Thermal boundary layer governing equations

I. Governing Equations and Rescaling in the Thermal Boundary-Layer Region... 

This is known as the thermal boundary-layer equation for this problem. Because we have obtained it by taking the limit Pe -> oo in the full thermal energy equation (9-222) with m = 1/3, we recognize that it governs only the first term in an asymptotic expansion similar to (9-202) for this inner region. 

The asymptotic formulation of the previous subsection has led not only to the important result given by (9-230) but also to a very considerable simplification in the structure of the governing equation in the thermal boundary-layer region. As a consequence, it is now possible to obtain an analytic approximation for 0. 

In Chap. 9, we considered the solution of this equation in the limit Re 1, where the velocity distribution could be approximated by means of solutions of the creeping-flow equations. When Pe 1, we found that the fluid was heated (or cooled) significantly in only a very thin thermal boundary layer of 0(Pe l/3) in thickness, immediately adjacent to the surface of a no-slip body, or () Pe l/2) in thickness if the surface were a slip surface with finite interfacial velocities. We may recall that the governing convection di ffusion equation for mass transfer of a single solute in a solvent takes the same form as (111) except that 6 now stands for a dimensionless solute concentration, and the Peclet number is now the product of Reynolds number and Schmidt number,... 

Thus, the governing equation for the leading-order term in an asymptotic expansion for this outer part of the thermal boundary layer is... 

There are at least two approaches that we can take to solve problems in which either the heat flux or the mixed-type condition is specified as a boundary condition. If it is desired to determine the temperature distribution throughout the fluid, then we must return to the governing thermal boundary-layer equation (11-6)- assuming that Re, Pe / - and develop new asymptotic solutions for large and small Pr, with either dT/dr] y 0 or a condition of the mixed type specified at the body surface. The problem for a constant, specified heat flux is relatively straightforward, and such a case is posed as one of the exercises at the end of this chapter. On the other hand, in many circumstances, we might be concerned with determining only the temperature distribution on the body surface [and thus dT /dr] [v from (11 98) for the mixed-type problem], and for this there is an even simpler approach that... 

We have considered the thermal boundary-layer problem in this chapter for an arbitrary 2D body with no-slip boundary conditions for Re 1 and Pr (or Sc) either arbitrarily large or small. If we assume that we have a body of the exact same shape, but the surface of which is a slip surface (e.g., it is an interface, so that the surface tangential velocity is not zero), the form of the correlation between Nusselt number and Pr will change for Pr 1. Solve this problem, i.e., derive the governing boundary-layer equation, and show that it has a similarity solution. What is the resulting form of the heat transfer correlation among Nu, Re, and Pr ... 

When two different fields have similar governing equations and boundary conditions they are called analogous. A majority of momentum and thermal boundary layers, however, are not analogous. For example, a pressure gradient in the momentum boundary layer or an energy generation in the thermal boundary layer, or an incompatibility between momentum and thermal boundary conditions, eliminates this analogy. 

In this section we derive (he governing equations of fluid flow in the boundary layers, lb keep the analysis at a manageable level, wc as.sume the flow to be steady and two-dimensional, and the fluid to be Newtonian with constant properties (density, viscosity, thermal conductivity, etc.). 

In Chapter 5, we learned the foundations of convection. Integrating the governing equations for laminar boundary layers, we obtained expressions for the heat transfer associated with forced convection over a horizontal plate and natural convection about a vertical plate. We also found analytically, as well as by the analogy between heat and momentum, that the thermal and momentum characteristics of laminar flow over a flat plate are related by... 

In this work, a numerical solver based on Finite Volume Method (FVM) is developed to solve the governing equations. The solver has been successfully applied in injection molding filling simulation [2]. Numerical experiments confirm the reliability and efficiency of the solver. Currently the proposed solver can handle tetra, hexa, prism, pyramid, and mixing elements. Prism layer element can also be used for analysis to improve thermal boundary resolution while without extensive refining of mesh. This is valuable in mold cooling analysis that may involve millions of elements. 

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