Strain rate, velocity gradient

[v_pino]

What is the difference between strain rate and velocity gradient of a Newtonian fluid?

 

[minger]

For a Newtonian fluid, they are the same, which is proportional to the the shear stress. They are proportional by the second viscosity coefficient. For example, the normal shear stress in the x-direction is given by:

$$\tau_{xx} = \lambda (\vec{\nabla}\cdot\vec{V})+2\mu\frac{\partial u}{\partial x}$$
You can see the velocity gradient term in there, with the leading coefficient being the proportional part. Do note that $$\lambda$$ is hard to measure, and this is where Stoke's Hypothesis (see number fudge so the equations can be solved) comes into play, where we just assume that:
$$\lambda = -\frac{2}{3}\mu$$

 

[oswaler]

Strain rate and velocity gradient are both important parameters in the study of fluid mechanics. However, they represent different aspects of a fluid's behavior.

Strain rate refers to the rate at which a fluid is deformed or stretched. It is a measure of the change in the fluid's shape or size over a given period of time. Strain rate is typically expressed in units of per second, such as 1/s or s^-1.

On the other hand, velocity gradient refers to the change in velocity of a fluid over a given distance. It is a measure of the fluid's shear or deformation rate. Velocity gradient is typically expressed in units of per meter, such as 1/m or m^-1.

The main difference between strain rate and velocity gradient is that they represent different types of deformation. Strain rate measures the overall deformation of a fluid, while velocity gradient measures the local deformation of a fluid. In other words, strain rate gives a global view of the fluid's deformation, while velocity gradient gives a more detailed view of how the fluid is deforming at a specific point.

For a Newtonian fluid, the strain rate and velocity gradient are directly proportional to each other. This means that as the strain rate increases, the velocity gradient also increases. This relationship is described by the Newtonian fluid equation: stress = viscosity x velocity gradient.

In summary, strain rate and velocity gradient are both important parameters in the study of fluid mechanics, but they represent different aspects of a fluid's behavior. While strain rate measures the overall deformation, velocity gradient measures the local deformation of a fluid.