Question

The divergence of the stress tensor is _____

(a) Scalar

(b) Vector

(c) 0

(d) 1

The question was asked in final exam.

I'm obligated to ask this question of Governing Equations topic in section Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

The correct answer is (b) Vector

To explain: Stress tensor is a square matrix given by

Τxy = \(\begin{bmatrix}

\tau_{xx} & \tau_{xy} & \tau_{xz} \\

\tau_{yx} & \tau_{yy} & \tau_{yz} \\

\tau_{zx} & \tau_{zy} & \tau_{zz}

\end{bmatrix}\)

The divergence of this will result in a vector

∇. Τ= \(\begin{bmatrix}

\frac{\partial \tau_{xx}}{\partial x}+\frac{\partial \tau_{yx}}{\partial y}+\frac{\partial \tau_{zx}}{\partial z} \\

\frac{\partial \tau_{xy}}{\partial x}+\frac{\partial \tau_{yy}}{\partial y}+\frac{\partial \tau_{zy}}{\partial z} \\  \frac{\partial \tau_{xz}}{\partial x}+\frac{\partial \tau_{yz}}{\partial y}+\frac{\partial \tau_{zz}}{\partial z}

\end{bmatrix}\)