Question

Prove that for a fluid at rest the stress tensor \(\sigma_{i j}\) is isotropic \(; \sigma_{i j}=\delta_{i j} \sigma_{k k} / 3\).

Short answer

For a fluid at rest, the stress tensor is isotropic: \\( \\sigma_{ij} = \\delta_{ij} \\sigma_{kk} / 3\\).

Step-by-step solution

Understanding Isotropic Stress Tensor

In an isotropic stress tensor, the stress is the same in all directions. For a fluid at rest, the stress tensor should be isotropic, implying the normal stresses are equal and shear stresses are zero.

Express Stress Tensor Components

The stress tensor \( \) is given by \( \sigma_{ij} = \begin{pmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \ \ \ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \ \ \ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{pmatrix}\). \ In fluids at rest, all off-diagonal terms like \( \sigma_{xy}, \sigma_{yz}, \sigma_{zx} \ \) are zero.

Identify Normal Stress for Fluids

When a fluid is at rest, the only stresses acting are the normal stresses. Since the stress tensor is isotropic for such a fluid, \( \sigma_{xx} = \sigma_{yy} = \sigma_{zz}\).

Express Isotropic Condition Using Kronecker Delta

The isotropic condition is expressed mathematically as \( \sigma_{ij} = \delta_{ij} \frac{1}{3} \sigma_{kk}\). \ Here, \( \delta_{ij} \ \) is the Kronecker delta, equal to 1 when \( \i = j\), and 0 otherwise.

Summarize with Trace of Stress Tensor

The trace of the stress tensor, \( \sigma_{kk}\), is the sum of the normal stresses, \( \sigma_{xx} + \sigma_{yy} + \sigma_{zz}. \ \) For isotropy, each diagonal component of \( \sigma_{ij}\) can be expressed as \( \sigma_{ii} = \frac{1}{3} \sigma_{kk}\).

Key concepts

Isotropic Conditions

Isotropic conditions are present when a material or fluid has physical properties that are identical in all directions. This uniformity ensures that no matter how you take a measurement in the material, the result is consistent. When we refer to a stress tensor as isotropic in the context of a fluid at rest, it means that the stress is evenly distributed across all planes.

• This indicates there are no shear stresses, implying the fluid isn't trying to 'slide' in any direction.
• The normal stresses, which act perpendicularly to the surface, are equal. This prevents deformation due to differential stress in different directions.

This solves the stress tensor problem by leading to a nice uniform state where all diagonal elements hold the same value while off-diagonal elements are zero. This is what makes pressure calculations, particularly for fluids at rest, significantly simplified.

Fluids at Rest

Fluids at rest, also known as hydrostatic conditions, describe a state where the fluid is completely still and no movement occurs within it. This scenario simplifies many fluid dynamics problems. Since a fluid at rest doesn't have any velocity gradients, there are a few unique characteristics to note:

• No shear forces exist within the fluid.
• Only normal stresses, otherwise known as pressure forces, act on a fluid element.

In this state, the stress tensor is described by the normal pressures acting equally in all directions. For example, in a body of water or air, every small volume of the fluid "feels" the same pressure from all directions, due to the isotropic nature of the stress tensor. This fundamentally stems from gravity's control, acting upon the weight of the fluid from above and around.

Kronecker Delta

The Kronecker delta, denoted as \( \delta_{ij} \), is a handy mathematical tool used in various fields, including fluid dynamics. Its value depends on the relationship between two indices (in this case, \( i \) and \( j \)):

• \( \delta_{ij} = 1 \) if \( i = j \) (meaning we are looking at the diagonal elements of a matrix).
• \( \delta_{ij} = 0 \) if \( i eq j \) (meaning we are addressing the off-diagonal elements).

When employed within the context of isotropic stress tensors for fluids at rest, the Kronecker delta ensures that only normal stresses (acting purely as pressures) are represented. This is because non-diagonal elements \( \sigma_{ij} \) for \( i eq j \) should indicate zero shear forces in a resting fluid. Therefore, it becomes critical for expressing isotropic conditions in stress tensors, where the stress \( \sigma_{ij} \) can be expressed as \( \sigma_{ij} = \delta_{ij} \frac{1}{3} \sigma_{kk} \). Here, \( \sigma_{kk} \) represents the trace of the tensor, or the sum of all the normal stresses.