Continuum Mechanics

This article is about ‘continuum mechanics’ and we first consider what is meant by this.

Continuum mechanics is a branch of mechanics that deals with the mechanical behaviour of materials modelled as a continuous mass rather than as discrete particles.  Augustin-Louis Cauchy was the first to formulate such models.

It may sound somewhat insignificant at first, but in the formulations of physics it makes a huge difference whether one understands the structure of matter as a continuum or as a lattice of discrete particles.  Ultimately, both formulations must merge in the limiting case of infinitesimal lattice distances.

Modelling an object as a continuum assumes that the substance of the object completely fills the space it occupies.  Modelling objects in this way ignores the fact that matter is made of atoms, and so is not continuous.  On scales much greater than that of inter-atomic distances, such models are highly accurate.

Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behaviour of such objects.

Continuum mechanics deals with physical properties of eg. fluids which are independent of any particular coordinate system in which they are observed. These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of a specific coordinate system.

We also need to introduce the concept of classical field theories.  What is this?

A classical field theory is a physical theory that predicts how physical fields interact with matter through field equations.  The term ‘classical field theory’ is reserved for describing those physical theories that describe electromagnetism and gravitation, two of the fundamental forces of nature.

A physical field can be thought of as the assignment of a physical quantity at each point of space and time.

Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum.  The mass continuity equation is a continuity equation, representing the conservation of mass.

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases.  The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy (also known as the First Law of Thermodynamics).

In addition to the above, fluids are assumed to obey the continuum assumption.  Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete.  Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at infinitesimally small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.

In addition to the mass, momentum and energy conservation equations, a thermodynamic equation of state that gives the pressure as a function of other thermodynamic variables is required to completely describe the problem.  An example of this would be the well-known equation of state of an ideal gas:

 p · V = n · R · T

$\mathsf{\text{With the molar mass M, which is the m in kg per number of moles, and the relation for the density}}\rho =\frac{m}{v}\mathsf{\text{we find for the pressure}}$ $$p=\rho ·\frac{R}{M}·T$$

where p is pressure, ρ is density, T the absolute temperature, while R is the gas constant and M is the molar mass for a particular gas.

• changes due to different field strengths,
• any time dependences of the field.

• the partial time derivative at a fixed location (local part)
• the change due to the movement of the fluid element because of which the temperature T must be evaluated at a “shifted” location (convective part)

Gauss’ Theorem

Let us return to fluid dynamics and look at these physical observables:

$\mathsf{\text{Gas flow (vector field)}}\overrightarrow{v}(x,y,z)$

$\mathsf{\text{Gas density }}\rho (x,y,z)$

The actual amount of gas inside a closed control volume V with the closed surface S is:

M = ∫ ρ(x, y, z) dxdydz  where the integral extends over the entire control volume.

$\mathsf{\text{We would like to know how the amount of gas changes over the time, i.e., what does }}\frac{dm}{dt}\mathsf{\text{look like?}}$

At this point we introduce the mass flow vector as follows:

$$\overrightarrow{\mathrm{\Phi }}(x,y,z)=\rho (x,y,z)\cdot \overrightarrow{v}(x,y,z)$$

$\mathsf{\text{The normal vector}}\overrightarrow{n}\mathsf{\text{ is perpendicular to the surface S at every point and has the length }}|\overrightarrow{n}|=1\mathsf{\text{ and always points away from the surface.}}$

$$\frac{dM}{dt}=-\oint \overrightarrow{\mathrm{\Phi }}\cdot \overrightarrow{n}\mathsf{\text{ dS where the integral again extends over the entire surface S}}$$

$\mathsf{\text{If }}\frac{dM}{dt}\ne 0\mathsf{\text{ we also see a change in the gas density }}\rho :$

$$\frac{dM}{dt}=\int \frac{\partial \rho (x,y,z)}{\partial t}dxdydz\mathsf{\text{ where the integral extends over the entire control volume.}}$$

Thus we find the relation for the mass conservation law in integral representation:

$$\int \frac{\partial \rho (x,y,z)}{\partial t}dxdydz=-\oint \overrightarrow{\mathrm{\Phi }}·\overrightarrow{n}dS$$

Gauss’ divergence theorem will not be derived here but applied to the above relation:

$$\oint \overrightarrow{\mathrm{\Phi }}·\overrightarrow{n}dS=\int (\overrightarrow{\mathrm{\Delta }}·\overrightarrow{\mathrm{\Phi }})dxdydz$$

This yields

$$\int \frac{\partial \rho (x,y,z)}{\partial t}dxdydz=-\int (\overrightarrow{\mathrm{\Delta }}·\overrightarrow{\mathrm{\Phi }})dxdydz$$

or

$$\frac{\partial \rho (x,y,z)}{\partial t}+\overrightarrow{\mathrm{\Delta }}·\overrightarrow{\mathrm{\Phi }}=0$$

