Question

Express the shear stress tensor(τ) of a three-dimensional fluid flow element in terms of the velocity vector(v).

(a) \(\tau=\mu\left\{(\nabla \vec{v})^T\right\}+\lambda(\nabla.\vec{v})I\)

(b) \(\tau=\mu\left\{(\nabla \vec{v})\right\}+\lambda(\nabla.\vec{v})I\)

(c) \(\tau=\mu\left\{(\nabla \vec{v})^T+(\nabla.\vec{v})^T\right\}\)

(d) \(\tau=\mu\left\{(\nabla \vec{v})^T+(\nabla.\vec{v})^T\right\}+\lambda(\nabla.\vec{v})I\)

This question was addressed to me during an online interview.

My question is based upon Governing Equations in division Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

The correct answer is (d) \(\tau=\mu\left\{(\nabla \vec{v})^T+(\nabla.\vec{v})^T\right\}+\lambda(\nabla.\vec{v})I\)

To explain: The shear stress tensor of a fluid element can be given by \(\tau=\mu\left\{(\nabla \vec{v})^T+(\nabla.\vec{v})^T\right\}+\lambda(\nabla.\vec{v})I\). This is not applicable for practical cases. However, common fluids like air and water are assumed to be Newtonian for using this relationship.