navier stokes equations

DERIVATION AND EXPLORATION OF THE

NAVIER-STOKES EQUATIONS Karina Zala

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Presentation Notes

Derive the Navier-Stokes equations and explain it clearly with necessary assumptions made in analysis.

HISTORICAL APPROACH TO THE NAVIER-STOKES EQUATIONS

Claude-Louis Navier

Sir George Gabriel Stokes

• Describes the motion of fluid flow • Based on Newton’s 2nd Law:

Sum of all forces = time rate of change of momentum

F = m a Sir Isaac Newton

• Assumes stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow

• Used in CFD simulations. • Describes the physics of many aspects of fluid

phenomena.

Presenter

Presentation Notes

These equations describe the physics of many things of academic and economic interest. Are known to be used to model the Weather ocean currents water flow in a pipe air flow around a wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars the study of blood flow the design of power stations the analysis of pollution, and many other applications The Navier–Stokes equations assume that the fluid being studied is a continuum Meaning, the matter in the body is continuously distributed and fills the entire region of space it occupies Historically, the Navier-Stokes equations are JUST the Momentum Equations however in modern literature the Navier-Stokes have been expanded to include the ENTIRE system of flow equations, being Continuity Momentum Energy Since, my colleagues are covering mass and energy, this presentation is solely focused on the original Navier-Stokes equations, being the Momentum Equations. Newton’s Laws An object at rest tends to stay at rest, and an object in motion tends to stay in motion, with the same direction and speed. The acceleration of an object produced by a net (total) applied force is directly related to the magnitude of the force, the same direction as the force, and inversely related to the mass of the object For every action (force) there is an equal and opposite reaction (force) Force (N) any influence that causes an object to undergo a certain change, either concerning its movement, direction, or geometrical construction i.e. a force can cause an object with mass to change its velocity (which includes to begin moving from a state at rest) like accelerate or to deform a flexible object. Can be push or pull Has magnitude and direction Mass (kg) is the resistance of a body during acceleration when there is a force applied mass = Force / acceleration Also is the amount of matter within an object Mass = density x volume� Acceleration ( 𝑚/𝑠 2 ) =F/m Rate at which the velocity of an object changes with time ∆𝑣 ∆𝑡 Has magnitude and direction

HOW CAN YOU APPLY THE NAVIER-STOKES EQUATIONS?

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CONTROL VOLUME OF THE FLUID ELEMENT

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Presentation Notes

Control Volume is a volume that is either fixed in space i.e. this room that we are in now, or a volume that moves in constant velocity, i.e. when you are travelling in a vehicle. In a CV, there are many particles within where one tiny particle is being focused on....

FORCES ACTING WITHIN A FLUID ELEMENT

FORCE

Surface

Pressure

Normal Stress

Shear Stress

Body

Gravity

Electric

Magnetic

Acts at a distance directly on the volumetric mass of the entire fluid

element

Body Force (BF): 𝐁𝐁 = 𝝆𝒇𝒙 𝒅𝒙𝒅𝒅𝒅𝒅

Always acts normal to the surface

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Presentation Notes

Normal Stress relates to the time rate of change of volume (normal) Shear Stress relates to the time rate of change of the shearing deformation (sliding) Both are dependent on velocity gradient Velocity gradient is defined as the change in velocity per unit of distance along the vertical velocity curve. Body Force in x-direction per unit mass. (General term)

GOVERNING EQUATIONS

1. Pressure (P) [Pa] or [bar]

2. Velocity for x-direction (u) [m/s]

3. Velocity for y-direction (v) [m/s]

4. Velocity for z-direction (w) [m/s]

5. Density (ρ) [kg/m3]

6. Temperature (T) [K][°C]

7. Viscosity (μ) [Ns/m2, Pa.s or kg/ms

Seven unknowns in determining the flow in a fluid

Seven equations: 1. Conservation of Mass

Momentum 2. 𝐹𝑥 = 𝑚𝑎𝑥 3. 𝐹𝑦 = 𝑚𝑎𝑦 4. 𝐹𝑧 = 𝑚𝑎𝑦 5. Perfect Gas 6. Energy 7. Stoke’s Hypothesis

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Presentation Notes

Pressure = 𝐹𝑜𝑟𝑐𝑒 𝐴𝑟𝑒𝑎 conversion 1bar = 100,000 Pascals is the ratio of force to the area over which that force is disturbed applied in a direction perpendicular to the surface of an object Velocity = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑖𝑚𝑒 Speed in a given direction; Distance per unit time; rate of change of position Changing velocity (speed, direction or both) means acceleration Density = 𝑚𝑎𝑠𝑠 𝑉𝑜𝑙𝑢𝑚𝑒 a physical property of matter, as each element and compound has a unique density associated with it. the measure of the relative "heaviness" of objects with a constant volume Refers to how closely "packed" or "crowded" the material appears to be Temperature Degree of heat present in the fluid Viscosity → 𝜏=𝜇 𝜕𝑢 𝜕𝑥 where 𝜏 = Force/ area and 𝜕𝑢 𝜕𝑥 = the local shear velocity measure of its resistance to gradual deformation by shear stress or tensile stress the state of fluid being thick, thin, sticky, and semi-fluid in consistency, due to internal friction.