$\mathsf{\text{With }}\overrightarrow{\mathrm{\Phi }}=\rho ·\overrightarrow{\mathrm{\Delta }}\mathsf{\text{ we eventually find the relation for mass conservation in}}\mathit{\text{ differential form:}}$

$$\frac{\partial \rho (x,y,z)}{\partial t}+\overrightarrow{\mathrm{\Delta }}·(\rho \overrightarrow{v})=0$$

Recall the definition of the substantial derivative

$$\frac{D}{Dt}≔\frac{\partial }{\partial t}+\overrightarrow{v}·\overrightarrow{\mathrm{\Delta }}$$

Applying this now to the density yields

$$\frac{D\rho }{Dt}=\frac{\partial \rho }{\partial t}+\overrightarrow{v}·\overrightarrow{\mathrm{\Delta }}\rho$$

Combining this with the relation for mass conservation in differential form is

$$\frac{D\rho }{Dt}=-\overrightarrow{\mathrm{\Delta }}·(\rho \overrightarrow{v})+\overrightarrow{v}·\overrightarrow{\mathrm{\Delta }}\rho$$ $$\frac{D\rho }{Dt}=-(\rho \overrightarrow{\mathrm{\Delta }}·\overrightarrow{v}+\overrightarrow{v}·\overrightarrow{\mathrm{\Delta }}\rho )+\overrightarrow{v}·\overrightarrow{\mathrm{\Delta }}\rho$$ $$\frac{D\rho }{Dt}=-\rho \overrightarrow{\mathrm{\Delta }}·\overrightarrow{v}$$

This relation is now the Lagrange version of the mass conservation law.

Euler Equation

We now need a relationship that describes the effects of the internal forces in the fluid in a control volume.

We make the assumption of an ideal liquid which means that with the exception of the pressure, we do not allow any other physical effects which may alter the momentum or the energy of the fluid i.e., no friction and no heat conduction.

Our fluid element is defined as a small volume e.g. in the form of a cube which feels pressure from all sides.

For the pressure change in x-direction we find:

The resulting force (note that pressure is defined a force per area) is then

$$p\delta y\delta z-(p+\frac{\partial p}{\partial x}\delta x)\delta y\delta z=-\frac{\partial p}{\partial x}\delta x\delta y\delta z$$

The components of the force in all three spatial dimensions is

$$d\overrightarrow{F}=\left(\begin{array}{r}d{F}_x\\ d{F}_y\\ d{F}_z\end{array}\right)=-\left(\begin{array}{r}\frac{\partial p}{\partial x}\\ \frac{\partial p}{\partial y}\\ \frac{\partial p}{\partial z}\end{array}\right)\delta x\delta y\delta z=-\overrightarrow{\mathrm{\Delta }}·\overrightarrow{p}\delta x\delta y\delta z$$

In a flow, the velocity is generally neither constant in time nor constant in space. The fluid therefore flows either slower or faster past the lateral surfaces of the fluid volume under consideration. Frictional forces occur, which are all the greater the more viscous the fluid. However, since we assume a frictionless flow and thus a non-viscous fluid, such shear forces do not occur in the flow.

Other forces that act on the fluid, but generally cannot be neglected, are field forces such as those caused by gravity. Air flow on earth or fluid flow in pipes, for example, are significantly influenced by gravity.

In addition to the pressure force F_p, field forces F_g also act on a fluid element (the weight force is only meant to be representative of other possible field forces at this point). The sum of both forces then ultimately corresponds to the resulting accelerating force F_acc, which influences the movement of the fluid element.

According to Newton’s second axiom, the accelerating force F_acc leads to a corresponding change in velocity (acceleration), which depends on the mass dm of the fluid element under consideration. This material acceleration is also called total or substantial acceleration a_sub. The control volume is abbreviated as dV:

$$\overrightarrow{a}=\frac{d\overrightarrow{F}}{dm}=\frac{d\overrightarrow{F_g}-\overrightarrow{\mathrm{\Delta }}·\overrightarrow{p}dV}{dm}=\frac{d\overrightarrow{F_g}}{dm}-\frac{\overrightarrow{\mathrm{\Delta }}·pdV}{dm}$$ $$\overrightarrow{a}=\overrightarrow{g}-\frac{1}{\rho }·\overrightarrow{\mathrm{\Delta }}·p$$

$\mathsf{\text{where we used }}\rho =\frac{dm}{dV}\mathsf{\text{ and }}\overrightarrow{g}=\frac{d\overrightarrow{F}}{dm}\mathsf{\text{ which is gravitational acceleration.}}$

The substantial or material acceleration of a fluid element has two reasons:

1. A flow and thus the speed of a fluid element changes not only in time but also in space.
2. The temporal change therefore causes a “local acceleration” and the local change, which is nothing other than a flow transport from one location to another one, causes a “convective acceleration”.

One could now conclude that in a stationary flow no acceleration acts on the fluid elements. But this is not the case!

