# Simple shear

**Simple**is a special case ofshear deformation of a fluid where only one component ofvelocity vectors has a non-zero value:$V\_x=f(x,y)$

$V\_y=V\_z=0$

And the

gradient of velocity is perpendicular to it:$frac\; \{partial\; V\_x\}\; \{partial\; y\}\; =\; dot\; gamma$,

where $dot\; gamma$ is the

shear rate and:$frac\; \{partial\; V\_x\}\; \{partial\; x\}\; =\; frac\; \{partial\; V\_x\}\; \{partial\; z\}\; =\; 0$

The

deformation gradient tensor $Gamma$ for this deformation has only one non-zero term:$Gamma\; =\; egin\{bmatrix\}\; 0\; \{dot\; gamma\}\; 0\; \backslash \; 0\; 0\; 0\; \backslash \; 0\; 0\; 0\; end\{bmatrix\}$

Simple shear with the rate $dot\; gamma$ is the combination of pure shear strain with the rate of $dot\; gamma\; over\; 2$ and

rotation with the rate of $dot\; gamma\; over\; 2$:$Gamma\; =egin\{matrix\}\; underbrace\; egin\{bmatrix\}\; 0\; \{dot\; gamma\}\; 0\; \backslash \; 0\; 0\; 0\; \backslash \; 0\; 0\; 0\; end\{bmatrix\}\backslash \; mbox\{simple\; shear\}end\{matrix\}\; =egin\{matrix\}\; underbrace\; egin\{bmatrix\}\; 0\; \{dot\; gamma\; over\; 2\}\; 0\; \backslash \; \{dot\; gamma\; over\; 2\}\; 0\; 0\; \backslash \; 0\; 0\; 0\; end\{bmatrix\}\; \backslash \; mbox\{pure\; shear\}\; end\{matrix\}+\; egin\{matrix\}\; underbrace\; egin\{bmatrix\}\; 0\; \{dot\; gamma\; over\; 2\}\; 0\; \backslash \; \{-\; \{\; dot\; gamma\; over\; 2\; 0\; 0\; \backslash \; 0\; 0\; 0\; end\{bmatrix\}\; \backslash \; mbox\{solid\; rotation\}\; end\{matrix\}$

An important example of simple shear is

laminar flow through long channels of constant cross-section (Poiseuille flow ).

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