THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

2.6 Conservation of Momentum 23⎛ ∂τzxτ zx + ⎞⎜ dz⎟dx ⋅ dy⎝ ∂z ⎠σ xx dy dzτ xy dy ⋅dzτ yz dx ⋅ dzτ xz dy ⋅ dz⎛ ∂σ zz ⎞⎜σzz+ dz⎟dx ⋅ dy⎛ ∂τzy⎞⎝ ∂z⎠⎜ τ zy + dz⎟dx ⋅dy⎝ ∂z⎠⎛ ∂τyxτ yx + ⎞⎜ dy⎟dx ⋅dz⎝ ∂y ⎠⎛ ∂σ yyσ yy + ∂ydy ⎞⎜⎟ dx ⋅dz⎝⎠⎛ ∂τ⎜τyz⎝+ yz ⎞dy⎟dx ⋅dz∂y ⎠σ yy dx ⋅ dzτ yx dx ⋅ dzτ zy dx ⋅ dy⎛ ∂τ⎜τxy⎝+ xy ⎞dx⎟dy ⋅dz∂x ⎠⎛ ∂σ xxσ xx + ⎞⎜dx⎟dy ⋅dz⎝ ∂x⎠⎛ ∂τ⎜τd⎝ xz + xz∂x dx ⎞⎟ dz⋅dy⎠zyσ zz dx ⋅ dyzx dx ⋅ dyxFigure 2.5. A fluid element acted on by normal and tangential stresses.(pressure) and tangential (shear) forces. A normal force is denoted by the symbolσ mm , where m denotes the direction of the normal. Because σ mm is dimensionallyexpressed in force per unit area, it must be multiplied by the area normal to it inorder to obtain the force.A shear stress acts along the plane of the surface. It is represented by the symbolτ mn , where the force produced by the shear is normal to coordinate m and parallelfor coordinate n, and either m or n may represent the principal coordinate x, y,orz, provided that m ≠ n.Ifm = n, then τ mm really represents the normal force σ mmand thus is no longer a tangential force. The shear stress is multiplied by the areait is acting on to yield the shear force. For example, a shear τ xy multiplied by area(dx dy) represents the shear force normal to the x-axis and parallel to the y-axis,as shown in Figure 2.5 for a fluid element displayed in Cartesian coordinates.In order to determine the net force F x in the x-direction, all of the forces in thex-direction must be summed. From Figure 2.5 we can write(dF x = ρg x dxdydz + −σ xx + σ xx + ∂σ xx+which simplifies to(τ yx + ∂τ yx∂y dx − τ yxdF x =(ρg x + ∂σ xx∂x)dxdz ++ ∂τ yx∂y)∂x dxdydz (τ zx + ∂τ zx∂z dx − τ zx)dxdz+ ∂τ )zxdxdydz (2.15)∂z