In a stationary flow, the velocity no longer changes in time at a fixed location. In general, a fluid element must permanently change its velocity when flowing, for example at narrow points in a pipe. Despite the lack of temporal change at a fixed location, the fluid element is nevertheless accelerated when it changes its location. This occurs through flow transport and is therefore referred to as “convective acceleration”.

Let us summarize at this point:

• The local acceleration is due to the time-varying flow velocity of a transient flow at a fixed location,
• the convective acceleration is due to the flow velocity changing from place to place,
• both components together result in the observable acceleration and are therefore referred to as substantial or material acceleration,
• the local component is omitted in the case of a stationary flow, as the velocity at a fixed location then no longer changes in time.

Energy Conservation

Let us consider a sample fluid element on a streamline:

In order to allow flowing in the indicated direction we postulate:

$$\frac{\partial p}{\partial s}<0\mathsf{\text{ and }}\frac{\partial p}{\partial r}>0$$

The total derivative of the velocity v of the fluid element is then

$$dv=\frac{\partial v}{\partial t}dt+\frac{\partial v}{\partial s}ds$$

Division by dt

$$\frac{dv}{dt}=\frac{\partial v}{\partial t}=\frac{\partial v}{\partial s}\frac{ds}{dt}$$

and eventually

$$\frac{dv}{dt}=\frac{\partial v}{\partial t}+\frac{\partial v}{\partial s}v$$

An accelerating tangential force acts on the fluid element of mass dm in direstion s:

$$F_{tangential}=dm·{\overrightarrow{a}}_{tangential}=dm·(\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial s})$$

This equation only describes the effect of a temporal or local change of the velocity, but not the cause for the acceleration which only stems from the external forces acting on the fluid element.

$\mathsf{\text{The end faces of the fluid elements are now subjected to compressive forces which decrease along the streamline, i.e., }}\frac{\partial p}{\partial s}<0:$

Note that

$$F_{p,left}=pd{A}_s\mathsf{\text{ and }}{F}_{p,right}=(p+\frac{\partial p}{\partial s}ds)d{A}_s$$ $$F_{pressure}=F_{p,left}-F_{p,right}=-\left(\frac{\partial p}{\partial s}ds\right)d{A}_s=-\frac{\partial p}{\partial s}dV$$

This is the effective force due to the pressure difference between the left and the right side of the fluid element.

Let us consider the different types of acceleration i.e., the local and convective ones:

$${\overrightarrow{a}}_{local}=\frac{\partial v}{\partial t}=\left(\begin{array}{r}\frac{d{v}_x}{dt}\\ \frac{d{v}_y}{dt}\\ \frac{d{v}_z}{dt}\end{array}\right)$$

The formulation of the convective acceleration in three dimensions is a bit more difficult beacuse each velocity component depends on three spatial coordinates:

$$a_{conv,x}=\frac{\partial v_x}{\partial x}{v}_x+\frac{\partial v_x}{\partial y}{v}_y+\frac{\partial v_x}{\partial z}{v}_z$$

which translates into the three-dimensional vector notation:

$${\overrightarrow{a}}_{convective}=\left(\begin{array}{r}{a}_{conv,x}\\ a_{conv,y}\\ a_{conv,z}\end{array}\right)=\left(\begin{array}{r}\frac{\partial v_x}{\partial x}{v}_x+\frac{\partial v_x}{\partial y}{v}_y+\frac{\partial v_x}{\partial z}{v}_z\\ \frac{\partial v_y}{\partial x}{v}_x+\frac{\partial v_y}{\partial y}{v}_y+\frac{\partial v_y}{\partial z}{v}_z\\ \frac{\partial v_z}{\partial x}{v}_x+\frac{\partial v_z}{\partial y}{v}_y+\frac{\partial v_z}{\partial z}{v}_z\end{array}\right)$$

or in a compact notation

$${\overrightarrow{a}}_{convective}=[\left(\begin{array}{r}{v}_x\\ v_y\\ v_z\end{array}\right)·\left(\begin{array}{r}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{array}\right)]\left(\begin{array}{r}{v}_x\\ v_y\\ v_z\end{array}\right)$$

and now using the Nabla-operator:

$${\overrightarrow{a}}_{convective}=(\overrightarrow{v}·\overrightarrow{\mathrm{\Delta }})\overrightarrow{v}$$

Thus the substantial derivative for the velocity reads:

$$a_{substantial}={\overrightarrow{a}}_{local}+{\overrightarrow{a}}_{convective}$$ $$a_{substantial}=\frac{\partial v}{\partial t}+(\overrightarrow{v}·\overrightarrow{\mathrm{\Delta }})\overrightarrow{v}$$

Recall

$$\overrightarrow{a}=\overrightarrow{g}-\frac{1}{\rho }·\overrightarrow{\mathrm{\Delta }}·p$$

