This paper presents two exact explicit solutionsfor the three dimensional dual-phase lag(DLP)heat conduc-tion equation,during the derivation of which the method oftrial and error and the authors' previous experiences are uti-lized.To the authors' knowledge,most solutions of 2D or3D DPL models available in the literature are obtained bynumerical methods,and there are few exact solutions upto now.The exact solutions in this paper can be used asbenchmarks to validate numerical solutions and to developnumerical schemes,grid generation methods and so forth.In addition,they are of theoretical significance since theycorrespond to physically possible situations.The main goalof this paper is to obtain some possible exact explicit solu-tions of the dual-phase lag heat conduction equation as thebenchmark solutions for computational heat transfer,ratherthan specific solutions for some given initial and boundaryconditions.Therefore,the initial and boundary conditionsare indeterminate before derivation and can be deduced fromthe solutions afterwards.Actually,all solutions given in thispaper can be easily proven by substituting them into thegoverning equation.
Analytical solutions of governing equations of various phenomena have their irre-placeable theoretical meanings. In addition, they can also be the benchmark solu-tions to verify the outcomes and codes of numerical solutions, and even to develop various numerical methods such as their differencing schemes and grid generation skills as well. A hybrid method of separating variables for simultaneous partial differential equation sets is presented. It is proposed that different methods of separating variables for different independent variables in the simultaneous equa-tion set may be used to improve the solution derivation procedure, for example, using the ordinary separating method for some variables and using extraordinary methods of separating variables, such as the separating variables with addition promoted by the first author, for some other variables. In order to prove the ability of the above-mentioned hybrid method, a lot of analytical exact solutions of two-buoyancy convection in porous media are successfully derived with such a method. The physical features of these solutions are given.
This paper presents two algebraically explicit analytical solutions for the incompressible unsteady rotational flow of Oldroyd-B type in an annular pipe. The first solution is derived with the common method of separation of variables. The second one is deduced with the method of separation of variables with addition developed in recent years. The first analytical solution is of clear physical meaning and both of them are fairly simple and valuable for the newly developing computational fluid dynamics. They can be used as the benchmark solutions to verify the applicability of the existing numerical computational methods and to inspire new differencing schemes, grid generation ways, etc. Moreover, a steady solution for the generalized second grade rheologic fluid flow is also presented. The correctness of these solutions can be easily proven by substituting them into the original governing equation.
CAI Ruixian1 & GOU Chenhua1,2 1. Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100080, China
