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Shearing stress relationships

Key Concepts: In a viscous fluid shearing stresses will act on each face of a differential element.

The shearing stress relationships may be expressed in differential form. Each component of

shearing stress depends on the dynamic viscosity, μ, of the fluid.

In a Nut Shell: For a viscous fluid each face of a fluid element will sustain a shearing stress. The table below lists the shearing stress on each face.

Shearing Stress Relationships

τ_xy = τ_yx = μ(∂u/∂y + ∂v/∂x)

τ_yz = τ_zy = μ(∂u/∂z + ∂w/∂y)

τ_zx = τ_xz = μ(∂w/∂x + ∂u/∂z)

where τ_xy = τ_yx are the shearing stresses on the x and y-faces of the element

τ_yz = τ_zy are the shearing stresses on the y and z-faces of the element

τ_zx = τ_xz are the shearing stresses on the z and x-faces of the element

u, v, w are the components of fluid velocity in the x, y, and z directions

μ is the dynamic viscosity of the fluid

(∂/∂x , ∂/∂y , and ∂/∂z are the partial derivatives in the x, y, and z - directions

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