Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 352 / 2 / c

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- Solution c

Solution

Write

g = \overset{γ}{˙}

and

L = \nabla u = 000 g 00 000, τ = τ_{xx} τ_{x y} 0 τ_{x y} τ_{yy} 0 00 τ_{zz} .

The flow and stresses are steady and homogeneous, so the material derivative vanishes. Direct multiplication gives

τ ▽ = - L τ - τ L^{T} = - 2 g τ_{x y} - g τ_{yy} 0 - g τ_{yy} 00 000

and

τ^{2} = τ_{xx}^{2} + τ_{x y}^{2} τ_{x y} (τ_{xx} + τ_{yy}) 0 τ_{x y} (τ_{xx} + τ_{yy}) τ_{x y}^{2} + τ_{yy}^{2} 0 00 τ_{zz}^{2} .

Since

\overset{γ}{˙}_{x y} = \overset{γ}{˙}_{y x} = g

, the four independent component equations are

τ_{xx} - 2 λ g τ_{x y} + \frac{α λ}{η} (τ_{xx}^{2} + τ_{x y}^{2}) = 0,

τ_{x y} - λ g τ_{yy} + \frac{α λ}{η} τ_{x y} (τ_{xx} + τ_{yy}) = η g,

τ_{yy} + \frac{α λ}{η} (τ_{x y}^{2} + τ_{yy}^{2}) = 0, τ_{zz} + \frac{α λ}{η} τ_{zz}^{2} = 0 .

The branch continuous from equilibrium has

τ_{zz} = 0

.

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