GOVERNING EQUATIONS

Governed for unsteady 3D compressible viscous flow. Non-conservation form

Substantial derivatives, represented by 𝐷𝐷𝐷

Physically, the time rate of change following a moving fluid

(Dynamic Fluid)

Conservation form Local derivatives, represented by 𝜕

𝜕𝐷

Physically, the time rate of change at a fixed point (Static Fluid)

FLUID ELEMENT : X DIRECTION

𝜏𝑥𝑥

Plane Direction

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Presentation Notes

Pressure is always pushing directly into the element Green an Blue are shear stresses – sliding on the surface Green top – tugging action; pull in +ve direction Green bottom – dragging action; push in –ve direction Blue – viscous stresses Red are normal stress – normal to surface, and pulling the element; viscous action, acts as suction retarding the motion Assume that the velocity increases in the positive direction. This means as the element moves from the origin into the x-direction the u-velocity increases. i.e. (Green and Blue) Top and bottom: Velocity below is slower than velocity above therefore, the velocity above pulls the flow along i.e. (Red) Sides: Velocity on the right is higher than velocity on the left therefore, sucks the element in the direction. + 𝜕 𝜏 𝑥𝑥 𝜕𝑥 𝑑𝑦 is the change over a certain distance that is added to the fluid element

BUILDING THE NAVIER-STOKES EQUATIONS SURFACE FORCE (SF)

SF = 𝑝 − 𝑝 + 𝜕𝜕𝜕𝑥𝑑𝑑 𝑑𝑑𝑑𝑑 + 𝜏𝑥𝑥 + 𝜕𝜏𝑥𝑥

𝜕𝑥𝑑𝑑 − 𝜏𝑥𝑥 𝑑𝑑𝑑𝑑

+ 𝜏𝑦𝑥 +𝜕𝜏𝑦𝑥𝜕𝑑

𝑑𝑑 − 𝜏𝑦𝑥 𝑑𝑑𝑑𝑑 + 𝜏𝑧𝑥 +𝜕𝜏𝑧𝑥𝜕𝑑

𝑑𝑑 − 𝜏𝑧𝑥 𝑑𝑑𝑑𝑑

Presenter

Presentation Notes

Add +ves, and subtract –ves

BUILDING THE NAVIER-STOKES EQUATIONS SURFACE FORCE (SF)

Or simplified:

SF = −𝜕𝑝𝜕𝑑

+𝜕𝜏𝑥𝑥𝜕𝑑

+𝜕𝜏𝑦𝑥𝜕𝑑

+𝜕𝜏𝑧𝑥𝜕𝑑

𝑑𝑑𝑑𝑑𝑑𝑑

Presenter

Presentation Notes

Simplify through cancellation dxdydz are common within all terms

BUILDING THE NAVIER-STOKES EQUATIONS FORCE (𝐹𝑥) = SURFACE + BODY

Surface Force Body Force

𝐹𝑥 = −𝜕𝑝𝜕𝑑




𝑑𝑑𝑑𝑑𝑑𝑑 + 𝜌𝑓𝑥 𝑑𝑑𝑑𝑑𝑑𝑑

↪ −𝜕𝜕𝜕𝑥

+ 𝜕𝜏𝑥𝑥𝜕𝑥

+ 𝜕𝜏𝑦𝑥𝜕𝑦

+ 𝜕𝜏𝑧𝑥𝜕𝑧

+ 𝜌𝑓𝑥 𝑑𝑑𝑑𝑑𝑑𝑑

Mass = matter within a CV

𝜌 · 𝑑𝑑𝑑𝑑𝑑𝑑 volume

BUILDING THE N-S EQUATIONS MASS AND ACCELERATION

Acceleration = velocity increase w.r.t time

𝑑 = 𝐷𝐷𝐷𝐷

; 𝑑 =𝐷𝐷𝐷𝐷

; 𝑑 =𝐷𝐷𝐷𝐷

NAVIER-STOKES EQUATIONS IN NON-CONSERVATION FORM (DYNAMIC FLUID)

For x-direction:

𝜌𝐷𝐷𝐷𝐷

= −𝜕𝑝𝜕𝑑




+ 𝜌𝑓𝑥

For y-direction:


= −𝜕𝑝𝜕𝑑

+𝜕𝜏𝑥𝑦𝜕𝑑

+𝜕𝜏𝑦𝑦𝜕𝑑

+𝜕𝜏𝑧𝑦𝜕𝑑

+ 𝜌𝑓𝑦

For z-direction:


= −𝜕𝑝𝜕𝑑

+𝜕𝜏𝑥𝑧𝜕𝑑

+𝜕𝜏𝑦𝑧𝜕𝑑

+𝜕𝜏𝑧𝑧𝜕𝑑

+ 𝜌𝑓𝑧

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Presentation Notes

Substantial derivative, which means it is physically the time rate of change following a moving fluid element therefore is the non-conservation form.

