The fluid exerts the tangential traction

- τ e_{x}

on the upper boundary and

+ τ e_{x}

on the lower boundary. Equivalently, the boundaries exert

+ τ e_{x}

and

- τ e_{x}

on the fluids, respectively.

ii

Solution

The viscous dissipation per unit boundary area is

D = \int_{- h}^{σ} μ_{+} (u_{+}^{'})^{2}, d y + \int_{σ}^{h} μ_{-} (u_{-}^{'})^{2}, d y = τ^{2} (\frac{d _{+}}{μ _{+}} + \frac{d _{-}}{μ _{-}}) .

The rate at which the upper and lower boundaries do work on the fluid, per unit area, is

W = τ U_{-} - τ U_{+} = τ (U_{-} - U_{+}) .

τ

gives

W = τ^{2} (\frac{d _{+}}{μ _{+}} + \frac{d _{-}}{μ _{-}}) = D,

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