From Reynolds Transport Theorem to Navier-Cauchy to Navier-Stokes Voyage: Part II

In part I of this blog, we have discussed the evolvement of the Conservation of Linear Momentum Equation from Newton’s Second Law of Motion using the Reynolds Transport Theorem. During the process we covered what is the need for Reynolds Transport Theorem, The Forcing & Stress Tensor Concept in Fluid: Fluid Kinematics Perspective, derived Cauchy Theorem, discussed the special property of the Second-Order Tensor, and finally arrived at an equation that is shown above and known as Navier-Cauchy Equation or Navier-Equation of Equilibrium in conservative form.

This is the well-known Navier-Cauchy Equation or Navier-Equation of Equilibrium. In a simple word, it is a mathematical representation of Newton’s Second Law of Motion written for a fluid material in a control volume space. On LHS we have the Unsteady & Convection term which if simplified (with the help of continuity equation) can be shown as a product of fluid density & total derivative of fluid acceleration which is nothing but the rate of change of fluid momentum that’s equal to RHS where we have Surface and Body Force which is what Newton’s Second Law of Motion talks about!

If you have by chance missed checking out the first part of the blog, don’t worry!! You just have to click on the below link, it will take you to the first part of the blog itself!

From Reynolds Transport Theorem to Navier-Cauchy to Navier-Stokes Voyage: Part I

Are we ready to solve Navier Cauchy Equation?

The answer to the above question is unfortunately No! Why? Let’s see.

The above Navier Cauchy Equation is valid for all types of fluid. It is a general form of Equation that governs the conservation of linear momentum for all types of fluids. If we closely look at the equation, the solution of Navier Cauchy Equation can be velocity field but there is one term on RHS that remains unknown and that term is Stress Tensor Term or Surface Force Term hence equation remains unclosed and it can’t be solved!

How do we demarcate one type of fluid to another? How do we differentiate Newtonian fluid from Non-Newtonian fluid?

To answer the above question we have to look closely into Stress Tenor Term, Tauij. Once we provide some type of appropriate mathematical description for Tauij We will be able to solve our Navier-Cauchy Equation and we will be able to demarcate one type of fluid from other.

What is Stress Tensor Tauij? Stress Tensor Term is a formal way of representing surface force on a fluid element. The effect of surface force acting on the Newtonian fluid must be different from the Non-Newtonian fluid. What is the effect of Surface force on a fluid in general? The surface force will start deforming a fluid element. Newtonian fluid will deform differently from Non-Newtonian fluid under the action of surface force. Hence we have to relate the Stress Tensor with the deformation of fluid. More formally we have to relate Stress tensor with the rate of deformation of the fluid as the fluid is a continuously deforming substance under the action of force. Now, We can have a couple of states of fluid and under those states, we can have various types of surface forces acting on a fluid. Let’s discuss that in the next section. 

Hence, We can say the mathematical description of the Stress Tensor must express and must be related to the application of surface force acting on a particular type of fluid, it should express the state of the stress/force acting on that fluid and how does it deform the fluid. It formally expresses the type of fluid. For example, Newtonian fluid relates its rate of deformation linearly to the stress tensor term.

Stress in Fluids due to its state?

There can be two states of the fluid, It can be static state or dynamic state. When fluid is at rest or at static condition then only stress acting on fluid is a normal type because fluid under rest can not sustain any shear. This state of stress is known as Hydrostatic stress.

When fluid is under motion, then the state of stress is hydrostatic stress + something which is responsible for the fluid deformation and this type of stress is known as Deviatoric Stress Component.

Deviatoric Stress Component is due to fluid’s deformation hence we must relate it with the rate of deformation of fluid or strain rate just like for solid, stress is directly proportional to strain. In general, we can write the rate of deformation as velocity gradient tensor which is a second-order tensor.

Velocity gradient tensor can be decomposed into two parts:-

Rotation of fluid element will not directly give rise to Deviatoric Stress Component but the rate of deformation component will do.

Now the question is how are they related? Is it a linear or non-linear relation?