NAVIER-STOKES EQUATIONS CONSERVATION FORM (STATIC FLUID)

For x:


= 𝜌𝜕𝐷𝜕𝐷

+ 𝜌𝑉 ∙ 𝛻𝐷

Expanding the derivative:

𝜕(𝜌𝐷)𝜕𝐷

= 𝜌𝜕𝐷𝜕𝐷

+ 𝐷𝜕𝜌𝜕𝐷

Rearranging the derivative:

𝜕(𝜌𝜌)𝜕𝐷

− 𝐷 𝜕𝜌𝜕𝐷

= 𝜌 𝜕𝜌𝜕𝐷

Presenter

Presentation Notes

Changing substantial derivative to local derivative because determining the conservation form where fluid is stationary Adding divergence term to measure the magnitude of the fluid element at a single point 𝑉∙𝛻= 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 + 𝜕𝑤 𝜕𝑧 Time rate of change of the volume of a moving fluid element , per unit volume Conservation sometimes also known as divergence form Expanding local derivative Rearrange the expanded local derivative

BUILDING THE NAVIER-STOKES EQUATIONS CONSERVATION FORM (STATIC FLUID)

Recalling divergence product: 𝛻 ∙ 𝜌𝐷𝑉 = 𝐷𝛻 ∙ 𝜌𝑉 + (𝜌𝑉) ∙ 𝛻𝐷

Or 𝛻 ∙ 𝜌𝐷𝑉 − 𝐷𝛻 ∙ 𝜌𝑉 = 𝜌𝑉 ∙ 𝛻𝐷 Substituting:


=𝜕(𝜌𝐷)𝜕𝐷

− 𝐷𝜕𝜌𝜕𝐷

− 𝐷𝛻 ∙ 𝜌𝑉 + 𝛻 ∙ 𝜌𝐷𝑉

=𝜕(𝜌𝐷)𝜕𝐷

− 𝐷𝜕𝜌𝜕𝐷

+ 𝛻 ∙ 𝜌𝑉 + 𝛻 ∙ 𝜌𝐷𝑉

Finally:

𝜌 𝐷𝜌𝐷𝐷

= 𝜕(𝜌𝜌)𝜕𝐷

+ 𝛻 ∙ 𝜌𝐷𝑉

Presenter

Presentation Notes

Divergent term measures the magnitude of a fluid element at a given point, in terms of a signed scalar. It represents the volume density of the outward flux (flow of a physical property in space, frequently also with time variation) of a vector field from an infinitesimal volume around a given point. Recalling the vector for divergence of the product of a scalar (has magnitude not direction) times a vector(quantity having direction and magnitude) Term in brackets is continuity equation, hence = 0.

With each direction there is ONE normal stress and TWO shear stresses acting on the fluid element.

𝜏𝑥𝑥 𝜏𝑥𝑦 𝜏𝑥𝑧𝜏𝑦𝑥 𝜏𝑦𝑦 𝜏𝑦𝑧𝜏𝑧𝑥 𝜏𝑧𝑦 𝜏𝑧𝑧

UNDERSTANDING STRESS IN A FLUID

UNDERSTANDING STRESS IN A FLUID

𝜏𝑥𝑥 = 𝜆 𝛻 ∙ 𝑉 + 2𝜇 𝜕𝜌𝜕𝑥

𝜏𝑦𝑦 = 𝜆 𝛻 ∙ 𝑉 + 2𝜇 𝜕𝑣𝜕𝑦

𝜏𝑧𝑧 = 𝜆 𝛻 ∙ 𝑉 + 2𝜇 𝜕𝑤𝜕𝑧

𝜏𝑥𝑦 = 𝜏𝑦𝑥 = 𝜇 𝜕𝑣𝜕𝑥

+ 𝜕𝜌𝜕𝑦

𝜏𝑥𝑧 = 𝜏𝑧𝑥 = 𝜇 𝜕𝜌𝜕𝑧

+ 𝜕𝑤𝜕𝑥

𝜏𝑦𝑧 = 𝜏𝑧𝑦 = 𝜇 𝜕𝑤𝜕𝑦

+ 𝜕𝑣𝜕𝑧

Assumption made by Stokes:

