Non-similar solution for the axisymmetric flow of a third-grade fluid over a radially stretching sheet

Summary The problem of axisymmetric flow of a third grade fluid over a radially stretching sheet is studied. By means of similarity transformation, the governing non-linear partial differential equations are reduced to a non-linear ordinary differential equation. The ordinary differential equation is analytically solved using homotopy analysis method (HAM). The solution for the velocity is obtained. The series solution is developed and the convergence of the results is discussed. Finally, the results are discussed with various graphs.     

Magnetofluiddynamic flow with a pressure gradient and fluid injection

Summary The nonlinear partial differential equation of motion for an incompressible fluid flowing over a flat plate under the influence of a magnetic field and a pressure gradient, and with or without fluid injection (or ejection) through the plate is transformed to a nonlinear, third order ordinary differential equation by using a stream function and a similarity transformation.The necessary boundary conditions are developed for flow with and without fluid injection (or ejection), and an example is presented to illustrate the solution to the flow problem.The controlling equation reduces to the well known Falkner-Skan equation when the magnetic field is zero, and if additionally the pressure gradient is zero, the equation reduces to the Blasius equation.     

非线性常微分方程高阶谐波平衡法傅里叶展开的简化

简化了一种求取非线性常微分方程高阶谐波解的近似解析计算方法。对平方和立方非线性项的傅里叶展开过程进行改进和简化,使计算过程变为两次矩阵运算即可完成展开过程,且两次矩阵运算过程一致,易于编程。以Duffing方程为算例,计算结果与数值方法一致,运算效率有所提高。     

Magnetohydrodynamic flow for a non-Newtonian power-law fluid having a pressure gradient and fluid injection

Summary The nonlinear partial differential equation of motion for an incompressible, non-Newtonian power-law fluid flowing over flat plate under the influence of a magnetic field and a pressure gradient, and with or without fluid injection or ejection, is transformed to a nonlinear third-order ordinary differential equation by using a stream function and a similarity transformation.The necessary boundary conditions are developed for flow with and without fluid injection (or ejection), and a solution for four different power-law fluids, including a Newtonian fluid, is presented.The controlling equation includes, as special cases, the Falkner-Skan equation and the Blasius equation.     

Integral transform solution of internal flow problems based on Navier–Stokes equations and primitive variables formulation

The generalized integral transform technique (GITT) is employed in the solution of incompressible laminar channel flows as formulated by the steady‐state Navier–Stokes and continuity equations under the primitive variables mathematical representation. A hybrid numerical–analytical solution is developed based on eigenfunction expansions in one space co‐ordinate and error‐controlled numerical solution of the resulting system of coupled ordinary differential equations in the remaining space direction. The approach is illustrated for developing flow between parallel‐plates with uniform and irrotational inlet flow condition. The conventional Poisson‐type equation for the pressure field with appropriate boundary conditions is also transformed and simultaneously solved with the momentum equation along the longitudinal direction, by considering eigenvalue problems for each of the two potentials, defined in the transversal direction. The transversal velocity component is then explicitly determined from the continuity equation. Numerical results of the longitudinal velocity component and friction factor fields are reported to illustrate the convergence behaviour and user prescribed error control inherent to the proposed hybrid approach. Critical comparisons with previous contributions on the same method that made use of the streamfunction‐only formulation are also provided. Copyright © 2006 John Wiley & Sons, Ltd.     

A finite element method for the general solution of ordinary differential equations

A finite element method is developed by which it is possible to obtain the general solution of an ordinary differential equation directly. The procedure consists of approximating the differential equation with a rectangular matrix equation and of solving the latter equation by using generalized matrix inversion. It is shown in the paper that the homogeneous and inhomogeneous solutions of the two systems correspond and that the approximate solutions produced form the complete general solution of the original differential equation.     

