Notes on Computational Fluid Dynamics: General Principles

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2.6 Forces at a surface

The next law to deﬁne is that of conservation of momentum, i.e. Newton’s second law of motion ( $\text{[math]}$ ) for ﬂuids. It involves forces within the ﬂuid so requires a description of forces at a surface $\text{[math]}$ bounding a volume $\text{[math]}$ .

$PICT\relax \special {t4ht=$

The force $\text{[math]}$ is in the direction of the traction vector with magnitude of (traction $\text{[math]}$ surface area) — compare with Eq. (2.3 )

df = tdS \relax \special {t4ht=

(2.16)

The tractions at a surface bounding some volume of ﬂuid (or any continuum, e.g. solid) depend on the orientation of the surface. Consequently, we cannot deﬁne forces within a ﬂuid as traction vectors at points within the ﬂuid.

$PICT\relax \special {t4ht=$

Instead, we require 3 traction vectors $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ , deﬁned in planes perpendicular to one another, i.e. $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ . This results in the stress tensor $\text{[math]}$ with 9 components, consisting of 3 traction vectors, each containing 3 components.

The traction can be calculated at a surface with any orientation by taking the inner product of the unit normal vector $\text{[math]}$ and the stress tensor $\text{[math]}$ such that

df = t dS = (n ☐☐☐)dS = dS ☐☐☐: \relax \special {t4ht=

(2.17)

Tensors

We have deﬁned the stress tensor ³, an entity with 9 component values, corresponding to our $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ axes (or more speciﬁcally base vectors of unit length aligned with our $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ axes).

In fact, the term “tensor” describes any entity with multiple component values corresponding to the dimensions of space — here 3. A tensor has rank $\text{[math]}$ , such that the number of component values for 3D space = $\text{[math]}$ .

In this book we use the term “tensor” to mean “tensor of rank 2” unless otherwise noted. A vector is a tensor of rank 1 and a scalar is rank 0.

The inner product of a vector $\text{[math]}$ and tensor $\text{[math]}$ produces a vector whose 3 components are evaluated as follows:

0 1 axTxx + ayTyx + azTzx a T = B@ axTxy + ayTyy + azTzy CA axTxz + ayTyz + azTzz \relax \special {t4ht=

(2.18)

This inner product tensor is commutative only if $\text{[math]}$ is symmetric since $\text{[math]}$ — see Eq. (2.24 ) for the transpose $\text{[math]}$ .

³More precisely the “Cauchy stress tensor”, introduced by Augustin-Louis Cauchy in De la pression ou tension dans un corps solide, 1827.

Notes on CFD: General Principles - 2.6 Forces at a surface