Description and Derivation of the Navier-Stokes Equations

Summary

This educational video script delves into the Navier-Stokes equations, crucial for understanding fluid dynamics. It simplifies the complex equations by illustrating their physical meaning and mathematical derivation from Newton's second law. The script explains how these equations, which describe the motion of fluid particles, are derived from the balance of forces including gravity, pressure differences, and viscosity. It also clarifies the distinction between local and convective accelerations, and how the equations are applied to different directions. The script demystifies the intimidating equations, emphasizing their fundamental role in expressing the principle that the sum of forces equals mass times acceleration.

Takeaways

• 📚 The Navier-Stokes equations are fundamental in fluid dynamics, expressing the balance of forces acting on a fluid element in motion.
• 🔍 These equations are derived from Newton's second law, which states that the sum of forces equals mass times acceleration, applied to an infinitesimal fluid element.
• 🧮 The equations are comprised of terms representing forces due to gravity, pressure differences, and fluid viscosity, all on a per unit volume basis.
• 📐 The script focuses on the x-direction for simplicity, but the principles extend to y and z directions, resulting in three-dimensional equations.
• 🕒 The acceleration term in the equations is broken down into local acceleration (rate of change of velocity at a point) and convective acceleration (due to the motion of the fluid element).
• 🌊 A physical interpretation of convective acceleration is provided through the example of fluid flow through a constriction, illustrating how velocity gradients lead to acceleration or deceleration.
• 📈 The forces on a fluid element are detailed, including gravity, normal stresses, and shear stresses, which are crucial for establishing the differential form of the equations.
• 🔗 The script explains the transition from the force balance equations to the Navier-Stokes equations through algebraic manipulations and the use of constitutive relations for Newtonian fluids.
• ⚖️ The importance of the continuity equation is highlighted, showing how it simplifies the Navier-Stokes equations, especially for incompressible fluids.
• 🔍 The final form of the Navier-Stokes equations is presented, emphasizing that despite their complexity, they essentially encapsulate the principle of force balance in fluid dynamics.

Q & A

What are the Navier-Stokes equations?

The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of fluid substances, including the effects of viscosity and gravity.

How are the Navier-Stokes equations derived?

The equations are derived from applying Newton's second law to a fluid element, considering the forces due to gravity, pressure differences, and viscosity.

What does the term 'local acceleration' refer to in the context of the Navier-Stokes equations?

The 'local acceleration' term refers to the rate of change of velocity at a specific point in the fluid, which is the time derivative of the velocity component.

What is meant by 'convective acceleration' in the script?

Convective acceleration refers to the acceleration of a fluid element as it is carried along by the bulk motion of the fluid, which is represented by the spatial derivatives of the velocity components.

How does the script explain the physical meaning of the terms in the Navier-Stokes equations?

The script explains that the terms in the Navier-Stokes equations represent the sum of forces (gravity, pressure, and viscosity) acting on a fluid element, equated to the mass times the acceleration of that element.

What is the significance of the differential element in the Navier-Stokes equations?

The differential element is an infinitesimally small volume of fluid used to model the fluid's behavior. It allows the equations to be applied locally, making the analysis more general and applicable to any point in the fluid.

What role does the continuity equation play in the Navier-Stokes equations?

The continuity equation, which states that the sum of the spatial derivatives of the velocity components is zero for incompressible fluids, simplifies the Navier-Stokes equations by eliminating certain terms.

How does the script simplify the forces acting on a differential element of fluid?

The script simplifies the forces by considering gravity, normal stresses, and shear stresses acting on the faces of the differential element and then equating the sum of these forces to the mass times the acceleration of the fluid element.

What is the significance of the constitutive relations for a Newtonian fluid in the Navier-Stokes equations?

The constitutive relations relate the normal and shear stresses in the fluid to its viscosity and velocity profiles. They are essential for transforming the equations of motion into the Navier-Stokes equations.

How does the script demonstrate the transition from the equations of motion to the Navier-Stokes equations?