Effect of the aspect ratio on the flow characteristics of magnetohydrodynamic (MHD) third grade fluid flow through a rectangular channel

The magnetohydrodynamic (MHD) flow of a third grade fluid through a rectangular channel, considering the effect of aspect ratio, has been investigated. The flow considered is steady, laminar, incompressible and hydro-dynamically fully developed. The equation, describing the flow, is a highly non-linear partial differential equation (PDE) with remote possibility of having an exact solution and even numerical solution also is very difficult to obtain. A combination of the homotopy perturbation method (HPM) and integral method (IM) has been employed to solve the non-linear PDE which is scarce in open literature. The results of the present study are compared with the results obtained by the least square method (LSM) of the MHD third grade fluid flow through a rectangular channel, without the effect of aspect ratio and are found to be in close agreement. The results indicate that the flow field is significantly affected by the aspect ratio which should be considered for practical applications. In all the available literatures of the third grade fluid flow, the aspect ratio effect is neglected and this simplifying assumption reduces the highly complicated non-linear PDE to a non-linear ordinary differential equation (ODE). The novelty of the subject work lies in the inclusion of the effects of aspect ratio in the governing equation describing the flow of a third grade fluid through a channel and solving this by a combined analytical method (HPM and IM). Further, the effects of the Hartmann number and non-Newtonian third grade fluid parameter on the flow filed are discussed.     

On the derivation of first integrals for similarity solutions

In two recent papers the authors have obtained a number of first integrals for similarity solutions of nonlinear diffusion and of general high-order nonlinear evolution equation. Such integrals exist only for special parameter values and are obtained via integration of the ordinary differential equation, which results when the functional form of the solution is substituted into the governing partial differential equation. In this paper we show that these special parameter values also occur in a natural way when we utilize the first order partial differential equation instead of the explicit functional form and we ask under what conditions can a first integral with respect to either of the independent variables x or t be deduced. This simple procedure generates all previous results and presents the idea of similarity solutions in an entirely new light. That is, the significant features of similarity solutions for partial differential equations are not necessarily the explicit functional form and subsequent reduction to an ordinary differential equation but rather that the solutions sort are common to two partial differential equations. The process is illustrated with reference to an extensive number of examples including nonlinear diffusion, general diffusion equations containing a number of parameters and high-order nonlinear evolution equations. In addition a new exact solution for nonlinear diffusion is obtained which is illustrated graphically.     

Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid

The problem of a magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid is considered for the analytical solution using homotopy analysis method (HAM). The non-linear partial differential equations are transformed to an ordinary differential equation first taking boundary layer approximations into account and then using the similarity transformations. The analytical solution is presented in the form of an infinite series. The recurrence formulae for finding the coefficients are presented and the convergence is established. The effects of the Deborah number and MHD parameter is discussed on the velocity profiles and the skin friction coefficients. It is found that the results are in excellent agreement with the existing results in the literature for the case of hydrodynamic flow.     

An efficient method of solving the Navier–Stokes equations for flow control

A new method of solving the Navier–Stokes equations efficiently by reducing their number of modes is proposed in the present paper. It is based on the Karhunen–Loève decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena, and consequently reduce the Navier–Stokes equation defined on a complicated geometry to a set of ordinary differential equations with a minimum degree of freedom. The present algorithm is well suited for the problems of flow control or optimization, where one has to compute the flow field repeatedly using the Navier–Stokes equation but one can also estimate the approximate solution space of the flow field based on the range of control variables. The low-dimensional dynamic model of viscous fluid flow derived by the present method is shown to produce accurate flow fields at a drastically reduced computational cost when compared with the finite difference solution of the Navier–Stokes equation. © 1998 John Wiley & Sons, Ltd.     

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.     

Non-orthogonal stagnation-point flow of a micropolar fluid

This paper considers the problem of steady two-dimensional flow of a micropolar fluid impinging obliquely on a flat plate. The flow under consideration is a generalization of the classical modified Hiemenz flow for a micropolar fluid which occurs in the boundary layer near an orthogonal stagnation point. A coordinate decomposition transforms the full governing equations into a primary equation describing the modified Hiemenz flow for a micropolar fluid and an equation for the tangential flow coupled to the primary solution. The solution to the boundary-value problem is governed by two non-dimensional parameters: the material parameter K and the ratio of the microrotation to skin friction parameter n. The obtained ordinary differential equations are solved numerically for some values of the governing parameters. The primary consequence of the free stream obliqueness is the shift of the stagnation point toward the incoming flow.     

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.     