We set the last two equations equal to each other

$$\overrightarrow{g}-\frac{1}{\rho }·\overrightarrow{\mathrm{\Delta }}·p=\frac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v}·\overrightarrow{\mathrm{\Delta }})\overrightarrow{v}$$

and eventually find the so-called Euler equation:

$$\frac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v}·\overrightarrow{\mathrm{\Delta }})\overrightarrow{v}+\frac{1}{\rho }·\overrightarrow{\mathrm{\Delta }}·p=\overrightarrow{g}$$

We now apply this equation to just streamline in one dimension:

$\mathsf{\text{Note that }}\alpha \mathsf{\text{ is the angle between g and }}{g}_s\mathsf{\text{ and hence }}{g}_s=-gcos\alpha andcos\alpha =\frac{dz}{ds}$

In one dimension we work with

$$\frac{\partial v}{\partial t}+\frac{\partial v}{\partial s}v+\frac{1}{\rho }\frac{\partial p}{\partial s}=-g\frac{dz}{ds}$$

$\mathsf{\text{For the stationary case we set }}\frac{\partial v}{\partial t}=0\mathsf{\text{ and simplify the relation above}}$

$$\frac{dv}{ds}v+\frac{1}{\rho }\frac{dp}{ds}=-g\frac{dz}{ds}$$

Multiplication with ds and ρ yields

ρ dv v + dp = -g ρ dz

or

dp + ρ dv v + g ρ dz = 0

Integration then gives

$$\int dp+\rho dvv+g\rho dz=constant$$

and eventually

$$p+\frac{1}{2}\rho v^2+g\rho z=constant$$

This describes the energy conservation and is no more than the Bernoulli equation.

The Navier-Stokes equation (for incompressible fluids)

Before we start to think about some generalizations of what we learned so far, it is worth while to take a look at the history of all these things.

Isaac Newton published his three-volume Principia with the laws of motion in 1686 and also defined the viscosity of a linear viscous fluid in the second book. This type of fluid is nowadays referred to as a Newtonian fluid.

In 1755, Euler derived the Euler equations from the laws of motion, with which the behaviour of viscosity-free fluids (liquids and gases) can be calculated.

French mathematician d’Alembert introduced the Eulerian approach, derived the local mass balance and formulated the d’Alembert paradox, according to which no force is exerted on a body in the direction of flow by the flow of viscosity-free fluids – which Euler had already proved earlier.

Because of this and other paradoxes of viscosity-free flows, it was clear that Euler’s equations of motion had to be supplemented.

Four researchers of that time ie. Navier, Poisson, de Saint-Venant and Stokes independently from each other formulated the momentum theorem for Newtonian fluids in differential form in the first half of the 19th century.

Navier (1827) and Poisson (1831) established the momentum equations after considering the effect of intermolecular forces.

In 1843, de Saint-Venant published a derivation of the momentum equations from Newton’s linear viscosity approach, two years before Stokes did so in 1845. However, the name Navier-Stokes equations became established for the momentum equations.

A significant advance in the understanding of viscous fluids was provided by Prandtl in 1904 with his boundary layer theory. From the middle of the 20th century, numerical fluid mechanics developed to such an extent that with its help solutions to the Navier-Stokes equations could be found for practical problems, which – as it turns out – correspond well with the real flow processes.

The Navier Stokes equation is a mathematical model of the flow of linear-viscous Newtonian liquids and gases referred to as fluids. We will see that the equations are an extension of the Euler equations of fluid mechanics by terms describing viscosity.

The Navier-Stokes equation hence is a universal formula to describe the flow of fluids.

Just to not discourage the reader it is worth to mention that the Navier-Stokes equation could not be solved in the three-dimensional case until today and is one of the millennium problems of mathematics.

Analytically, the Navier-Stokes equation can so far only be solved under simplifying assumptions which we will also use in our article.

In the two-dimensional case, the existence of such solutions has already been successfully investigated. However, the Navier-Stokes equation can be solved numerically, which is used for the simulation of fluids, for example.

Especially in physics, the Navier-Stokes equations means the momentum equation for flows. In a broader sense i.e., in fluid mechanics, this momentum equation is extended by the continuity equation and the energy equation and then forms a system of non-linear partial differential equations of second order.

This is the fundamental mathematical model of fluid mechanics. In particular, the equations also represent turbulence and boundary layers.

If the Navier-Stokes equations are formulated by using dimensionless quantities, various dimensionless ratios such as the Reynolds number or the Prandtl number can be derived.

In what follows we will derive the Navier-Stokes equation for incompressible fluids, which just means at this point that the density is constant and doesn’t change across the three-dimensional movement of the fluid through a pipe.  We will also look at the difference between compressible and incompressible fluids and how to simplify the Navier-Stokes equation for certain systems.