The script demonstrates this transition through a series of algebraic manipulations, including substitutions and differentiations, that incorporate the effects of gravity, pressure, and viscosity into the equations.

What is the final form of the Navier-Stokes equations presented in the script?

The final form of the Navier-Stokes equations presented in the script is an expression of the sum of forces (gravity, pressure, and viscosity) equal to the density of the fluid times its acceleration, including both local and convective components.

Outlines

00:00

🔍 Introduction to the Navier-Stokes Equations

The video script begins with an exploration of the Navier-Stokes equations, which are fundamental in fluid dynamics. These equations are presented in a way that emphasizes their physical meaning, showing that they are essentially a statement of Newton's second law, which equates the sum of forces to mass times acceleration. The script clarifies that the equations are written for an infinitesimally small fluid element, and it explains the three forces involved: gravity, pressure differences, and fluid viscosity. The script then focuses on the x-direction for simplicity, discussing the mathematical derivation of the equations using the chain rule and the concept of acceleration, which includes both local and convective components. The physical interpretation of convective acceleration is illustrated through an example of fluid flowing through a constriction, where the fluid element's velocity changes due to the constriction.

05:00

📐 Analyzing Forces and Stresses on a Fluid Element

In the second paragraph, the script delves into a more detailed analysis of the forces acting on a differential element of fluid. It discusses the forces due to gravity, normal stresses, and shear stresses, and how these forces are represented mathematically. The forces are broken down into components acting on the faces of the fluid element, and the script uses the notation Sigma and Tau to represent these stresses. The forces are then equated to the mass times the acceleration of the fluid element in the x-direction. The script simplifies the equation by canceling out terms and eventually arrives at a differential form that represents the sum of forces due to gravity, normal forces, and shear stresses. This leads to the equations of motion for a fluid, which are a precursor to the Navier-Stokes equations.

10:01

🧩 Deriving the Navier-Stokes Equations

The final paragraph of the script focuses on the derivation of the Navier-Stokes equations from the equations of motion. It mentions the need for constitutive relations to relate the stresses to the fluid's viscosity and velocity profiles, which are specific to Newtonian fluids. The script outlines a series of algebraic manipulations that lead to the Navier-Stokes equations. It emphasizes that the equations are a statement of the sum of forces equal to mass times acceleration, and it simplifies the equations by incorporating the local and convective components of acceleration. The script concludes by reiterating that despite their complexity, the Navier-Stokes equations can be summarized as an expression of Newton's second law for fluid dynamics.

Keywords

💡Navier-Stokes Equations

The Navier-Stokes Equations are a set of nonlinear partial differential equations that describe the motion of fluid substances. In the script, they are introduced as a fundamental representation of Newton's second law for fluid dynamics, stating that the sum of forces acting on a fluid element is equal to the mass times its acceleration. The equations are derived from considering the forces due to gravity, pressure differences, and fluid viscosity, and are essential for understanding fluid behavior in various contexts, such as weather prediction and engineering applications.

💡Newton's Second Law

Newton's second law of motion states that the force acting on an object is equal to the mass of that object multiplied by its acceleration. In the context of the video, this law is applied to fluid dynamics, where the forces on a fluid element include gravity, pressure, and viscous forces. The script explains how the Navier-Stokes equations are a direct application of this law to fluid systems, emphasizing the importance of understanding the acceleration of fluid elements in different directions.

💡Differential Element

A differential element refers to an infinitesimally small portion of a larger system, used in the analysis of fields such as fluid dynamics. In the script, the Navier-Stokes equations are written for a differential element of fluid, which simplifies the complex behavior of the fluid into a manageable mathematical model. This approach allows for the analysis of fluid properties at a local level, which is crucial for understanding how larger fluid systems behave.

💡Fluid Viscosity

Fluid viscosity is a measure of a fluid's resistance to deformation at a given rate. It is a key property in fluid dynamics that affects the fluid's flow characteristics, especially in relation to the shear stress. In the script, viscosity is one of the forces considered in the Navier-Stokes equations, and it is integral to understanding how different fluids react under various conditions, such as the flow of oil versus water.

