# Shear rate

The shear rate (older, non-DIN-compliant terms: shear gradient, shear rate, velocity gradient, symbol (gamma point); previously: D, dimension T −1 ) is a term from kinematics that describes the spatial change in the flow velocity in liquids . Since there are frictional forces in real liquids, a shear of a fluid means a transmission of force just like a solid body . In rheology , the shear rate is used as a measure of the mechanical load to which a sample is subjected during a rheological measurement. ${\ displaystyle {\ dot {\ gamma}}}$ ## Measurement of viscosity

The shear rate is in the rheology of the definition of the viscosity used η that the proportionality factor between shear stress and shear rate: . A stratified flow between two plates is considered as shown in the picture. The shear rate is then calculated from the ratio between the speed difference between two adjacent liquid layers and their distance : ${\ displaystyle \ tau}$ ${\ displaystyle \ tau = \ eta {\ dot {\ gamma}}}$ ${\ displaystyle \ mathrm {d} u}$ ${\ displaystyle \ mathrm {d} y}$ ${\ displaystyle {\ dot {\ gamma}} = {\ frac {\ mathrm {d} u} {\ mathrm {d} y}} \ ,.}$ With a small distance between the plates, a linear velocity distribution over the height can be assumed, as in the figure, and the shear rate is the speed of the upper plate divided by the distance between the plates. At the limit crossing the derivative of the speed u arises according to the coordinate y. ${\ displaystyle \ mathrm {d} y \ to 0}$ In more complex flows, shear can also be caused by changing the vertical velocity component v in the horizontal x-direction. Because both directions are equal, the generalization is obvious

${\ displaystyle {\ dot {\ gamma}} = {\ frac {\ mathrm {d} u} {\ mathrm {d} y}} + {\ frac {\ mathrm {d} v} {\ mathrm {d} x}}}$ on. In the case of the stratified flow, the second term with the velocity v perpendicular to the plates can be neglected. In axially symmetrical flows, a cylinder or spherical coordinate system is advantageously used as a basis, in which the radial velocity disappears at the walls.

## general definition

Expressed mathematically, the shear rate is determined from the components of the rate gradient , which is a second order tensor :

${\ displaystyle \ operatorname {grad} {\ vec {v}} = {\ begin {pmatrix} {\ frac {\ partial v_ {x}} {\ partial x}} & {\ frac {\ partial v_ {x} } {\ partial y}} & {\ frac {\ partial v_ {x}} {\ partial z}} \\ {\ frac {\ partial v_ {y}} {\ partial x}} & {\ frac {\ partial v_ {y}} {\ partial y}} & {\ frac {\ partial v_ {y}} {\ partial z}} \\ {\ frac {\ partial v_ {z}} {\ partial x}} & {\ frac {\ partial v_ {z}} {\ partial y}} & {\ frac {\ partial v_ {z}} {\ partial z}} \ end {pmatrix}} \ ,.}$ The speed components relate to a Cartesian coordinate system with the coordinates x, y and z. The shear rate is calculated with the symmetrical part of the gradient, the strain rate tensor${\ displaystyle v_ {x, y, z}}$ ${\ displaystyle \ mathbf {D}: = {\ frac {1} {2}} [\ operatorname {grad} {\ vec {v}} + (\ operatorname {grad} {\ vec {v}}) ^ { \ top}] = {\ frac {1} {2}} {\ begin {pmatrix} 2 {\ frac {\ partial v_ {x}} {\ partial x}} & {\ frac {\ partial v_ {x} } {\ partial y}} + {\ frac {\ partial v_ {y}} {\ partial x}} & {\ frac {\ partial v_ {x}} {\ partial z}} + {\ frac {\ partial v_ {z}} {\ partial x}} \\ & 2 {\ frac {\ partial v_ {y}} {\ partial y}} & {\ frac {\ partial v_ {y}} {\ partial z}} + {\ frac {\ partial v_ {z}} {\ partial y}} \\ {\ text {sym.}} && 2 {\ frac {\ partial v_ {z}} {\ partial z}} \ end {pmatrix} } \ ,,}$ The superscript stands for the transposed matrix . In continuum mechanics , the small d is also used as a designation because this tensor is formulated in Euler's approach . The shear rate in a plane, which is spanned by two mutually perpendicular vectors of length one, then results from the product ${\ displaystyle \ top}$ ${\ displaystyle {\ hat {g}} _ {1,2}}$ ${\ displaystyle {\ dot {\ gamma}} = 2 {\ hat {g}} _ {2} \ cdot \ mathbf {D} \ cdot {\ hat {g}} _ {1} \ ,.}$ In the case of the stratified flow above, the vectors are parallel to the x or y direction and this results when the flow takes place in the x direction and in the xy plane as shown in the picture ${\ displaystyle {\ hat {g}} _ {1,2}}$ ${\ displaystyle {\ dot {\ gamma}} = 2 {\ begin {pmatrix} 0 \\ 1 \ end {pmatrix}} \ cdot {\ frac {1} {2}} {\ begin {pmatrix} 2 {\ frac {\ partial v_ {x}} {\ partial x}} & {\ frac {\ partial v_ {x}} {\ partial y}} + {\ frac {\ partial v_ {y}} {\ partial x} } \\ {\ frac {\ partial v_ {x}} {\ partial y}} + {\ frac {\ partial v_ {y}} {\ partial x}} & 2 {\ frac {\ partial v_ {y}} {\ partial y}} \ end {pmatrix}} \ cdot {\ begin {pmatrix} 1 \\ 0 \ end {pmatrix}} = {\ begin {pmatrix} 0 \\ 1 \ end {pmatrix}} \ cdot { \ begin {pmatrix} 2 {\ frac {\ partial v_ {x}} {\ partial x}} \\ {\ frac {\ partial v_ {x}} {\ partial y}} + {\ frac {\ partial v_ {y}} {\ partial x}} \ end {pmatrix}} = {\ frac {\ partial v_ {x}} {\ partial y}} + {\ frac {\ partial v_ {y}} {\ partial x }} = {\ frac {\ partial v_ {x}} {\ partial y}} \ ,,}$ because the term with the velocity perpendicular to the plates can be neglected as I said. ${\ displaystyle v_ {y}}$ 