hydrostatic stress

In the particular case of an incompressible fluid,

the thermodynamic pressure coincides with the mechanical pressure (i.e. the opposite of the hydrostatic stress):

$p\; =\; -\; \backslash sigma\_h\; =\; -\; \backslash frac\; 1\; 3\; \backslash operatorname\{tr\}\; (\backslash boldsymbol\; \backslash sigma)$

In the general case of a compressible fluid, the thermodynamic pressure $p$ is no more proportional to the isotropic stress term (the mechanical pressure), since there is an additional term dependent on the trace of the strain rate tensor:

$p\; =\; -\; \backslash frac\; 1\; 3\; \backslash operatorname\{tr\}\; (\backslash boldsymbol\; \backslash sigma)\; +\; \backslash zeta\; \backslash operatorname\{tr\}\; (\backslash boldsymbol\; \backslash epsilon)$

where the coefficient $\backslash zeta$ is the bulk viscosity. The trace of the strain rate tensor corresponds to the flow compression (the divergence of the flow velocity):

$\backslash operatorname\{tr\}\; (\backslash boldsymbol\; \backslash epsilon)\; =\; \backslash operatorname\{tr\}\; \backslash left(\backslash frac\; 1\; 2\; (\backslash nabla\; \backslash mathbf\; u\; +\; (\backslash nabla\; \backslash mathbf\; u)^T)\; \backslash right)\; =\; \backslash nabla\backslash cdot\backslash mathbf\{u\}$

So the expression for the thermodynamic pressure is usually expressed as:

$p\; =\; -\; \backslash sigma\_h\; +\; \backslash zeta\; \backslash nabla\backslash cdot\backslash mathbf\{u\}\; =\; \backslash bar\; p\; +\; \backslash zeta\; \backslash nabla\backslash cdot\backslash mathbf\{u\}$

where the mechanical pressure has been denoted with $\backslash bar\; p$.

In some cases, the second viscosity $\backslash zeta$ can be assumed to be constant in which case, the effect of the volume viscosity $\backslash zeta$ is that the mechanical pressure is not equivalent to the thermodynamic pressureLandau & Lifshitz (1987) pp. 44–45, 196 as stated above.

$$\backslash bar\{p\}\; \backslash equiv\; p\; -\; \backslash zeta\; \backslash ,\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{u\}\; ,$$

However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves,White (2006) p. 67. where second viscosity coefficient becomes important) by explicitly assuming $\backslash zeta\; =\; 0$. The assumption of setting $\backslash zeta\; =\; 0$ is called as the Stokes hypothesis.Stokes, G. G. (2007). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory;Vincenti, W. G., Kruger Jr., C. H. (1975). Introduction to physical gas dynamic. Introduction to physical gas dynamics/Huntington. for other gases and liquids, Stokes hypothesis is generally incorrect.

Its magnitude in a fluid, $\backslash sigma\_h$, can be given by Stevin's Law:

:$\backslash sigma\_h\; =\; \backslash displaystyle\backslash sum\_\{i=1\}^n\; \backslash rho\_i\; g\; h\_i$

where

• {{mvar|i}} is an index denoting each distinct layer of material above the point of interest;
• $\backslash rho\_i$ is the density of each layer;
• $g$ is the gravitational acceleration (assumed constant here; this can be substituted with any acceleration that is important in defining weight);
• $h\_i$ is the height (or thickness) of each given layer of material.

For example, the magnitude of the hydrostatic stress felt at a point under ten meters of fresh water would be

:$\backslash sigma\_h\; =\; \backslash rho\_w\; g\; h\_w\; =1000\; \backslash ,\backslash text\{kg\; m\}^\{-3\}\; \backslash cdot\; 9.8\; \backslash ,\backslash text\{m\; s\}^\{-2\}\; \backslash cdot\; 10\; \backslash ,\backslash text\{m\}\; =9.8\; \backslash cdot\; \{10^4\}\; \backslash text\{\; kg\; m\}^\{-1\}\; \backslash text\{s\}^\{-2\}\; =9.8\; \backslash cdot\; 10^4\; \backslash text\{\; N\; m\}^\{-2\}$

where the index {{mvar|w}} indicates "water".

Because the hydrostatic stress is isotropic, it acts equally in all directions. In tensor form, the hydrostatic stress is equal to

:$\backslash sigma\_h\; \backslash cdot\; I\_3\; =$

\sigma_h \left[ \begin{array}{ccc}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1 \end{array} \right] =

\left[ \begin{array}{ccc}

\sigma_h & 0 & 0 \\

0 & \sigma_h & 0 \\

0 & 0 & \sigma_h \end{array} \right]

where $I\_3$ is the 3-by-3 identity matrix.

Hydrostatic compressive stress is used for the determination of the bulk modulus for materials.

• Stress tensor
• Volumetric strain
• Deviatoric stress tensor
• Flow velocity
• Pressure
• Bulk viscosity
• Isotropy

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Category:Continuum mechanics

Category:Orientation (geometry)