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On the Solution of Micro-fluid Boundary Layer Equation

Author: LiLong
Tutor: ZhanHuaShui
School: Jimei University
Course: Applied Mathematics
Keywords: Prandtl boundary layer Micro - flow boundary layer Degenerate parabolic equation Classical solutions
CLC: O175.26
Type: Master's thesis
Year: 2010
Downloads: 35
Quote: 0
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Abstract


The Prandtl equations derived from the boundary layer theory of fluid mechanics is important for revealing the nature of small viscous fluid movement , however, the classic Prandtl boundary layer theory apparently did not consider the nature of the side wall boundary layer flow characteristics , the theory on solid wall and the water flow of the interaction involved in the engineering practice , especially fine water flow problems , it is difficult to conduct a full and satisfactory explanation , and sometimes appear contradictory and wrong conclusions , therefore , Prandtl boundary layer theory is not comprehensive . consider solid surface is relatively strong adsorption of water molecules , according to the experimental results show that : the micro - flow characteristics of the fluid boundary layer embodies precisely, we can use the micro - flow boundary layer equations to describe the system [ 1] . Oleinik proved Prandtl equation group local classical solution exists only under certain initial and boundary conditions [2] we are very natural to consider the following questions : how initial and boundary conditions microfluidic local classical solution of the boundary layer system exists and only we can learn from the the Oleinik processing Prandtl boundary layer ideological paper consists of two parts . discussed in the first part of the micro - flow boundary layer equations local classical solution (X given , t sufficiently small ). first, the use of Crocco transformation microfluidic the boundary layer equations transform into a degenerate parabolic partial differential equations , and then using the the Oleinik linearization method to the parabolic equations are transformed into ordinary differential equations that meet a series of a priori estimator solution of ordinary differential equations and solution of ordinary differential equations line sexual expansion for the solution of the parabolic equation , and finally return to the original micro - flow boundary layer equations to prove the existence and uniqueness of local classical solution . idea of this proof comes from the Oleinik processing Prandtl system of thinking ; However, as discussed in this article micro flow in the boundary layer , the coefficients of the partial derivatives of the low-level items are no longer only contain linear terms , Prandtl system there are essentially difficulties in the proof process , we equation nonlinear linear processing is skillful . in the second part , we show that the micro - flow boundary layer equations for any t gt ; 0 , X is sufficiently small , the existence and uniqueness of local classical solution . similar to the idea of proof in the first part , but the specific calculation process , including some of the structure of the corresponding function is different .

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations > Parabolic equation
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