We want to understand how the Navier-Stokes equation is derived from Newton’s law.  To be able to describe the flow of a volume element ΔV with a mass of Δm = ρ · ΔV we start from the well-known 2^nd law of Newton:

$$\overrightarrow{F}=\mathrm{\Delta }m·\ddot{\overrightarrow{r}}=\rho ·\mathrm{\Delta }V·\dot{\overrightarrow{v}}$$

As we derived earlier, the expression for the material derivative of velocity is:

$$\dot{\overrightarrow{v}}=\frac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v}·\overrightarrow{\mathrm{\nabla }})\overrightarrow{v}$$

To repeat the meaning we consider the temporal change of  at the same location and the temporal change when the fluid flows from one location to another.

Therefore, the term is also called substantial acceleration.

We know the right hand side of Newton’s 2^nd law and next we need to know the forces acting on a volume element ΔV of the fluid. As we will see there are various forces which need to taken into consideration. We recall the formula for the force due to the pressure differences between two different locations.

$${\overrightarrow{F}}_{pressure}=-\overrightarrow{\mathrm{\nabla }}p\mathrm{\Delta }V$$

We now also consider friction forces which also contribute to the right hand side of Newton’s 2^nd law.  So, if the friction within a liquid is not negligible, a force or pressure is necessary to move a liquid uniformly in relation e.g. to a pipe system. How great the necessary thrust force is depends on the viscosity of the fluid.

Considering the fluid as a stack of layers, the movement of the layers will cause friction between them if the velocity of the layers is different. If, for example, a cover glass is placed on a drop of liquid and moved slowly and evenly along the base on the thin layer of liquid, a force F is required to maintain the movement. This force is proportional to the area A of the glass, the speed v of the movement and the viscosity; moreover, the force is inversely proportional to the thickness Δx of the liquid layer. Altogether, therefore, the force F necessary to overcome friction is

$$F=\eta ·A·\frac{v}{\mathrm{\Delta }x}$$

More generally speaking we expose the liquid to two plates which are moving with different velocities v1 and v2

Thus we find for the force F to overcome the friction

$$F=\eta ·A·\frac{v_2-v_1}{\mathrm{\Delta }x}\equiv \eta ·A·\frac{d{v}_z}{dx}$$

A volume element dv experiences frictional forces at the top and the bottom.  The area A becomes dy · dz

F₁ is the force acting on top of dV at position x:

$$F_1=-\eta ·dy·dz·\frac{d{v}_z}{dx}$$

F₂ is the force acting on the bottom of dV at position x+dx

$$F_2=-\eta ·dy·dz·\frac{d{v}_z}{dx}$$

The resulting force F₁ + F₂ is then

$$F_z=-\eta ·dx·dy·dz·\frac{\frac{d{v}_z}{dx}(atx+dx)-\frac{d{v}_z}{dx}(atx)}{dx}$$

Note that

$$F_z=-\eta ·dx·dy·dz·\frac{\frac{d{v}_z}{dx}(atx+dx)-\frac{d{v}_z}{dx}(atx)}{dx}$$

is the second derivative of v_z, with respect to dx

$${\partial }^2{v}_z\partial x^2$$

The resulting force for F₂ with regard to variations on x-direction now becomes

$$F_z=\eta ·dV·\frac{{\partial }^2{v}_z}{\partial z^2}$$

However, this is not yet the entire force component in the z-direction because we also need to include changes of velocity in y- and z-directions:

Similar expressions can be derived for F_x and F_y and we introduce at this point the Laplace operator. This will allow us to write the formula for friction forces in a compact manner.

$$\mathrm{\Delta }=\overrightarrow{\mathrm{\nabla }}·\overrightarrow{\mathrm{\nabla }}=\left(\frac{{\partial }^2}{\partial x^2}\frac{{\partial }^2}{\partial y^2}\frac{{\partial }^2}{\partial z^2}\right)$$

The expression for the friction force thus becomes:

$${\overrightarrow{F}}_{friction}=\eta ·\mathrm{\Delta }\overrightarrow{v}·dV$$

There are many other forces which may contribute and to include them in our calculation we recall

$$\overrightarrow{F}=\mathrm{\Delta }m·\ddot{\overrightarrow{r}}=\rho ·dV·\overrightarrow{a}$$ $$$$ $\mathsf{\text{where }}\overrightarrow{a}\mathsf{\text{ denotes an arbitrary acceleration, sometimes also referred to as force density with regard to the mass.}}$

As an example we consider the gravitational acceleration which may play a role in many of our applications.  Then our expression for external forces takes the form

$${\overrightarrow{F}}_{external}=\rho ·dV·\overrightarrow{g}$$

We now consider the total expression for any type of forces in accordance with Newton’s 2^nd Law:

$${\overrightarrow{F}}_{total}={\overrightarrow{F}}_{pressure}+{\overrightarrow{F}}_{friction}+{\overrightarrow{F}}_{external}=-\overrightarrow{\mathrm{\nabla }}p\mathrm{\Delta }V+\eta ·\mathrm{\Delta }\overrightarrow{v}·dV·\overrightarrow{g}$$