💡Shear Stress

Shear stress is the component of stress that causes deformation in a fluid or solid. In the context of the video, shear stress is discussed as one of the forces acting on the faces of a differential element of fluid, contributing to the overall force balance described by the Navier-Stokes equations. The script explains how shear stresses at different faces of the fluid element are calculated and their importance in fluid dynamics.

💡Normal Stress

Normal stress is the stress component acting perpendicular to a surface. In the video script, normal stress is mentioned as the force acting on the left and right sides of a differential fluid element, which is crucial in the derivation of the Navier-Stokes equations. Understanding normal stress is essential for analyzing the forces that fluids exert on the boundaries of their containers or channels.

💡Acceleration

Acceleration in fluid dynamics refers to the rate of change of velocity of a fluid element. The script discusses how the acceleration of a fluid element is calculated by considering both local and convective components. This concept is central to the Navier-Stokes equations, as it relates directly to the forces acting on the fluid and how these forces result in changes in the fluid's velocity.

💡Convective Acceleration

Convective acceleration is the change in velocity of a fluid element as it moves through the fluid. In the script, it is described as part of the acceleration term in the Navier-Stokes equations, which accounts for the change in velocity due to the fluid's motion. The video uses the example of fluid flowing through a constriction to illustrate how convective acceleration can be positive or negative, depending on the direction of the velocity gradient.

💡Continuity Equation

The continuity equation in fluid dynamics is a conservation law that states the product of density and velocity is constant along a streamline for an incompressible fluid. In the script, the continuity equation is mentioned in the context of simplifying the Navier-Stokes equations, particularly when considering the sum of velocity gradients, which equals zero for incompressible fluids. This equation is fundamental to understanding fluid flow and ensuring mass conservation.

💡Constitutive Relations

Constitutive relations are mathematical equations that describe the relationship between material properties, such as stress and strain, in a material. In the context of the video, constitutive relations for a Newtonian fluid are mentioned as a way to connect the stresses in the fluid to its viscosity and velocity profiles. These relations are essential for transforming the general force balance into the specific form of the Navier-Stokes equations.

Highlights

The Navier-Stokes equations are derived from Newton's second law, expressing the balance of forces as mass times acceleration.

The equations are written for an infinitesimally small differential element of fluid, representing fluid dynamics at a local level.

Three forces are considered in the Navier-Stokes equations: gravity, pressure differences, and fluid viscosity.

The force terms are expressed per unit volume, emphasizing the local nature of the forces acting on the fluid.

The derivation begins by focusing on the x-direction to simplify the explanation, with the understanding that similar logic applies to y and z directions.

The acceleration of the fluid is described using the time derivative of the velocity component, incorporating the chain rule for partial derivatives.

Local acceleration is the rate of change of velocity at a point, while convective acceleration accounts for the movement of fluid elements.

A physical interpretation of convective acceleration is provided through the example of fluid flowing through a constriction.

The forces due to gravity are calculated by considering the mass of the fluid element and the gravitational acceleration component.

Normal and shear stresses on the faces of the fluid element are considered, contributing to the forces acting on the fluid.

The sum of forces is equated to the mass times the acceleration, leading to an expression that includes gravity, pressure, and viscous forces.

The derivation simplifies to a differential form as the volume of the fluid element approaches zero, resulting in the Navier-Stokes equations.

The continuity equation plays a crucial role, showing that the sum of the velocity gradients in all directions is zero for incompressible fluids.

Constitutive relations for a Newtonian fluid are used to relate stresses to the viscosity and velocity profiles, although not detailed in this screencast.

Algebraic manipulations are performed to transform the force balance equation into the familiar form of the Navier-Stokes equations.

The final form of the Navier-Stokes equations expresses the sum of forces due to gravity, pressure, and viscosity as equal to the mass times the fluid's acceleration.

The Navier-Stokes equations, despite their complexity, fundamentally describe the relationship between forces and acceleration in fluid dynamics.

