Analytical Solution Of N-S Equation For Steady-state Flow Field Of Functional Fluids

The numerical solution of both the rheological fluid and the magnetic fluid is one of the challenging research topics.Its external electric field or magnetic field will show a special fluid properties and flow field orientation.These two functional fluids are used in the fields of mechanical damping,automation control,mechanical manufacturing,mechanical transmission,robot simulation,sealing,medical devices,sound conditioning.Electrorheological fluid and magnetic fluid are non-Newtonian fluids,and their shear stress and shear rate are non-linear relationship.N-S equations as a model to describe the fluid movement,there are still many problems to be solved,the most important cause of these problems is the obstruction of nonlinear terms.Although the classic Sobolev space can solve the moment of confusion,but such as a variable exponential growth conditions of the nonlinear term shows its limitations.Therefore,in order to solve such nonlinear problems,the variable exponential Lebesgue space plays an important role.It is necessary to study the N-S equation of variable exponential function space theory on Clifford algebra.In this paper,the analytical situation of the NS equations in bounded confined spaces and semi-spaces is studied.Firstly,the research status and development trend of the NS equations are systematically summarized.NS equations are studied in bounded confined spaces and in the half space,the expression of the solution are given.The numerical analysis and the numerical data of the Fluent software are compared with each other.The advantages and limitations of the analytical solution are pointed out.This paper provides important theoretical results for revealing the flow mechanism and simulation logistics processes of these two functional fluids,and provides a new method for the study of the velocity of functional fluid flow field,which provides the performance prediction for the rheological fluid and magnetic fluid flow field The basis.(1)We establish a theory of the variable exponential Clifford value function space in bounded and semi-spaces.For the first time,the weighted variable exponential Clifford function Lebesgue space and the weighted variable exponent Clifford function Sobolev space are introduced.The basic properties of the semi-space are discussed,such as completeness,reflexivity,compactness,separability Embedded theorem.The operator theory of the semi-space is studied,such as the Teodorescu operator.We prove the straight sum decomposition of an improved Clifford function Lebesgue space in a semi-space and bounded region.(2)For the electrorheological fluid,the unstokes flow field of the non-Newtonian steady-state Stokes equations and the N-S equations are studied in the bounded exponential Clifford function space.According to the theory of operator and the properties of straight and decomposed,it is proved that there exists a unique solution in the space W0l,p(x)(?Cln)×Lp(x)(?)of the viscoelastic Stokes equations and gives the expression of the solution.Then,we study the viscoelastic NS equations to construct an iteration.Each step requires the existence and uniqueness of the solution of the corresponding Stokes equations,and then the convergence of the iteration is obtained by using the principle of compression mapping.estimate.It is proved that there is a unique solution in the space W0l,p(x)(?Cln)×Lp(x)(?)when the N-S system of the viscous steady state satisfies a certain electric field condition,and the iterative expression of the solution is given.(3)For the magnetic fluid,the Stokes equations,N-S equations and N-S equations with heat conduction term in the unsteady flow field of the Newtonian steady-state flow field in the Clifford-valued function space are studied.It is proved that there exists a unique solution in the space W0l,p(x)(R+n,Cln)×LP(x)(R+n,R)of the Stokes equations and the solution is obtained.Continue to generalize to the N-S equation,and when it satisfies certain initial conditions,it obtains its analytical solution and proves to be the only convergence.Finally,it is proved that there is a unique solution in the space W0l,p(x)(R+n,Cln)xLp(x)(R+n,R)of N-S equations with heat conduction term under appropriate conditions.(4)Ansys Fluent 14.0 software was used to analyze the momentum fluid and fluid fluid in the plane channel.The relationship between the analytical solution of the formula,the numerical solution of the Fluent software and the experimental data is compared,and it is pointed out that the discontinuity is generated when the velocity is plotted due to the existence of the imaginary solution.The analytical solution and the numerical solution are compared with the experimental data respectively.It is found that the analytic solution is better than the experimental data.In order to provide theoretical basis for more functional fluids in the future,the theoretical results of the iterative solutions of the N-S equations are verified by the error estimates using the analytical solutions of the two cases.The numerical results show stable and convergent.