At one point in a flow of water

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity

Fluid Dynamics, Ch The stress tensor completed: constitutive law and fluid properties

Problem

At one point in a flow of water, with μ=1.00×10−3 Pa s\mu=1.00\times10^{-3}\,\mathrm{Pa\,s} and μb=2.8×10−3 Pa s\mu_b=2.8\times10^{-3}\,\mathrm{Pa\,s}, the velocity gradient ∂ui/∂xj\partial u_i/\partial x_j (row ii, column jj) is ∂ui∂xj=(04000−10000−20)s−1.\frac{\partial u_i}{\partial x_j}=\begin{pmatrix}0&40&0\\0&-10&0\\0&0&-20\end{pmatrix}\mathrm{s^{-1}}. ((a)) Split the gradient into eije_{ij} and Ωij\Omega_{ij}, and find ∇⋅u\nabla\cdot\mathbf u. Is the fluid being compressed or expanded here? ((b)) Find every non-zero component of the viscous stress τij=σij+p δij\tau_{ij}=\sigma_{ij}+p\,\delta_{ij}, and say which part of the velocity gradient has no effect on it. ((c)) Find pˉ−p\bar p-p, and explain its sign. ((d)) Find the dissipation function Φ\Phi from the sum of squares, and check it by forming τijeij\tau_{ij}e_{ij} directly. What fraction of Φ\Phi comes from the bulk viscosity, and what would Stokes's hypothesis μb=0\mu_b=0 have predicted?

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