The left hand side of Newton’s 2^nd Law corresponds to the total force:

$$\overrightarrow{F}=\mathrm{\Delta }m·\ddot{\overrightarrow{r}}=\rho ·\mathrm{\Delta }V·\dot{\overrightarrow{v}}=(\frac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v}·\overrightarrow{\mathrm{\nabla }}))\overrightarrow{v}$$

which implies this complicated looking expression:

$$\rho ·dV·(\frac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v}·\overrightarrow{\mathrm{\nabla }})\overrightarrow{v})=-\overrightarrow{\mathrm{\nabla }}pdV+\eta ·\mathrm{\Delta }\overrightarrow{v}·dV+\rho ·dV·\overrightarrow{g}$$

We divide this by dV and arrive at the Navier-Stokes equation for for incompressible fluids then

$$\rho ·(\frac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v}·\overrightarrow{\mathrm{\nabla }})\overrightarrow{v})=-\overrightarrow{\mathrm{\nabla }}p+\eta ·\mathrm{\Delta }\overrightarrow{v}+\rho ·\overrightarrow{g}$$

To better understand the meaning of all these terms it therefore makes sense to illustrate the individual summands with an example i.e., we apply the Navier-Stokes logic to a crowd of people.

What do all the summands mean in this case?

$\begin{array}{rl}& \rho ·\frac{\partial \overrightarrow{v}}{\partial t}\mathsf{\text{The velocity of the people changes in time (people just move at different velocities)}}\\ & \rho ·(\overrightarrow{v}·\mathrm{\nabla })\overrightarrow{v}\mathsf{\text{People are moved to an area where other people move faster (areas with different velocities i.e., the velocity}}\\ & \mathsf{\text{not only changes in time = first term, but also depends on the location of the flow which itself may also change in time)}}\\ & \rho ·\overrightarrow{g}\mathsf{\text{People also move downhill following gravity (external gravitational force)}}\\ & \eta ·\mathrm{\Delta }\overrightarrow{v}\mathsf{\text{People are carried away by the people flowing past them (friction)}}\end{array}$

We can easily see here the analogies to the movement of particles or infinitesimally small volume elements in a liquid or gas flow, just replace ‘people’ by ‘fluid elements’.

If we set herein η = 0 we are back to frictionless movements of the liquid and the Navier-Stokes equation becomes the Euler equation which we derived before and which is prevailingly used for calculations of flows in pipe systems:

$$\rho ·(\frac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v}·\overrightarrow{\mathrm{\nabla }})\overrightarrow{v})=-\overrightarrow{\mathrm{\nabla }}p+\rho ·\overrightarrow{g}$$

We can further simplify this equation if we consider very viscous fluids where the time dependence of the velocity field and the inertia term can be neglected i.e.,

$$\frac{\partial \overrightarrow{v}}{\partial t}=0\mathsf{\text{ and }}(\overrightarrow{v}·\mathrm{\nabla })\overrightarrow{v}=0$$

which then leads us to the simpler form:

$$0=-\overrightarrow{\mathrm{\nabla }}p+\eta ·\mathrm{\Delta }\overrightarrow{v}+\rho ·\overrightarrow{g}$$

This is the so called Stokes Equation which is used in the case of very viscous fluids.

Before we continue the discussion of the Navier-Stokes equation, we need to take a closer look at another partial differential equation which is of relevance for our considerations, the continuity equation.

The Continuity Equation

In physics, the ‘law of conservation’ is the formulation of the observed fact that the value of a quantity, called a conservation quantity, does not change in certain physical process.  In a closed system, conservation variables do not change.  Typical conservation quantities are mass, electrical charge, energy, momentum, angular momentum and others.

$\mathsf{\text{If, in fluid dynamics, the mass density changes }}\rho (\overrightarrow{x},t)\mathsf{\text{ because the fluid flows with velocity }}\overrightarrow{v}(\overrightarrow{x},t)=\frac{d\overrightarrow{x}}{dt}\mathsf{\text{ along the path curves }}\overrightarrow{x}(t),$

the corresponding current density is defined as

$$\overrightarrow{ȷ}=\rho \overrightarrow{v}$$

A continuity equation is a particular partial differential equation associated with a conservation variable (see above).  It links the temporal change of the density belonging to this conservation variable with the spatial change of its current density.

$$\frac{\partial \rho }{\partial t}+\overrightarrow{\mathrm{\nabla }}·\overrightarrow{ȷ}=0$$ $$\frac{\partial \rho }{\partial t}+\rho (\overrightarrow{\mathrm{\nabla }}·\overrightarrow{v})+(\overrightarrow{\mathrm{\nabla }}\rho )·\overrightarrow{v}$$ $$\frac{\partial \rho }{\partial t}+(\overrightarrow{\mathrm{\nabla }}\rho )·\frac{\partial \overrightarrow{x}}{\partial t}=-\rho (\overrightarrow{\mathrm{\nabla }}·\overrightarrow{v})$$

