Velocity gradient

Consider a velocity field like a fluid flowing through a pipe, here we can observe that the layer of fluid in contact with the pipe tends to be in rest with respect to the pipe. This is called no slip condition. If the velocity difference of fluid layers at the centre of the pipe and at the sides of the pipe is sufficiently small then, the fluid flow is observed in the form of continuous layers. This type of flow is called laminar flow

The flow velocity difference between adjacent layers can be measured in terms of velocity gradient. It is given by ∆u/∆y

∆u = Flow velocity difference between the layers

∆y = Distance between the layers

Relevance

The study of velocity gradients is useful in analysing path dependent materials and in the subsequent study about stresses and strains, e.g. Plastic deformation of metals

Dimensional formula

We know that the dimensional formula of velocity= M⁰L¹T⁻¹ From the definition of velocity gradient

Velocity Gradient = velocity/distance.

Therefore Dimensional Formula of Velocity Gradient = M⁰L⁰T⁻¹ From the above dimensional formula SI unit of Velocity Gradient is s⁻¹

Relationship between shear stress and the velocity field

Shear stress is directly proportional to the velocity gradient as proposed by Sir Isaac Newton

$\tau =\mu {\frac {\partial u}{\partial y}}$.

Here the constant of proportionality is called dynamic viscosity or for general use coefficient of viscosity.

References

1.http://faculty.poly.edu/~rlevicky/Handout2_6333.pdf
2.http://www.infoplease.com/encyclopedia/science/viscosity-the-velocity-gradient.html
3.http://www.answers.com/topic/velocity-gradient
4.http://www.continuummechanics.org/cm/velocitygradient.html
5.http://www.azformula.com/physics/dimensional-formulae/what-is-dimensional-formula-of-velocity-gradient/
6.https://en.wikibooks.org/wiki/Fluid_Mechanics/Fluid_Properties