幂律型非牛顿流体能量边界层本构方程

The constitutive equation for energy boundary layer in
power law non-Newtonian fluids
Liancun Zheng 1, Xinxin Zhang 2
1Department of Mathematics and Mechanics, University of Science and Technology Beijing,
Beijing 100083, China, e-mail: liancunzheng@
2Mechanical Engineering School, University of Science and Technology Beijing,
Beijing 100083, China, e-mail: xxzhang@
Abstract: A new energy boundary layer equation model for power law non-Newtonian fluids is established first time by assuming that the thermal diffusivity a is characterized as a power law function of temperature gradient. The Prandtl number is characterized by a relationship of velocity gradient, temperature gradient, and the power law index. Furthermore, a new similarity number are derived by supposing that the heat boundary layer equation existing similarity solution.
Keywords: Power law fluids, heat transfer, similarity solution, nonlinear boundary value problem. AMS Subject Classification: 34B15, 76D10
1. Introduction
Recently, considerable attention has been devoted to the problem of how to predict the drag force behavior of non-Newtonian fluids. The main reason for this is probably that fluids(such as molten plastics, pulps, slurries, emulsions), which do not obey the Newtonian postulate that the stress tensor is directly proportional to the deformation tensor, are produced industrially in increasing quantities, and are therefore in some cases just as likely to be pumped in a plant as the more common Newtonian fluids. Understanding the nature of this force by mathematical modeling with a view to predicting the drag forces and the associated behavior of fluid flow has been the focus of considerable research work. In addition, the mathematical model considered in the present paper has significance in studying many problems of engineering [1-3, 6-16].
2. Boundary Layer Governing Equations
When a fluid flows past a solid body at high Reynolds number , a thin viscous boundary layer is known to form at least along the forward portion of the solid surface. Historically, the boundary layer flow past a flat plate was first example considered by Blasius to illustrate the application of Prandtl’s boundary layer theory. Schowalter R [2] applied the boundary layer theory to power law pseu-doplastic fluids and developed the two-dimensional and three dimensional boundary layer equations for the momentum transfer. Acrivos and Shah [3] considered the momentum and heat transfer for a non-Newtonian fluids past
arbitrary external surfaces. Following the discussion by Schowalter and Acrivos, the similarity equation of momentum boundary layer has been known as
0)('' )())'('')(''(1 =+−ηηηηf f f f n (1)
Eqs.(1) has been used to describe the momentum transfer in power law fluids boundary layer for more than 40 years [2-20]. However, the similarity equation for thermal boundary layer has not been established up to now. This paper investigates the applicability of boundary layer theory for the flow of power law fluids.
A special emphasis is given to the formulation of boundary layer equations, which provide similarity solutions.
Consider a semi-infinite plate aligned with a uniform power law flow of constant speed U at uniform wall temperature. The laminar boundary layer equations expressing conservation of mass, momentum and energy should be written as follows: ∞∂∂∂∂U X V Y
+=0 (2) Y
Y U V X U U XY ∂∂=+τρ∂∂∂∂1 (3) (Y T a Y Y T V X T U ∂∂∂∂=+∂∂∂∂ (4)
where the and axes are taken along and perpendicular to the plate, and V are the velocity
components parallel and normal to the plate, X Y U 1−∂∂=n Y
U γν(γ) is the kinematic viscosity, the thermal diffusivity may be defined as ρ/K =a 1−∂∂=n Y T ω0<n a with and as positive constant. The case
corresponds to a Newtonian fluid and the case is “power law” relation proposed as being descriptive of pseudo-plastic non-Newtonian fluids and n describes the dilatant fluid. The appropriate boundary conditions are:
γω1<1>1=n ∞+∞======U U V
U Y Y Y ,0 ,000 (5) ,0∞+∞====T T T T Y w Y (6)
3. Nonlinear Boundary Value Problem.
The dimensionless variables, the stream function ),( y x ψ, the similarity variable ηand the dimensionless temperature function are introduced as )(ηw [12-14], we arrive at the nonlinear boundary value problems of the form:
0)('' )())'('')(''(1 =+−ηηηηf f f f n (7)
1)(' ,0)0(' ,0)0(===+∞=ηηf f f (8)
0)(')())'(')('(1=+−ηηηηw f N w w Zh n (9)
1)( ,0)0(==+∞=ηηw w (10)
Eqs.(7)-(10) are the similarity equations for both momentum and thermal boundary layer in non-Newtonian fluids. It is clearly that when , Eqs.(7)-(10) reduce to the Falkner-Skan’s equations for Newtonian fluid.
1=n Where the similarity number defined
as Zh N ω
⋅−=∞∞Re )(N T T L U N W n Zh (11) Assuming the solution of Eqs.(7)-(10) possesses a positive second derivative in and ( it is closely related to boundary conditions). Defining the general Crocco variable transformation as:
)(ηf ′′) ,0(∞+0)(=+∞′′f []n
f t
g )( )(η′′=,φ, (12) )()(ηw t =)('ηf t =where is the dimensionless tangential velocity, is the dimensionless shear force, φ is the dimensionless temperature. Substituting (12) into Eqs.(7)-(10) and applying the chain rule yield the following singular nonlinear boundary value problems:
t )(t g )(t 10 , )()(1
<<−=′′−t t tg t g n (13)
0=(1) ,0)0(g g =′ (14)
0)()())'())('((=′′+t g t N t g t zh n φφ (15)
1)1( ,0)0(==φφ (16)
The momentum equation and the energy equation are decoupled since the fluid is incompressible. As the positive solutions of Eqs.(13)-(14) is concerned, Zheng et al.[12-14] discussed some general cases of power law fluid boundary layer equations for . Sufficient conditions for existence, non-uniqueness, uniqueness and analyticity of positive solutions to the problems were established utilizing the perturbation and shooting techniques. It was shown that for special parameters of , Eqs.(13)-(14) have an analytical solution which may be represented by a power series for at t (i.e.,10≤<n )(t g n 0=∑∞
==0)(!)0()(i i i t i g t g ) and converges at . 1=t The nonlinear differential equations (7)-(8)(momentum equation) and (9)-(10)(energy equation) are solved for the dependent variables and as a function of . Clearly, the nonlinear boundary value problems
(7)-(8) are de-coupled and can be discussed firstly. The solutions then may be used immediately for solving the nonlinear boundary value problems (9)-(10).
f w ηUtilizin
g the solutions of momentum equations (8)-(9), the solutions of energy equations (10)-(11) can be solved by a shooting technique. For all the results are qualitatively agree very well wit
h that of the classical Blasius problems for Newtonian Fluids which have been discussed by many authors 1=n [1].
4. Conclusions
The new energy boundary layer model are developed which can be characterized by a power law relationship between shear stress and velocity gradient. A new similarity number are derived by supposing that the heat boundary layer equation existing similarity solution. The solutions may be presented numerically by using the standard Runge-Kutta formulas and a shooting technique and the associated transfer characteristics are discussed in detail.
Acknowledgement: The work is supported by the National Natural Science Foundations of China ( No. 50476083).
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