It all dependent all depends on the type of fluid. For several fluids, the deviatoric stress component and rate of deformation are linearly related. Those types of fluids are known as Newtonian Fluid.

Heading into special form of Fluids: Newtonian Fluid

Now we will keep our discussion to Newtonian Fluid and proceed further. We had seen in the first part of the blog that Second-Order tensor maps vector onto another vector using Cauchy Theorem.

Similarly, our objective is to relate a second-order tensor (deviatoric stress component) to another second-order tensor vector (rate of deformation). Hence we require a 4th Order tensor for this mapping.

Just like a second-order tensor maps a vector onto another vector, a 4th Order tensor maps a second-order tensor (deviatoric stress component) on another second-order tensor (rate of deformation), a 4th Order tensor is specified by 4 indices.

So with the above discussion, we need to find 81 independent components in order to define the deformation behavior of the Newtonian Fluid. But in reality, we do not need these many constants, we can simplify the situation. 

We will consider that fluid is Homogenous and Isotropic. Isotropy means Rotation invariance and Homogeneity means positional invariance as if constitutive behavior and property of fluid are independent of the positional and rotational direction change in the coordinate system.

Now it all depends on how we use the homogeneity and isotropy property of the fluid in a mathematical sense to describe the behavior of fluid and reduce those 81 independent components. This is where the little mathematical exercise comes into the picture.

Let’s look into Isotropy first, If the fluid is isotropic, we can try forming an isotropic scalar. To do that we can consider 4 vectors A, B, C & D. Each vector will consist of 1 index. Each vector can be combined with the i, j, k & l index of our Cijkl 4th order tensor.  So we can write :-

Remember we want to represent the isotropic property of the fluid in a mathematical sense i.e. Rotational Invariance. We can do that by taking 2 vectors at a time and doing a dot product between them because the dot product depends on the cosine of the angle between 2 vectors as if we rotate 2 vectors, the angle between them should not change. Keeping that in mind we can write those dot product combination :-

Let’s consider the above-mentioned combination is linear. So we can multiply each dot product with some kind of scalar coefficient and obtain the expression. 

the dot product of 2 vector means, their corresponding component multiplied with each other, ith component of A multiplied with ith component of B where i is the repeated index and it is summed over i = 1 to 3. Hence we can write as following :-

Now to compare LHS & RHS of the equation, we must have AiBjCkDl term on RHS also. For that we will use Kronecker delta Tensor.

We can use the Kronecker delta Tensor on the RHS of our expression to do some mathematical manipulation and then we can obtain :-

Because our fluid is homogenous

Using the property of homogeneity and isotopy, We have reduced 81 constants to 3 independent constants which are alpha, beta and gamma. We can reduce them further!

Now we are left with two independent components by using the symmetry of the stress tensor matrix. We can write :-

alpha and beta should have some physical implication? What are those? Let’s discuss!

We know that for the Newtonian fluid, the deviatoric stress tensor is linearly proportional to the rate of deformation. If we represent that proportionality constant with single material property, that property turns out to be the Viscosity of the fluid.

alpha is coupled with ekk term which is nothing but :-

Finally we can write physical implication of alpha and beta term in our deviatoric stress component expression.

In most of the textbook alpha is written as lambda.

Now we have the mathematical representation of the deviatoric stress component term in terms of fluid deformation, What is left to express mathematically is the Hydrostatic stress component term in order to have a full representation of the Fluid Stress Tensor.

The hydrostatic stress component is due to the static nature of the fluid. When fluid is in a static state, It will have stress due to the hydrostatic pressure field. When fluid is at rest, the state of force/stress generated in fluids is such that at each and every point in the fluid, the stresses are directed towards that point from all directions and are of the same magnitude. It’s like a compressive nature of stress which is formally known as Hydrostatic Pressure. Fluid can also exert normal force on any contacting surface and is negative to what we call pressure. Hydrostatic Pressure generates compressive stress on the static fluid that’s why the negative convention is used to represent that. Hydrostatic stress component mathematically expressed using hydrostatic pressure and Kronecker delta. Kronecker delta helps us to ensure that the effect of the hydrostatic component comes into the picture while looking into the normal direction (i = j), there is no shear component of hydrostatic stress as pressure always acts normal to the surface.  

p is hydrostatic pressure. We can write stress tensor expression as :-

We have finally expressed stress tensor in terms of fluid deformation or in terms of primitive variables which are velocity gradients. We must not forget the assumptions behind the expression which are Fluid is Homogenous, Isotropic & Newtonian.

