parenleftbigg u x u x parenrightbigg 2 μu x 338a τyy μ parenleftbigg v y v y

Parenleftbigg u x u x parenrightbigg 2 μu x 338a

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parenleftbigg ∂u ∂x + ∂u ∂x parenrightbigg = 2 μu x , (3.38a) τ yy = μ parenleftbigg ∂v ∂y + ∂v ∂y parenrightbigg = 2 μv y , (3.38b) τ zz = μ parenleftbigg ∂w ∂z + ∂w ∂z parenrightbigg = 2 μw z . (3.38c) The basic physical idea is stretching (or compressing) the fluid element. In the x direction the only non-rotational contribution to this comes from x -component gradients of the velocity component u . In the presence of this, u is, in general, changing with x throughout the fluid element, so ∂u/∂x is different on the two x faces; thus, these two contributions must be added. But as the element size is shrunk to zero these approach equality, leading to the result in Eq. (3.38a). Analogous arguments hold for the other two coordinate directions as indicated in Eqs. (3.38b) and (3.38c). We remark that 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 we had formally averaged, e.g. , u y and v x in arriving at the formula for τ yx . The factor 1 / 2 did not appear in the final shear stress formulas because of the preceding arguments. Construction of the Matrix T We now again recall that the matrix T must carry the same information as does the surface force vector F S . We have not previously given much detail of this except to note that F S could be resolved into normal and tangential forces. But we can see from Eq. (3.33) that this vector must also be viewed in terms of its separate components each of which is acting as the force in a momentum balance associated with the corresponding individual velocity components. Furthermore, we see from Eq. (3.35) that these force contributions come from taking the divergence of T , implying that the first column of T should provide the force to balance time-rate of change of momentum associated with the u component of velocity ( i.e. , the x -direction velocity), etc . Thus, if we recall Fig. 3.9 we see that the information in the first column of T should consist of τ xx , τ yx , τ zx and the contribution of pressure p acting on the x face, all of which lead to forces (per unit area) acting in the x direction. With regard to the last of these we note that pressure is a scalar quantity, and as such it has no direction dependence. On the other hand, changes in pressure can be different in the different directions, and we will soon see that it is, in fact, only changes in pressure that enter the equations of 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, all such terms will have minus signs in the formulas to follow. The arguments given above for the first column of the matrix T apply without change to the remaining two columns, so we arrive at the following:
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78 CHAPTER 3. THE EQUATIONS OF FLUID MOTION T = p + τ xx τ xy τ xz τ yx p + τ yy τ zy τ zx τ zy p + τ zz = p 1 0 0 0 1 0 0 0 1 + τ xx τ xy τ xz τ xy τ yy τ yz τ xz τ yz τ zz (3.39) = p I + τ .
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