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Shear stress plays an important role in flow dynamics so for a better understanding of dynamics we should understand shear stress and its role in depth. Let's do that!
Table of Contents :
Introduction Scalars Vectors and Tensors :
The basic rule for the flow of any fluid to take place is, there must be some shear stress acting on it. It doesn’t matter how small or large in magnitude it is, but it is the most fundamental reason for flow.
Fluids are generally defined on this basis only, fluids are the materials in which shear stress is directly proportional to the strain rates.
Shear stress is a Tensor quantity which means it requires three things to describe its existence. Similarly, strain is also a tensor quantity of second order.
In simple words what we can say is, a scalar quantity requires only magnitude, a vector quantity requires magnitude, as well as direction, and a tensor quantity, requires magnitude, direction, and orientation for its complete description.
It is well known that scalars are also called zero order tensors, vectors are called first order tensors, and so on. The order is nothing but the number of indices required to define a quantity fully.
Stress and components on the fluid element :
Let’s look at the following fluid element and identify the different components of shear stress acting on it. We will denote shear stress by using the symbol ‘’ called ‘tau i j’. we will look at the meaning going ahead into the blog.
As shown in figure 1, there is a fluid element on which shear stress is acting. Consider planes CDQR and ABOP then planes ABCD and PORQ, then ADPQ and BCRO. These planes are special planes in the Cartesian coordinate system. But why? Because, when we observe them carefully we can notice that the orientation of the plane is special. The area vector for planes CDQR and ABOP is along the x-direction and similarly, for planes ABCD and PORQ the area vector is along the y-direction and for planes, ADPQ and BCRO area vector is along the z-direction.
What is an area vector? Area vector is a mathematical quantity using which we can define the orientation of a given surface area. So for example consider any arbitrary shape as shown below then the area vector is perpendicular to that plane in which the surface area exists.
Here OP will be an area vector, it shows the orientation of the plane surface and this vector will be of magnitude equal to the magnitude of the area of surface which it represents. So now we have understood the concept of area vector and some special planes in Cartesian coordinate systems.
As we can see in the figure that the components of the stresses are named Tau(xx), Tau(xy)
and so on and these are the names with very specific meaning, let’s look at the meaning now. So consider a component Tau(ij) where i = 1, 2, and 3 will represent x, y and z-direction respectively. This component will represent shear stress on i plane in j direction. So if we consider Tau(xx) it represents shear stress on the x plane in the x-direction. Similarly, now you can deduce the meaning for all other components.
Matrix Representation of shear stress :
This total shear stress on an element can be represented using 3 by 3 matrix also in the following way.
The matrix representation of shear stress can be viewed as diagonal and non-diagonal elements, where diagonal elements are actually normal stresses and non-diagonal elements are shear stresses or tangential stresses respectively.
So far we have understood that stress is a tensor quantity and how the different components are named to represent the stress. Now the next and important task is to figure out how to express stress in terms of strain.
Stress-Strain relationship :
We have seen earlier that the stress is directly proportional to the strain rates. Strain rate is expressed in terms of velocity components in a field. Here we have to make an assumption before going into further analysis.
Newton’s Law for fluids states that the shear stress is a linear function of the strain rates, so we assume the Newtonian fluids i.e. fluids which obey Newton’s Law.
Now we need something which can connect a second order tensor (stress) to another second order tensor (strain) linearly and that something must be a fourth order tensor. It’s like mapping one vector on another using some second order tensor.
Before going ahead we should keep in mind that there is a component of stress which will act on fluid if it is at the rest or not and that is “pressure”. When fluid is at rest there will be pressure and when it is flowing also there will be pressure.
Stress as the sum of deviatoric and hydrostatic components :
Hence any stress component can be written in two different terms i.e. as a sum of two terms and those terms are 1) Hydrostatic stress 2) Deviatoric stress. The hydrostatic term is directly equal to pressure and it has nothing to do with the strain or motion of the fluid. So Deviatoric stresses are the ones that are to be modeled using strain rates.
Consider Tau(ij) as Deviatoric stress and e(kl) as a strain. So let’s consider that C(ijkl) is that fourth order tensor that can connect them so we can write using Newton’s Law for fluids
Where
Where C(ijkl) will need 81 material constants (i = 1 to 3, j = 1 to 3, k= 1 to 3 and l = 1 to 3 so 3*3*3*3 i.e. 81) for its complete description.
If we apply the principle of conservation of angular momentum i.e.then the required number of material constants becomes 36. Still, it is quite difficult to handle the 36 different material constants to model the shear stress. Then we assume the fluid to be homogeneous and Isotropic so that the required number of constants will reduce.
History :
The relationship development between shear stress and shear strain started by Claude-Louis Navier who was a French mechanical engineer and was specialized in continuum mechanics and fluid mechanics. Along with him, Sir George Stokes worked on developing the equations of motion for viscous forces. Their famous work Navier-Stokes equations contributed to the field of fluid mechanics with huge impact.
Navier-Stokes equations are nonlinear coupled partial differential equations that can describe the flows where viscous forces, pressure forces, and gravity play a major role. There is no analytical or general solution available for it and it is one of the Million Dollar problems which can be solved to receive the greatest prize in science i.e. The Nobel Prize.
Summary :
Area vector is a mathematical quantity using which we can define the orientation of a given surface area.
Newton’s Law for fluids states that the shear stress is a linear function of the strain rates,
Scalars are zero ordered tensors, vectors are first ordered tensors and stresses are second order tensors.
References :
1) Prof Micheal Rosen Stanford University course work for physics Autumn http://large.stanford.edu/courses/2007/ph210/rosen2/ 2007.
2) Textbook of Fluid Mechanics By Frank White seventh edition.
In the next article, we will look at the conversion of fourth order tensor into the scalars for Homogeneous, Isotropic, and Newtonian Fluids.