Recall the form of the substantial derivation which represents the total differential

$$\frac{d}{dt}\rho (t,\overrightarrow{x}(t))=-\rho (\overrightarrow{\mathrm{\nabla }}·\overrightarrow{v})$$

$\mathsf{\text{So along a trajectory }}\overrightarrow{x}(t)\mathsf{\text{ the density changes with the divergence of the flow }}\overrightarrow{v}.\mathsf{\text{ The flow is incompressible if the density remains constant along the path:}}$ $$\frac{d}{dt}\rho (t,\overrightarrow{x}(t))=0$$

$\begin{array}{rl}& \mathsf{\text{The divergence of a vector field }}\overrightarrow{v}\mathsf{\text{ is a scaler field that indicates at each point how much the vectors diverge in a small vicinity of a point.}}\\ & \mathsf{\text{If one interprets the vector field }}\overrightarrow{v}\mathsf{\text{ as the flow field of a quantity for which the continuity equation applies, then the divergence is the }}\\ & \mathsf{\text{source density. Sinks have negative divergence. If the divergence is equal to zero everywhere, the field is called source-free.}}\end{array}$

This means in our case that if there are no sources in the flow then we don’t see any changes in density. That’s the case of incompressible fluids. The situation is different if we look at compressible fluids.

In fluid mechanics, the continuity law for (incompressible) fluids follows from the continuity equation.

The continuity law states (in integral form) that the mass flow of a fluid in a pipe is independent of where it is measured.

The differential form is the continuity equation.

It applies in both frictionless and frictional cases for steady-state ie. time-independent and for transient flows of incompressible fluids, but not for transient flows of compressible fluids.

For incompressible (non-compressible) fluids, continuity also applies to the volume flow.

Again, for illustration purposes, for incompressible fluids we note that the same volume emerges from a pipe section that simultaneously enters at the other end.

The entering volume is V₁ = A₁ · v₁, the leaving volume is V₂ = A₂ · v₂. Because of V₁ = V₂ we in the narrower part of the pipe the displacement x₂ is greater than x₁ by the same factor as the cross-section A₁ is greater than A₂. The same applies to the flow velocities averaged over the cross-section

$v=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}.\mathsf{\text{ This is the phenomenon described by the Bernoulli equation.}}$

On the other hand, for compressible and non-stationary fluids that can change their density, the following applies to the mass flow rate:

$$\dot{m}=\rho ·\dot{V}=\rho ·A·\dot{x}=\rho ·A·v=constant$$

The mass flow rate is conservation variable i.e., the mass which enters from one side has to leave on the other side, but compared with the incompressible case, we now have to consider also the density which may be subject to change, i.e.,

$${\rho }_1·A_1·v_1={\rho }_2·A_2·v_2$$

For example, the density of the fluid can change if the temperature of the fluid changes between the beginning and the end of the pipe. If the density decreases, a larger volume must come out in the same time.

As an illustration we look at cars in a traffic jam due to narrowing lanes.  They also behave according to the law of continuity when the lane narrows. The distance between the cars may be considered as density.  If the cross-section is large, the density is low, the speed is high and the traffic flows freely.  In the congestion before the narrowing, the density is high and the speed is low.  In the constriction, the cross-section is small, the speed and density are medium, and the vehicle throughput is the same in all cases – provided that no car leaves or joins the road.

Navier-Stokes equation for compressible fluids and nondimensionalisation

This section is the most difficult one to understand and is added for the sake of a complete presentation of the topic.

It remains to consider the Navier-Stokes equation for compressible fluids.  Essentially, the extension of the equation derived so far consists in the addition of a further term, the derivation of which we will spare ourselves here:

$$\rho ·(\frac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v}·\overrightarrow{\mathrm{\nabla }})\overrightarrow{v})=-\overrightarrow{\mathrm{\nabla }}p+\eta ·\mathrm{\Delta }\overrightarrow{v}+(\gamma +\eta )\mathrm{\nabla }(\mathrm{\nabla }·\overrightarrow{v})+\rho ·\overrightarrow{f}$$

$\begin{array}{rl}& \mathsf{\text{Here }}\rho \mathsf{\text{ is the density, p is the (static) pressure, }}\overrightarrow{v}\mathsf{\text{ is the velocity of a fluid element in the flow, }}\frac{\partial \overrightarrow{v}}{\partial t}\mathsf{\text{ the partial derivative with respect to}}\\ & \mathsf{\text{time when the fluid element is fixed in place, · the (formal) scalar product with the Nabla operator }}\mathrm{\nabla }and\mathrm{\Delta }\mathsf{\text{ the Laplace operator.}}\end{array}$

To the left of the equals sign is the substantial acceleration of the fluid elements and the term formed with the Nabla operator represents their convective part. The vector f→ stands for a volume force density such as gravitation g→ or the Coriolis force, in each case related to the unit volume. The parameters η and γ are the dynamic viscosity and the first Lamé constant.

