, these two parameters also enter the formulation of the problem through algebraic equations. The two-point boundary value problem is solved with the aid of the shooting method by satisfying the boundary conditions at the end of the soluton interval and these normalizing conditions by means of a minimization procedure. We outline the numerical scheme and report selective numerical results. The computations are the first performed with the exact equations of the Jenkins–Savage model; they permit delineation of the conditions of applicability of the model and thus prove to be a useful tool for the granular flow modeller.     

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid are investigated under plane strain and plane stress conditions. In each case, by expanding the plastic fields and the governing equations in the coordinate y, the problem is reduced to solving a system of nonlinear ordinary differential equations which is similar to that of mode III derived by Achenbach and Z.L.Li. An approximate solution for small values of x is obtained and matched with the elastic field of a blunt crack at the elastic-plastic boundary. The crack growth criterion of critical strain is employed to determine the value of K _II of the far-field that would be required for a steadily growing crack.     

Thermal interaction between laminar film condensation and forced convection along a conducting wall

Summary A theoretical analysis is presented to investigate the thermal interaction between laminar film condensation of a saturated vapor and a forced convection system separated by a heat conducting wall. In this work, the effect of the wall thermal resistance is considered. It is assumed that the countercurrent boundary layer flow is formed on the two sides. Governing boundary layer equations together with their corresponding boundary conditions for film condensation and forced convection are all cast into dimensionless forms by using the non-similarity transformation. The resulting system of equations is solved by using the local non-similarity method in conjunction with the fourth order Runge-Kutta method in conjunction with the Nachtsheim-Swigert iteration scheme. The total heat flux through the wall and the wall temperature distribution are determined. The present results show that the effect of the forced convection Prandtl number Pr _c is not negligible for large values of the thermal resistance ratioA ^*, and the effect ofA ^* and Pr _c on the overall heat transfer through the wall is more pronounced than that of the Jakob number and film Prandtl number.     

Stability of free convection with uniform suction or injection from a vertical flat plate subjected to a constant wall heat flux

Summary A theoretical study is presented for the stability characteristics of the laminar free convection boundary layer flow along a vertical porous (permeable) flat plate subjected to a constant heat flux. The disturbance equations are solved numerically on the basis of the linear stability theory for a wide range of values of the modified Grashof number,G, and some values of the suction or injection parameterX when the Prandtl number, Pr, is 0.73 (air). These solutions indicate the important role of the parametersG andX on the flow and heat transfer characteristics. It is found that the present results are in very good agreement with those from the open literature.     

A seventh-order numerical method for solving boundary value nonlinear ordinary differential equations

A numerical method for solving two-point boundary value problems associated with systems of first-order nonlinear ordinary differential equations is described. It needs three function evaluations for each sub-interval and is of order O(h⁷), where h is the space chop. Results of computational experiments comparing this method with other known methods are given.     

On consistent finite difference formulae for ordinary differential equations

This paper presents a finite difference method for two-point boundary value problems described by fourth-order ordinary differential equations which results in consistency of truncation errors. It is demonstrated that the order of the formulae used to approximate the boundary conditions must be higher than those used for similar derivative terms in the differential equation. A generalization of the method to differential equations of order n is discussed. The procedure is illustrated with a numerical example.     

Mathematical model of heat and mass transfer processes in evaporative condensers

This study presents a new mathematical model of heat and mass transfer processes in evaporative condensers. The model consists of four ordinary differential equations with their boundary conditions and some associated algebraic equations. The model was formulated for steady-state heat and mass transfer conditions. A simulation computer program based on the model was written. It was devised for heat calculations in condensers built from bare tubes. The quality of the model was calculated by comparing the results obtained by running the program with experimental results achieved by other authors. The computed results show a good degree of conformity with experimental results. The differences are less than 20% (but in one case, 30%). The computer program may be used to determine heat performance of evaporative condensers of horizontal in-line and staggered bundle systems (if S_q > 2d_z).     

Vibration analysis of stepped rectangular plates using the extended Kantorovich method

In this study, vibration behaviors of stepped plates are investigated based on the variational principle of minimum total energy and the extended Kantorovich method. The out-of-plane displacement is represented by a separable function of parameters x and y. A set of governing equations, boundary conditions, and continuity conditions in the form of ordinary differential equations are derived from the energy condition. The natural frequency and out-of-plane displacement function can be determined numerically from the derived equations and conditions. Solutions from the proposed approach are in good agreement with results from past studies and those of the finite element method.     

Similarity solutions of unsteady boundary layer equations of a special third grade fluid

Two-dimensional, unsteady, laminar boundary layer equations of a special model of non-Newtonian fluids are considered. The fluid can be considered as a special type of power-law fluid. The problem investigated is the flow over a moving surface, with suction of injection. Two different type of ordinary differential equations system are found using the transformations. Using scaling and translation transformations, equations and boundary conditions are transformed into a partial differential system with two variables. Using translation and a more general transformation, the boundary value problem is transformed into an ordinary differential equations system. Finally, we numerically solve two different ordinary differential equations, separately.     

Network numerical analysis of magneto-micropolar convection through a vertical circular non-Darcian porous medium conduit

The fully developed, mixed convection heat transfer of a magneto-micropolar fluid in a Darcy–Forchheimer porous medium containing heat sources contained in a vertical circular conduit is investigated in this article. The conservation equations for mass, linear momentum, micro-inertia, angular momentum (micro-rotation) and energy are presented in a cylindrical coordinate system (r, θ, z) with appropriate boundary conditions. A Darcy–Forchheimer drag force model is employed to simulate the effects of bulk linear porous impedance and second order porous resistance. The governing partial differential equations are non-dimensionalized into a set of ordinary differential equations in a single independent variable (η) and solved using the Network Simulation Method. Benchmark solutions are compared with earlier computations using the finite element method, showing excellent agreement. The influence of Darcy number, Forchheimer number, Grashof number, Hartmann number, geometric scale ratio (conduit radius to length ratio), Eringen parameter (ratio of vortex viscosity to Newtonian viscosity) and heat source/sink parameter on the linear velocity, angular velocity (micro-rotation) and temperature functions are studied in detail. Flow i.e. linear (translational) velocity, f, is seen to be inhibited with increasing magnetic field (Hartmann number), Forchheimer number and Eringen parameter, but accelerated with increasing Darcy number. Micro-rotation (g) is decreased with increasing Forchheimer number and Hartmann number, but increased with a rise in Grashof number, Darcy number, geometric scale ratio and Eringen parameter. Both velocity (f) and micro-rotation (g) are increased in the presence of a heat source but decreased with a heat sink. Several special cases of the flow regime are also documented. Applications of the problem include the cooling of porous combustion chambers, geophysical transport in electrically-conducting zones, exhaust nozzles of porous walled flow reactors, hydromagnetic control processes in nuclear engineering and magnetic materials processing (ceramic foams).