Transcripts

these are the navier-stokes equations as

they're commonly written in this

screencast we examine their physical

meaning and perform a simple

mathematical derivation based on

Newton's second law while these

equations may look intimidating and

complicated to a lot of people all they

really are is a statement that the sum

of forces is equal to the mass times the

acceleration to make it a little bit

more apparent let's flip the equations

about the equal sign so what we have in

the first equation is the sum of forces

in the x-direction is equal to the mass

times the acceleration in the

x-direction in the second and third

equations of the some forces in the Y

and Z Direction equal to their

respective accelerations these equations

are written for a differential element

of fluid which is infinitesimally small

so as small as we can possibly imagine

the three forces we're concerned with of

forces due to gravity forces due to

differences in pressure and forces due

to the viscosity of the fluid keep in

mind that each of these terms is on a

per unit volume basis so typically we

would say the force due to gravity might

be the weight of something if I took the

weight and divided by the volume what

I'm left with is the density M over V

times gravity we see this occurring for

gravity most explicitly on the

right-hand side of the equation if we

have mass times acceleration if we were

to divide that by the volume of the

fluid we would be left with the density

times the acceleration so we're looking

at the sum of forces is equal to MA on a

per unit volume basis for this

screencast let's deal only with the

x-direction the same mathematics would

apply for the Y in the Z directions but

we'll leave it with two the X direction

to save time let's examine the

right-hand side of the X component the X

component of velocity for a fluid we'll

call it lower case U and strictly

speaking you could be a function of X Y

Z and time the X component of

acceleration for the fluid is equal to

the time derivative of U is a derivative

of U with respect to time but because U

is not simply a function of time it's

also a function of XY and Z we need to

use the chain rule to perform this

differentiation

so I'm going to do the partial you with

respect to time plus D u DX times DX DT

Plus D u dy dy DT Plus D u DZ times DZ

DT if I use the definition that DX DT

simply equal to u dy DT is equal to

lowercase V and DZ DT is equal to

lowercase W we're left with something

that looks an awful lot like the

right-hand side of the equation above

the first term on the right hand side is

known as a local acceleration and the

remaining three terms are known as the

convective acceleration to think about

what this means physically let's

consider some fluid that's flowing

steadily from left to right through this

two dimensional constriction and let's

consider a differential element of fluid

which will be infinitesimally small so

I'll make it a real small cube and let's

place it right here to begin with if you

think about the motion of this element

to fluid as it flows through it's

flowing steadily from left to right it's

slow in this region but then it begins

to accelerate because of the

constriction where the fluid is moving

very rapidly here and now when it

reaches the right hand side it slows

down again and it recovers to a steady

velocity as it moves towards the exit if

the flow of fluid is at steady state

then the velocity of the differential

element to fluid at points one two and

three we'd see no change over time but

let's examine the constricting region

we've got U is a positive quantity it's

moving from left to right and D U DX is

also a positive quantity so this term of

the convective acceleration is greater

than zero so we see in the highlighted

region that the fluid element is

accelerating from left to right

conversely in the expanding region

although U is positive D u DX is less

than zero it's a negative quantity the

fluid is slowing down within that region

so the convective acceleration term is

less than zero or it would be the

acceleration would be to the left in the

highlighted read

let's examine the forces acting on the

differential element to fluid a little

bit more carefully call this point XY Z

in our differential element two flew it

has length DX a height dy and a depth DZ

the first force will consider is gravity

and typically when you draw a free body

diagram gravity will be acting downward

in the Y direction but let's do an

arbitrary case where a component of

gravity could for example could act in

the X direction so the force due to

gravity is the mass of our differential

element two fluid times the X component

of gravity let's rewrite the mass is

equal to the density times the volume of

our differential element two fluid DX dy

DZ so again have the mass times gravity

in the X direction let's examine forces

acting on the left and the right sides

of our differential element we could