Heat and mass transfer processes in two phase He II/vapor

Modeling heat and mass transfer characteristics of two phase He II is discussed. The case considered assumes that the channel flow is one-dimensional and stratified, with mass exchange between the two phases. Two specific examples are considered in some detail. The first is the heat and mass transfer characteristics for small liquid flow rate. Use of several simplifying assumptions allows the problem to be reduced to solution of a one-dimensional ordinary differential equation. The result is a non-dimensional expression for the liquid level or void fraction along the channel. A set of dimensionless parameters are defined that establish the relative contributions of vapor mass transport and counterflow in the He II. The model also predicts the temperature profile and vapor mass flow rate. The second case concerns the flow of liquid under nearly isothermal conditions with relatively small vapor mass flow rate. Under these conditions, the flow may be modeled using classical hydrodynamics taking into consideration the unique characteristics of the He II. Results of these models are compared to experimental data for heat and mass transfer in a two phase He II/vapor flow.     

Some generalized similarity solutions for plastic flow

Summary In kinematically determined regimes, the velocity equations for axially symmetric flow of a perfectly plastic solid which obeys Tresca's yield condition and associated flow rule possess many symmetries. These equations apply also to the flow of granular materials according to the double-shearing theory of Spencer [1], [2]. Using Lie group methods, five classes of generalized self-similar solutions are identified. Special cases are the two types of solution due to Lippmann [3] [4] and the flow past a cone found by Spencer [5]. For each class of solution, determination of the stream function requires the solution of a second-order ordinary differential equation, which can in each case be reduced to the analysis of a first-order equation. Examples of the flow fields and corresponding streamlines for three of the four newly determined cases are computed numerically.     

Local consequences of a global model for mountain pine beetle mass attack

A coupled partial differential equation model for interaction between mountain pine beetles (MPBs) and lodgepole pine is reviewed. An asymptotic cone-tree' solution is examined and the time-scale of the chemical response discussed. The equations are decoupled making an adiabatic assumption for MPB chemotaxis, and a 'local' projection is made using the leading eigenfunction for the MPB density equation. This projection yields a system of ordinary differential equations for the spatio-temporal response ui a single tree. These equations are analyzed, nnd their behavior compared with observations.     

Exact solutions for Couette and Poiseuille flows for fourth grade fluids

Summary Exact solutions for four types of flows between two parallel plates are presented, viz. Couette flow, plug flow, Poiseuille flow and generalized Couette flow. The nonlinear second-order ordinary differential equation for the velocity field is solved exactly in each case. These solutions are compared to those found by perturbation and homotopy analysis methods by Siddiqui et al. [1].     

The application of shape-preserving splines for the solution of differential equations

This paper investigates the use of shape-preserving interpolants based on Bernstein polynomials for the numerical solution of differential equations. A simple and accurate algorithm is presented for the integration of the initial value ordinary differential equation, as well as for the partial differential equation of the Burger type.     

An iterative procedure for the oscillatory laminar boundary layer

The problem of small sinusoidal oscillations imposed on a stagnation point flow is considered. This leads to an ordinary differential equation which, in the high frequency limit, has a simple solution. An iterative method (involving a linear integral operator) is developed, starting from this basic solution. This scheme, performed numerically, is ‘under-relaxed’ and this renders it applicable to as wide a frequency range as possible: convergence criteria are discussed in terms of the relaxation parameter. The streamwise velocity profiles computed are compared with those given by an independent method.     

On Kaplun limits and the generalized boundary layer theory

In the present work, Kaplun limits will be used to derive a generalized version of the boundary layer theory. The intermediate variable technique will be applied to the Navier-Stokes equations and, through the concept of principal equation proposed by Kaplun, a set of partial differential equations will be obtained which represents the asymptotic limit of the momentum equations as the Reynolds number approaches infinite. These equations combine the inviscid flow formulation and the classical boundary layer equations into a single and more general theory that disregards the needs for any type of viscid-inviscid interaction. The proposed formulation will be used to study the flow over a flat plate, and a quasi-similar ordinary differential equation will be deduced which represents an extension of the Blasius equation. The domain of validity of the quasi-similar equation will be analyzed and the asymptotic character of the Blasius solution, which is valid only as Reynolds number tends to infinity, will be studied.