Today, we will do better. Consider two fluid particles, indicated below in red and blue. At time 1, they are separated by a small displacement $\\Delta \\vec{x}^1$. After some short time, the fluid velocity carries the particles to their new locations at time 2, where they are separated by a new displacement $\\Delta \\vec{x}^2$:<\/p>\n\n\n
![\"\"](\"https:\/\/oceanofmath.blog\/wp-content\/uploads\/2026\/05\/displacement.jpg\")
<\/figure>\n<\/div>\n\n\nThe two displacements will usually differ because the fluid velocity at $\\vec{x}_1^1$ will be slightly different than the velocity at $\\vec{x}_2^1$, so the two particles will move along different paths at different speeds. On the other hand, if our particles were part of a rigid object that was being moved, the distance between the two particles would stay constant throughout the motion. So the rate of change of the mutual displacement tells us something about the non-rigidity of the fluid.<\/p>\n\n\n\n
The quantity of interest is the rate of change of the separation $\\vec{r} = \\Delta \\vec{x}$, per unit length of that separation. Let:<\/p>\n\n\n\n
$$\\displaystyle M = \\frac{\\frac{d}{dt} |\\vec{r}|}{|\\vec{r}|} $$<\/p>\n\n\n\n
This quantity is clearly related to strain, i.e. deformation relative to initial size, and is also clearly a rate. So we are justified in expecting a “strain rate tensor” to arise from it. Evaluating some terms:<\/p>\n\n\n\n
$$\\displaystyle \\begin{aligned} M = \\frac{\\frac{d}{dt} |\\vec{r}|}{|\\vec{r}|} &= \\frac{\\frac{d}{dt} \\sqrt{\\vec{r}\\cdot\\vec{r}}}{|\\vec{r}|} \\\\ &= \\frac12 \\frac{\\frac{d}{dt} (\\vec{r}\\cdot\\vec{r}) }{\\vec{r}\\cdot\\vec{r}} \\end{aligned} $$<\/p>\n\n\n\n
Note the $\\frac12$ factor that has emerged naturally. Now for the derivative in the numerator, keep in mind that $\\vec{r} = \\Delta \\vec{x}$ is a difference between two positions, so $\\frac{d\\vec{r}}{dt} = \\Delta \\vec{v}$ is a difference of velocities.<\/p>\n\n\n\n
$$\\displaystyle M = \\frac12 \\frac{(\\Delta v)^T \\vec{r} + \\vec{r}^T (\\Delta\\vec{v})}{\\vec{r}^T \\vec{r}} $$<\/p>\n\n\n\n
I have switched to writing the dot products as products of row and column vectors, because the transposes are about to become important. The velocity difference can be expressed in terms of the deformation tensor $D_{ij}$, aka the velocity Jacobian:<\/p>\n\n\n\n
$$\\displaystyle D_{ij} = \\frac{\\partial u_i}{\\partial x_j} $$<\/p>\n\n\n\n
$$\\displaystyle \\Delta v_i = D_{ij} r_j + \\mathcal{O}(|\\vec{r}|^2) $$<\/p>\n\n\n\n
Substituting this relation into our expression for $M$ and leaving out the higher-order terms for clarity:<\/p>\n\n\n\n
$$\\displaystyle \\begin{aligned} M &= \\frac{1}{2\\vec{r}^T\\vec{r}} \\left( (D\\vec{r})^T\\vec{r} + \\vec{r}^T (D\\vec{r}) \\right) \\\\ &= \\frac{1}{\\vec{r}^T\\vec{r}}\\vec{r}^T \\left[ \\frac12\\left( D + D^T \\right)\\right] \\vec{r} \\\\ &= \\hat{r}^T S \\hat{r} \\end{aligned}$$<\/p>\n\n\n\n
where we have defined $\\hat{r} = \\vec{r}\/|\\vec{r}|$ the unit vector in the direction of the separation $\\vec{r}$. The higher-order terms vanish as $\\vec{r} \\to 0$.<\/p>\n\n\n\n
So the rate of change of distance, per unit distance between two particles, (i.e. the strain rate) is related to the symmetric part of the velocity Jacobian. As a side note, this means that the antisymmetric part describes the part of the fluid motion that is effectively rigid.<\/p>\n","protected":false},"excerpt":{"rendered":"
In fluid dynamics, the “strain rate tensor” is declared to be: $$\\displaystyle S_{ij} = \\frac12\\left( \\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i} \\right) $$ This quantity is usually derived using a diagram of a square fluid element in 2D and considering the rates of increase of two angles as the square deforms into a parallelogram. … Read More “The Strain Rate Tensor”<\/span> »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2,18],"tags":[],"class_list":["post-531","post","type-post","status-publish","format-standard","hentry","category-continental-shelf","category-fluid-dynamics"],"_links":{"self":[{"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=\/wp\/v2\/posts\/531","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=531"}],"version-history":[{"count":5,"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=\/wp\/v2\/posts\/531\/revisions"}],"predecessor-version":[{"id":545,"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=\/wp\/v2\/posts\/531\/revisions\/545"}],"wp:attachment":[{"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=531"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=531"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/oceanofmath.blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=531"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}