parenleftbigg∂u∂x+∂u∂xparenrightbigg= 2μux,(3.38a)τyy=μparenleftbigg∂v∂y+∂v∂yparenrightbigg= 2μvy,(3.38b)τzz=μparenleftbigg∂w∂z+∂w∂zparenrightbigg= 2μwz.(3.38c)The basic physical idea is stretching (or compressing) the fluid element. In thexdirection the onlynon-rotational contribution to this comes fromx-component gradients of the velocity componentu.In the presence of this,uis, in general, changing withxthroughout the fluid element, so∂u/∂xisdifferent on the twoxfaces; thus, these two contributions must be added. But as the element size isshrunk to zero these approach equality, leading to the result in Eq. (3.38a). Analogous argumentshold for the other two coordinate directions as indicated in Eqs. (3.38b) and (3.38c). We remarkthat this same argument leads to account of shear stresses from opposite faces of the fluid element.In particular, recall that we treated only one face in arriving at Eqs. (3.37), but in this process wehad formally averaged,e.g.,uyandvxin arriving at the formula forτyx. The factor 1/2 did notappear in the final shear stress formulas because of the preceding arguments.Construction of the MatrixTWe now again recall that the matrixTmust carry the same information as does the surfaceforce vectorFS. We have not previously given much detail of this except to note thatFScould beresolved into normal and tangential forces. But we can see from Eq. (3.33) that this vector must alsobe viewed in terms of its separate components each of which is acting as the force in a momentumbalance associated with the corresponding individual velocity components.Furthermore, we seefrom Eq. (3.35) that these force contributions come from taking the divergence ofT, implyingthat the first column ofTshould provide the force to balance time-rate of change of momentumassociated with theucomponent of velocity (i.e., thex-direction velocity),etc. Thus, if we recallFig. 3.9 we see that the information in the first column ofTshould consist ofτxx,τyx,τzxand thecontribution of pressurepacting on thexface, all of which lead to forces (per unit area) acting inthexdirection.With regard to the last of these we note that pressure is a scalar quantity, and as such it hasno direction dependence. On the other hand, changes in pressure can be different in the differentdirections, and we will soon see that it is, in fact, only changes in pressure that enter the equationsof motion. Moreover, we view the associated forces as compressive, as indicated in Fig. 3.9;viz.,they act in directions opposite the respective outward unit normals to the fluid element. Hence, allsuch terms will have minus signs in the formulas to follow.The arguments given above for the first column of the matrixTapply without change to theremaining two columns, so we arrive at the following:
78CHAPTER 3. THE EQUATIONS OF FLUID MOTIONT=−p+τxxτxyτxzτyx−p+τyyτzyτzxτzy−p+τzz=−p100010001+τxxτxyτxzτxyτyyτyzτxzτyzτzz(3.39)=−pI+τ.
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