When we wrote Navier-Cauchy Equation in terms of Tauij, 6 components of Tauij (Tauij is a symmetric matrix, so a total of 6 independent components) were unknown to us. So we needed to have 6 additional equations (each one for stress tensor component) in order to close the system of equations and match the number of unknowns to the number of equations. Top of Continuity and Momentum equation solving 6 additional equations definitely computationally expensive. Now we are able to express those 6 Stress Tensor term compoents in terms of primitive variables which are velocity gradient and pressure.

Back to Stress Tensor Matrix: Linear & Shear Stress Physical effect on Newtonian Fluid

Let’s spend some time focusing on each and every term in our Stress Tensor Matrix. We will try to understand the physical effect on fluid deformation of each and every term in occurring in the Stress Tensor Matrix. We have six independent components coming up in our Stress Tensor Matrix as shown below.

Let’s put the definition of indices as per the stress tensor matrix component and we will look at the off-diagonal terms first.

Now let’s look at the diagonal terms.

In this way, we can see how maths talk about physics. The effect such as Linear Deformation, Rigid Body translation, angular and volumetric deformation of a fluid element can be seen embedded inside the Stress Tensor matrix. Viscosity coupled with velocity gradient gives rise to angular, linear and volumetric deformation.

Mechanical & Thermodynamic Pressure, The Stokes Hypothesis!

Let’s find the arithmetic mean of normal stresses.

Let’s spend some time discussing Thermodynamic and Mechanical Pressure. The molecules of fluid are in a state of random molecular motion (by the kinetic theory of gases), due to which molecules interact with each other by random collisions that cause a change in momentum of these molecules. The rate of change of the momentum of the molecules results in a normal force per unit area which is known as pressure. Hence Pressure no matter if it is thermodynamic or mechanical, is inherently an outcome of intermolecular interaction. Molecules possess some energy which can be classified as Translation Energy, Rotation Energy & Vibrational Energy. There are fluids for which we have all 3 forms of energy and there are fluids for which we have a restricted mode of energy.

Mechanical Pressure – Fluid persists only translation mode of energy in the molecule for example Dilute Monoatomic gas. Mechanical pressure can also be defined as the arithmetic average of normal stresses in the fluid.

Thermodynamic Pressure – Fluid persists all 3 modes of energy i.e. translation, rotational and vibrational. Thermodynamic pressure satisfies equation of state.

There can be several situations where we can have Mechanical Pressure and Thermodynamic pressure to be equal. For Dilute Monoatomic gas Mechanical Pressure is equal to Thermodynamic pressure as Dilute Monoatomic gas only has a Translation Mode of Energy.

Let’s consider a process where we are changing the thermodynamic state of the system. We are heating the system to increase its temperature. Now the pressure will try to adjust to that change. The question is how fast or how slow the system pressure can adjust to that change in temperature. It depends on the characteristic response time scale of the system as compared to the time scale of imposed disturbance on the system. If the temperature of the system is changing at a rapid rate such that the system at every instance is not able to achieve local thermodynamic equilibrium by responding to that change then the thermodynamic and mechanical pressure can not be equal. System response time is a threshold amount of time. If the disturbance imposition time scale is faster than the system response time scale and the system can not adjust immediately then it can not equilibrate in terms of mechanical pressure and thermodynamic pressure.

If the disturbance is imposed on the system at a slower rate such that we allow the system to have sufficient time to have all modes of energy manifested in terms of corresponding Translation mode of energy then thermodynamic pressure will be equal to the mechanical pressure.

