031013

031013 - Chapter 3 Laminar ows 3.1 Assumptions Flow...

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Chapter 3 Laminar flows 3.1 Assumptions Flow equations discussed in Chapter 2 provide analytical solutions only in some special cases. In this chapter we shall consider the equations for incompressible flow: (2.4), (2.22) and (2.74), assuming that all the coefficients are constant: u i i 0 (3.1) ˙ u i u k u i k ν u i kk ρ 1 p i g i (3.2) ρ c p ˙ T u i T i kT ii u j i τ ij (3.3) Definition 3.1.1 Laminar flow Let’s make an assumption of laminar flow which states that the time scale of changes in the flow can not be lower than the time-scale of the motion of the boundary or any external sources. In other words, if there is any repeatability in the motion of the boundary or in the external forces then the frequencies associ- ated with either factors can not be lower than the frequencies of the flow motion. It means that neither the boundaries not external forces can induce any addi- tional frequencies in the flow. In the limit case of non-moving boundaries and non-changing forces the flow should not depend on time, which means that all the dependent variables should become functions of spatial coordinates only. The conditions of laminar flow defined by (3.1.1) are realized when the con- tribution of the non-linear term in the momentum equation (3.2) is small, or when 51
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52 CHAPTER3.LAMINARFLOWS Figure 3.1: Flow between parallel plates: the lower plate is at rest, the upper plate is moving with velocity U . the contribution of the viscous term ν u i kk dominates. This is usually the case when non-dimensional Reynolds number (2.109): (3.4) R e LU ν is small. In practical situations the ”smallness” of R e corresponds to R e 100 . 3.2
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031013 - Chapter 3 Laminar ows 3.1 Assumptions Flow...

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