Capillary Surface - The Stress Balance Equation - The Stress Tensor

The Stress Tensor

The stress tensor is related to velocity and pressure. Its actual form will depend on the specific fluid being dealt with, for the common case of incompressible Newtonian flow the stress tensor is given by

sigma_{ij} &=
end{pmatrix} +
mu begin{pmatrix}
2 frac{partial u}{partial x} & frac{partial u}{partial y} + frac{partial v}{partial x} & frac{partial u}{partial z} + frac{partial w}{partial x} \
frac{partial v}{partial x} + frac{partial u}{partial y} & 2 frac{partial v}{partial y} & frac{partial v}{partial z} + frac{partial w}{partial y} \
frac{partial w}{partial x} + frac{partial u}{partial z} & frac{partial w}{partial y} + frac{partial v}{partial z} & 2frac{partial w}{partial z}
end{pmatrix} \
&= -p I + mu (nabla mathbf{v} + (nabla mathbf{v})^T)

where is the pressure in the fluid, is the velocity, and is the viscosity.

Read more about this topic:  Capillary Surface, The Stress Balance Equation

Other articles related to "the stress tensor, tensor":

Derivation Of The Navier–Stokes Equations - The Stress Tensor
... elements, so that a quantity called the stress tensor appears naturally in the Cauchy momentum equation ... Since the divergence of this tensor is taken, it is customary to write out the equation fully simplified, so that the original appearance of the stress tensor is lost ... However, the stress tensor still has some important uses, especially in formulating boundary conditions at fluid interfaces ...
Buoyancy - Forces and Equilibrium
... by some outer field on the fluid, and σ is the stress tensor ... In this case the stress tensor is proportional to the identity tensor Here is the Kronecker delta ... The force exerted on the body can be calculated by integrating the stress tensor over the surface of the body which is in contact with the fluid The surface integral can be transformed into a ...

Famous quotes containing the word stress:

    Like all weak men he laid an exaggerated stress on not changing one’s mind.
    W. Somerset Maugham (1874–1966)