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1.
This work looks at the heat transfer effects on the flow of a second grade fluid over a radially stretching sheet. The axisymmetric flow of a second grade fluid is induced due to linear stretching of a sheet. Mathematical analysis has been carried out for two heating processes, namely (i) with prescribed surface temperature (PST case) and (ii) prescribed surface heat flux (PHF case). The modelled non-linear partial differential equations in two dependent variables are reduced into a partial differential equation with one dependent variable. The resulting non-linear partial differential equations are solved analytically using homotopy analysis method (HAM). The series solutions are developed and the convergence is properly discussed. The series solutions and graphs of velocity and temperature are constructed. Particular attention is given to the variations of emerging parameters such as second grade parameter, Prandtl and Eckert numbers. 相似文献
2.
M. Sajid I. Ahmad T. Hayat M. Ayub 《Communications in Nonlinear Science & Numerical Simulation》2009,14(1):96-108
The problem of unsteady boundary layer flow of a second grade over a stretching sheet is investigated in this paper. The governing equations of motion are reduced into a partial differential equation with two independent variables by using similarity transformations. The heat transfer analysis has been also carried out for two heating processes namely the prescribed surface temperature (PST case) and prescribed surface heat flux (PHF case). The series solutions of the problem are developed by employing homotopy analysis method (HAM). Convergence of the obtained series solutions are analyzed. It is noted that the present solutions of a second grade are valid for all dimensionless times. Finally, the results are obtained and discussed through graphs for various parameters of interest. 相似文献
3.
In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching
surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed
exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled
partial differential equations of Jeffrey fluid flow along with the energy equation are simplified by using similarity transformation
techniques based on boundary layer assumptions. The reduced similarity equations are then solved analytically by the homotopy
analysis method (HAM). The convergence of the HAM series solution is obtained by plotting (h/2p)\hbar-curves for velocity and temperature. The effects of physical parameters on the velocity and temperature profiles are examined
by plotting graphs. 相似文献
4.
5.
Bohdan Krajewski 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1961,12(3):231-242
Summary This is an analysis of the axially symmetric rotational flow which takes place (amongst other things) in turbomachinery. The problem can be reduced to a partial differential equation with given boundary conditons. Bearing in mind that the differential equation considered corresponds to a variational problem, the Ritz method is used for a numerical solutions. This requires a few additional assumptions to facilitate the computation procedure and enables us to correlate various design parameters. The velocity field being thus obtained, it is possible to determine the form of the blades. 相似文献
6.
Exact solutions of Calgero–Bogoyavlenskii–Schiff equation using the singular manifold method after Lie reductions
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Calgero–Bogoyavlenskii–Schiff (CBS) equation is analytically solved through two successive reductions into an ordinary differential equation (ODE) through a set of optimal Lie vectors. During the second reduction step, CBS equation is reduced using hidden vectors. The resulting ODE is then analytically solved through the singular manifold method in three steps; First, a Bäcklund truncated series is obtained. Second, this series is inserted into the ODE, and finally, a seminal analysis leads to a Schwarzian differential equation in the eigenfunction φ(η). Solving this differential equation leads to new analytical solutions. Then, through two backward substitution steps, the original dependent variable is recovered. The obtained results are plotted for several Lie hidden vectors and compared with previous work on CBS equation using Lie transformations. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
7.
通过引入一个变换式,克服了Sakiadis流动中半无限大流动区域以及无穷远处渐近边界条件所带来的数学处理上的困难.基于泛函分析中的不动点理论,采用不动点方法求解了变换后的非线性微分方程,获得了Sakiadis流动的近似解析解.该近似解析解用级数的形式来表达并在整个半无限大流动区域内一致有效. 相似文献
8.
C. Ahmed Basha G. RahamathunissaS. Sivakumar Chang Woo Lee 《Applied Mathematical Modelling》2012,36(2):618-625
The analytical solution for a nonlinear ordinary differential equation was obtained but was only applicable for lower input parameters. A numerical solution for the same equation which provided good results for all input parameters was also obtained. Since the analytical solution contained an infinite series, the solution depended on the number of terms used in the series, resulting in failure for certain input parameters at low time values. 相似文献
9.