Please note that this is the already known Navier-Stokes equation plus a term which caters for the effects of being now a compressible fluid.

As it was mentioned before so far nobody was able to solve this equation analytically; only numerical procedures could be developed following proper approximation schemes by truncating polynomials, but it remains a numerical solution only and not a thorough one.

What can we do to make our life simpler now?

In order to make this very confusing equation a little more “digestible”, we resort to a common method in mathematics, the nondimensionalisation of the Navier-Stokes equation. We do not deal with this in detail, but apply it without questioning or even deriving the individual steps.

But a little bit of background will be helpful:

Nondimensionalisation is the partial or complete removal of dimensioned quantities such as units of measurement from a physical equation by a suitable substitution of all variables. With the help of dimensionless variables and dimensioned constants, effects resulting from the choice of the system of units can be eliminated and intrinsic constants of the system, such as characteristic lengths, times or frequencies, can be found. The technique is therefore suitable for simplifying systems of differential equations. That’s exactly what we need now!

The Navier-Stokes equations can be nondimensionalised with characteristic measures of the entire flow area for the length L, the velocity v_∞ and the density ρ_∞. This gives rise to the dimensionless quantities ie. we replace the variables (parameters without an asterisk) by these ones:

$$\begin{array}{rl}& {\overrightarrow{x}}^{\ast }=\frac{\overrightarrow{x}}{L}\\ & {\mathrm{\nabla }}^{\ast }=L\mathrm{\nabla }\\ & {\mathrm{\Delta }}^{\ast }=L^2\mathrm{\Delta }\\ & {\overrightarrow{v}}^{\ast }=\frac{\overrightarrow{v}}{v_{\mathrm{\infty }}}\\ & t^{\ast }=\frac{v_{\mathrm{\infty }}}{L}\\ & {\rho }^{\ast }=\frac{\rho }{{\rho }_{\mathrm{\infty }}}\\ & p^{\ast }=\frac{\rho }{{\rho }_{\mathrm{\infty }}{v}_{\mathrm{\infty }}^2}\\ & {\overrightarrow{f}}^{\ast }=\frac{\overrightarrow{f}L}{{\rho }_{\mathrm{\infty }}{v}_{\mathrm{\infty }}^2}\end{array}$$

This eventually leads to the dimensionless Navier-Stokes equation which looks very similar to the previous version but exclusively works with dimensionless (or scaled) variables:

$${\rho }^{\ast }·(\frac{\partial {\overrightarrow{v}}^{\ast }}{\partial t}+({\overrightarrow{v}}^{\ast }·{\overrightarrow{\mathrm{\nabla }}}^{\ast }){\overrightarrow{v}}^{\ast })=-{\overrightarrow{\mathrm{\nabla }}}^{\ast }{p}^{\ast }+\frac{1}{Re}·{\mathrm{\Delta }}^{\ast }{\overrightarrow{v}}^{\ast }+\frac{1}{3Re}\overrightarrow{\mathrm{\nabla }}·({\overrightarrow{\mathrm{\nabla }}}^{\ast }·\overrightarrow{v})+{\overrightarrow{f}}^{\ast }$$

with

$$Re=\frac{{\rho }_{\mathrm{\infty }}{v}_{\mathrm{\infty }}}{\eta }L$$

This is the Reynolds number, a dimensionless parameter, which characterizes the flow in terms of the ratio of inertial to shear forces. In the following we omit the indices which were just introduced to outline the nondimensionalization procedure.

The Reynolds number is thus defined as

$$Re=\frac{\rho v}{\eta }L$$ $\begin{array}{rl}& \mathsf{\text{where }}\rho \mathsf{\text{ is the density of fluid in }}\frac{kg}{m^3},\mathsf{\text{v is the mean flow velocity of the fluid relative to the body in }}\frac{m}{s},\mathsf{\text{ and L is the “characteristic }}\\ & \mathsf{\text{length” of the body in m, and η is the dynamic viscosity in}}\frac{kgm}{s}.\end{array}$

The characteristic length L, or reference length, is to be defined for the respective problem.

For flow bodies, the length of the body in the direction of flow is usually chosen. For resistance bodies, the width or height transverse to the direction of flow is usually taken as the characteristic length, for pipe flows the radius or diameter of the pipe and for flumes the depth or width at the flume surface.

The kinematic viscosity ν (Greek: nu) of the fluid differs from the dynamic viscosity η = ν ρ by the factor ρ. Hence the Reynolds number can also be expressed in terms of the kinematic viscosity ν:

$$Re=\frac{v}{\nu }L$$

If the Reynolds number exceeds a (problem-dependent) critical value, a previously laminar flow becomes susceptible to the smallest disturbances. Accordingly, for  Re > Re_critical a change from laminar to turbulent flow is to be expected.  In ideal fluids (incompressible, no friction) there is no viscosity and the Reynolds number is infinite because the quotient of velocity and viscosity becomes infinite.