1. The Cauchy stress in an incompressible fluid is given by T= -pI + 2µD, where u is a material constant. (a). For the homogeneous motion X1 = X1 + u(X2, t), x2 = X2, X3 = X3, derive the Cauchy stress. (b). Express the balance of linear momentum equations in terms of p and u. (d). Consider the special case where u(X2, t) momentum equations are satisfied at the absence of body force if and k(t) is linear in t. K(t)X2. Show that the linear = const

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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1. The Cauchy stress in an incompressible fluid is given by
T = -pI + 2µD,
where
is
a material constant.
(a). For the homogeneous motion
= X1+ u(X2,t),
x2 = X2,
X3 = X3,
derive the Cauchy stress.
(b). Express the balance of linear momentum equations in terms of p and u.
(d). Consider the special case where u(X2, t) = k(t)X2. Show that the linear
momentum equations are satisfied at the absence of body force if p = const
and k(t) is linear in t.
Transcribed Image Text:1. The Cauchy stress in an incompressible fluid is given by T = -pI + 2µD, where is a material constant. (a). For the homogeneous motion = X1+ u(X2,t), x2 = X2, X3 = X3, derive the Cauchy stress. (b). Express the balance of linear momentum equations in terms of p and u. (d). Consider the special case where u(X2, t) = k(t)X2. Show that the linear momentum equations are satisfied at the absence of body force if p = const and k(t) is linear in t.
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