Based on homotopy, which is a basic concept in topology, a general analytic method (namely the homotopy analysis method) is proposed to obtain series solutions of nonlinear differential equations. Different from perturbation techniques, this approach is independent of small/large physical parameters. Besides, different from all previous analytic methods, it provides us with a simple way to adjust and control the convergence of solution series. Especially, it provides us with great freedom to replace a nonlinear differential equation of order n into an infinite number of linear differential equations of order k , where the order k is even unnecessary to be equal to the order n . In this paper, a nonlinear oscillation problem is used as example to describe the basic ideas of the homotopy analysis method. We illustrate that the second-order nonlinear oscillation equation can be replaced by an infinite number of (2κ)th-order linear differential equations, where κ≥ 1 can be any a positive integer. Then, the homotopy analysis method is further applied to solve a high-dimensional nonlinear differential equation with strong nonlinearity, i.e., the Gelfand equation. We illustrate that the second-order two or three-dimensional nonlinear Gelfand equation can be replaced by an infinite number of the fourth or sixth-order linear differential equations, respectively. In this way, it might be greatly simplified to solve some nonlinear problems, as illustrated in this paper. All of our series solutions agree well with numerical results. This paper illustrates that we might have much larger freedom and flexibility to solve nonlinear problems than we thought traditionally. It may keep us an open mind when solving nonlinear problems, and might bring forward some new and interesting mathematical problems to study. 相似文献
10.
《Communications in Nonlinear Science & Numerical Simulation》2010,15(7):1843-1857
A mathematical analysis has been carried out to study magnetohydrodynamic boundary layer flow, heat and mass transfer characteristic on steady two-dimensional flow of a micropolar fluid over a stretching sheet embedded in a non-Darcian porous medium with uniform magnetic field. Momentum boundary layer equation takes into account of transverse magnetic field whereas energy equation takes into account of Ohmic dissipation due to transverse magnetic field, thermal radiation and non-uniform source effects. An analysis has been performed for heating process namely the prescribed wall heat flux (PHF case). The governing system of partial differential equations is first transformed into a system of non-linear ordinary differential equations using similarity transformation. The transformed equations are non-linear coupled differential equations which are then linearized by quasi-linearization method and solved very efficiently by finite-difference method. Favorable comparisons with previously published work on various special cases of the problem are obtained. The effects of various physical parameters on velocity, temperature, concentration distributions are presented graphically and in tabular form. 相似文献
11.
In this paper, we develop a set of differential equations describing the steady flow of an Oldroyd 6-constant magnetohydrodynamic fluid. The fluid is electrically conducting in the presence of a uniform transverse magnetic field. The developed non-linear differential equation takes into account the effect of the material constants and the applied magnetic field. We presented the solution for three types of steady flows, namely,
- (i)
- Couette flow
- (ii)
- Poiseuille flow and
- (iii)
- generalized Couette flow.
12.
A model of a series of price increments with jumps is constructed based on a linear stochastic differential equation with a Poisson component. Some estimates of unknown parameters of the model and SDE are obtained by using the method of moments. A statistical simulation algorithm for solving an SDE with a Poisson component in general form is proposed. Results of numerical experiments are given. 相似文献
13.
Analytical solutions for two-dimensional oscillatory flow on free convective-radiation of an incompressible viscous fluid, through a highly porous medium bounded by an infinite vertical plate are reported. The Rosseland diffusion approximation is used to describe the radiation heat flux in the energy equation. The resulting non-linear partial differential equations were transformed into a set of ordinary differential equations using two-term series. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. The free-stream velocity of the fluid vibrates about a mean constant value and the surface absorbs the fluid with constant velocity. Expressions for the velocity and the temperature are obtained. To know the physics of the problem analytical results are discussed with the help of graph. 相似文献
14.
15.