have a normal stress acting directly to

the right on the right face and a stress

acting to the left outward from the left

face the notation we'll use for these

stresses is Sigma xx and we're going to

evaluate Sigma xx at X plus a distance

DX and on the left face we have Sigma X

X evaluated at X on the top and the

bottom faces we could have a shear

stress acting to the right on the top

face and a shear stress acting to the

left on the bottom face the notation

we'll use for these stresses is tau YX

evaluated at y plus dy for the top of

the cube and tau YX evaluated at Y for

the bottom of the cube and additionally

we could have stresses acting to the

right on the front of the cube and a

stress acting to the left at the back of

the cube for the back of the cube we'll

use the notation tau ZX evaluated at Z

and at the front of the queue we'll use

the notation tau ZX evaluated at Z plus

DZ so continue writing the sum forces in

x-direction I've got any X component of

gravity plus a normal stress Sigma X X

evaluated at X plus DX multiplied by the

surface area of the right side of the

cube which is equal to dy DZ because it

acts to the left I'll subtract Sigma X X

evaluated at X multiplied by the same

area dy DZ

then I'll add the force due to the shear

stress at the top of the cube tau YX

evaluated at y plus dy multiplied by its

area which is DX times DC I'll subtract

the shear stress acting on the bottom

face multiplied by its area then I'll

add the force due to the shear stress at

the front of the face and subtract off

the shear stress acting at the rear face

the sum of these forces will equal the

mass times the acceleration in the X

direction or I could write mass is equal

to Rho times DX dy DZ times the

acceleration in the X direction you have

cleaned up that equation divided by the

volume of the differential element and

what I immediately see is that DX dy DZ

Zwilling out and in some terms if I

simplify a dy and DZ will cancel out DX

cancels out as well as DZ in this term

and so forth as I cancel out terms as I

continue to simplify am left on the neck

with this expression and in the limit of

DX dy and DZ are approaching zero this

turns into a differential form and the

resulting equation represents the sum of

forces in the x-direction due to gravity

due to the normal forces acting on the

left and the right side of the

differential element the shear stresses

acting on the top and the bottom of the

element and the shear stresses acting in

the front in the rear faces of the

element so these are equal to the

density times the acceleration of the

differential element in the X direction

and if I expand the

raishin into its local and convective

components were left with the an

expression on the right hand side if I

did the same analysis for the Y and the

Z directions I would come up with these

three equations which are known as the

equations of motion for a fluid but to

get from these equations to the

navier-stokes equations we need a way to

relate the normal and the shear stresses

to the viscosity of the fluid and the

velocity profiles and this is done using

these equations which are the

constitutive relations for a Newtonian

fluid which we won't get into here but

if we accept them as being true for this

screencast becomes a series of algebraic

manipulations to arrive at the

navier-stokes equation the first

substitution into the left hand side

gives me the expression at the bottom if

I differentiate the three terms I'm left

with this expression and some additional

manipulations leaves me with this

expression in which I've split this term

into two parts and I've switched the

order in which I differentiate these two

terms so I'm switching dy/dx and DD and

this expression as I continue to

simplify and rearrange terms I'm left

with this expression but what's

interesting is the sum of these three

terms D u DX plus DV dy plus DW DZ is

equal to zero by way of the continuity

equation so that whole term on the right

is identically zero for an

incompressible fluid so what I'm left

with the sum is three forces the force

due to gravity force due to any pressure

differences in the fluid plus all the

forces due to viscosity the sum of these

three is equal to the density of the

fluid multiplied by the X component of

its acceleration and expanding a X out

into its local and convective components

of acceleration I've just arrived at the

first the X component for the

navier-stokes equation so we could do

the exact same thing for the Y and the Z

directions it will arrive at the

navier-stokes equations for those

directions as well if I flip the order

of the equations you're left with the

navier-stokes equations as you'll

commonly see them

and although they may look intimidating

and complicated to begin with if someone

asks you to sum up what the

navier-stokes equations are in words

just simply tell them they're an

expression of the sum of forces is equal

to the mass times the acceleration