If we impose the disturbance at a rapid rate for example we have a bubble that is expanding and contracting alternatively at a rapid rate such that before the bubble comes to a new equilibrium state, another excitation force has come. The time scale of imposition of the disturbance is faster/shorter than the time scale required by the system to come into equilibrium. In this situation, Mechanical Pressure is not equal to thermodynamic pressure.

For most of the practical cases that we deal in reality, usually fluid relaxes to a state of local equilibrium very fast so if we impose a force on the fluid, fluid will relax to a new equilibrium state rapidly, all degrees of freedom instantaneously converted into the translation mode and that’s how it comes to the equilibrium. Then we have Mechanical pressure equal to the Thermodynamic Pressure.

Hence mechanical pressure is equal to the thermodynamic pressure for most real-life practical cases. Usually, the disturbance is not imposed at a very rapid rate, at least not at a rate that is faster than the characteristic response time scale of the system.

This is the hypothesis, it is a justification only with an argument for most of the cases.

The time scale of imposed disturbance on the system is slower than the time scale required by the system to achieve thermodynamic equilibrium. This is known as Stokes Hypothesis. Fluids that follow Stokes Hypothesis are known as Stokesian Fluid.

Stokes Hypothesis

Physical Significance of Second Coefficient of Viscosity (Lambda)

Let’s look into lambda more closely and try to understand its physical effect on the fluids.

  Let’s look at the normal stress component of the stress tensor matrix.  

That means for a fluid element that already stretching due to the normal stress component tau11. The proportional enhancement of stress to further expand that fluid element is actually not an enhancement but a reduction!

The Voyage end! Arrival to Navier-Stokes Equation!!

Initially, we had 81 unknown variables to describe the behavior or deformation of fluid. After that, we have used the assumption of Isotropic and homogenous fluid. We were able to reduce 81 unknown variables to 3 variables, namely alpha, beta and gamma. Then we have used the symmetric property of the deformation tensor matrix and we were able to reduce 3 variables into 2, namely alpha and gamma. Then proposed alpha has the physical property of the fluid as viscosity and gamma as the second coefficient of viscosity. Finally, we have used the stokes hypothesis to build up viscosity & the second coefficient of viscosity relation which is valid for Stokesian fluid.   

Let’s write the final Linear Momentum governing equation. Starting with Navier Cauchy Equation.

Let’s take the differentiation of stress tensor matrix expression.

Our governing equation become :-

Navier Stokes Equation in Conservative form for Compressible flow

The above equation is the Navier Stokes Equation in Conservative form for Compressible flow. Conservation form of equation directly talks about conservation of momentum in the mathematical sense as density is inside the derivative hence our dependent variable is flux variable, momentum flux in our case.

Conservative Form to Non-Conservative form conversion

Let’s expand LHS Unsteady & Convection term derivative by applying the product rule of differentiation.

We can write our equation as: –

Navier Stokes Equation in Non-Conservative form for Compressible flow

The time rate of change of the volume of a moving fluid element, per unit volume is zero for incompressible flow.

Let’s look into total acceleration term!

We can write our final equation.

Non-Consevative form Navier Stokes Equation for Incompressible flow

This is the Newton’s Second Law of Motion for a control volume expressed in differential form

LHS & RHS of Navier Stokes Equation are balanced. Both have units of Force per unit Volume.

Special Characteristic of Navier-Stokes Equation!

Navier Stokes Equation is Second Order Non-Linear Partial Differential Equation which has mixed characteristics.

Simplified Forms of Navier-Stokes Equation!

Navier Stokes Equation for Stokesian & Newtonian Fluid – Compressible Flow
Navier Stokes Equation for Stokesian & Newtonian Fluid – Incompressible Flow
Stokes Equation for Stokesian, Newtonian Fluid – Creepy Flow [Negligible Inetria/Convective Strength in Flow]
Euler Equation for Inviscid Flow [Negligible Viscous Strength in Flow, High Speed Flow]
Equation of Fluid Static

HAPPY COMPUTATION !!!!!

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