Summary. This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate
elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous
media. In this paper we consider a finite element approximation for this system. The elliptic equation for the pressure and
velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated
by a Galerkin finite element method. A fully discrete approximation is analyzed. Sharp error estimates in energy norms are
obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error
estimates are derived directly from the degenerate equation. Also, the analysis does not impose any restriction on the nature
of degeneracy. Finally, it respects the minimal regularity on the solution of the differential system.
Received March 9, 1998 / Revised version received July 17, 2000 / Published online May 30, 2001 相似文献
16.
A demonstration of how the point symmetries of the Chazy equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy equation under a generalized transformation, and find the point symmetries of the Chazy equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy equation which is given by a right Painlevé series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a left Painlevé series and one given by a right Painlevé series where the leading-order behaviors and the resonances are explicitly those of the Chazy equation. 相似文献
17.
《Nonlinear Analysis: Real World Applications》2008,9(4):1269-1282
The problem dealing with the two-dimensional flow of an incompressible viscoelastic Maxwell fluid past an infinite porous plate is investigated. It is assumed that the suction velocity is normal to the plate and oscillates about a mean value. The external free-stream velocity varies periodically in time. The resulting differential equation subject to the relevant boundary and initial conditions is numerically solved by means of a numerical technique, in which a coordinate transformation is employed to transform the semi-infinite physical space to a bounded computational domain. The effects of various values of the emerging parameters, e.g. the elasticity parameter, the oscillation amplitude and frequency of the external flow and the suction velocity, on the time series of velocity, especially on the boundary-layer structure near the plate, are discussed. The nature of the shear stress engendered due to the flow is also investigated. 相似文献
18.
The behavior of meromorphic solutions of differential equations has been the subject of much study. Research has concentrated on the value distribution of meromorphic solutions and their rates of growth. The purpose of the present paper is to show that a thorough search will yield a list of all meromorphic solutions of a multi-parameter ordinary differential equation introduced by Hayman. This equation does not appear to be integrable for generic choices of the parameters so we do not find all solutions—only those that are meromorphic. This is achieved by combining Wiman-Valiron theory and local series analysis. Hayman conjectured that all entire solutions of this equation are of finite order. All meromorphic solutions of this equation are shown to be either polynomials or entire functions of order one. 相似文献
19.
I. Kenneth Johnpillai James M. Hill 《Journal of Mathematical Analysis and Applications》2005,301(1):135-157
The Airy stress function, although frequently employed in classical linear elasticity, does not receive similar usage for granular media problems. For plane strain quasi-static deformations of a cohesionless Coulomb-Mohr granular solid, a single nonlinear partial differential equation is formulated for the Airy stress function by combining the equilibrium equations with the yield condition. This has certain advantages from the usual approach, in which two stress invariants and a stress angle are introduced, and a system of two partial differential equations is needed to describe the flow. In the present study, the symmetry analysis of differential equations is utilised for our single partial differential equation, and by computing an optimal system of one-dimensional Lie algebras, a complete set of group-invariant solutions is derived. By this it is meant that any group-invariant solution of the governing partial differential equation (provided it can be derived via the classical symmetries method) may be obtained as a member of this set by a suitable group transformation. For general values of the parameters (angle of internal friction ? and gravity g) it is found there are three distinct classes of solutions which correspond to granular flows considered previously in the literature. For the two limiting cases of high angle of internal friction and zero gravity, the governing partial differential equation admit larger families of Lie point symmetries, and from these symmetries, further solutions are derived, many of which are new. Furthermore, the majority of these solutions are exact, which is rare for granular flow, especially in the case of gravity driven flows. 相似文献
20.
M.A.A. Hamad I. Pop A.I. Md Ismail 《Nonlinear Analysis: Real World Applications》2011,12(3):1338-1346
In this paper, we discuss similarity reductions for problems of magnetic field effects on free convection flow of a nanofluid past a semi-infinite vertical flat plate. The application of a one-parameter group reduces the number of independent variables by 1, and consequently the governing partial differential equation with the auxiliary conditions to an ordinary differential equation with the appropriate corresponding conditions. The differential equations obtained are solved numerically and the effects of the parameters governing the problem are discussed. Different kinds of nanoparticles were tested. 相似文献