𝜆 = −23𝜇

where: 𝜇 = dynamic (shear) viscosity coefficient 𝜆 = second viscosity coefficient

Normal Stresses

Shear Stresses

Presenter

Presentation Notes

Dynamic (shear) Viscosity: its resistance to shearing (sliding) flows, where layers next to each other move parallel to each other with different speeds. Also called absolute viscosity SI units Pa.s or Ns/m^2 or kg/ms Second Viscosity: When a compressible fluid is compressed or expanded evenly, without shear, it may still exhibit a form of internal friction that resists its flow. These forces are related to the rate of compression or expansion. Mean free path: the average distance travelled by a particle between collisions with other particles. Related to Stoke’s Hypothesis: 𝜏=𝜇 𝜕𝑢 𝜕𝑥 μ and λ are proportionality constants associated with the assumption that stress depends on strain linearly λ produces a viscous effect associated with volume change, is very difficult to determine When taken nonzero, the most common approximation is λ ≈ - ⅔ μ (𝜆+ 2 3 𝜇)

STOKES HYPOTHESIS

If pressure is defined as:

𝑃� ≡ −13

(𝜏𝑥𝑥 + 𝜏𝑦𝑦+𝜏𝑧𝑧) = 𝑝 − 𝜆 + 23𝜇 𝛻 ∙ 𝑉

Unless 𝜆 + 23𝜇 𝑜𝑜 (𝛻 ∙ 𝑉) = 0

Mean pressure ≠ thermodynamic pressure

Therefore, Stokes assumed that 𝜆 + 23𝜇 =0

So transposing to make λ the subject,

𝜆 = −23𝜇

For x: 𝜕(𝜌𝐷)𝜕𝐷

+𝜕(𝜌𝐷𝐷)𝜕𝑑



= −𝜕𝑝𝜕𝑑

+ 𝜕𝜕𝑥

− 23𝜇𝛻 ∙ 𝑉 + 2𝜇 𝜕𝜌

𝜕𝑥+ 𝜕

𝜕𝑦𝜇 𝜕𝑣

𝜕𝑥+ 𝜕𝜌

𝜕𝑦+ 𝜕

𝜕𝑧𝜇 𝜕𝜌

𝜕𝑧+ 𝜕𝑤

𝜕𝑥

+𝜌𝑓𝑥

BUILDING THE NAVIER-STOKES EQUATIONS

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Presentation Notes

With each direction there is one normal stress and two shear stress acting on the fluid element. Force = Surface force ( -Pressure gradient + Normal stress + Shear stresses) + Body force

For y: 𝜕(𝜌𝐷)𝜕𝐷




= −𝜕𝑝𝜕𝑑

+ 𝜕𝜕𝑥

𝜇 𝜕𝑣𝜕𝑥

+ 𝜕𝜌𝜕𝑦

+ 𝜕𝜕𝑦

− 23𝜇𝛻 ∙ 𝑉 + 2𝜇 𝜕𝑣

𝜕𝑦+ 𝜕

𝜕𝑧𝜇 𝜕𝑤

𝜕𝑦+ 𝜕𝑣

𝜕𝑧

+𝜌𝑓𝑦



For z: 𝜕(𝜌𝐷)𝜕𝐷




+= −𝜕𝑝𝜕𝑑

+ 𝜕𝜕𝑥

𝜇 𝜕𝜌𝜕𝑧

+ 𝜕𝑤𝜕𝑥

+ 𝜕𝜕𝑦

𝜇 𝜕𝑤𝜕𝑦

+ 𝜕𝑣𝜕𝑧

+ 𝜕𝜕𝑧

− 23 𝜇𝛻 ∙ 𝑉 + 2𝜇 𝜕𝑤

𝜕𝑧

+𝜌𝑓𝑧

NAVIER-STOKES EQUATIONS CONSERVATION FORM (STATIC FLUID)

For x-direction: 𝜕(𝜌𝐷)𝜕𝐷

+ 𝛻 ∙ (𝜌𝐷𝑉) = −𝜕𝑝𝜕𝑑




+ 𝜌𝑓𝑥

For y-direction: 𝜕(𝜌𝐷)𝜕𝐷

+ 𝛻 ∙ (𝜌𝐷𝑉) = −𝜕𝑝𝜕𝑑

+𝜕𝜏𝑥𝑦𝜕𝑑

+𝜕𝜏𝑦𝑦𝜕𝑑

+𝜕𝜏𝑧𝑦𝜕𝑑

+ 𝜌𝑓𝑦

For z-direction: 𝜕(𝜌𝐷)𝜕𝐷

+ 𝛻 ∙ (𝜌𝐷𝑉) = −𝜕𝑝𝜕𝑑

+𝜕𝜏𝑥𝑧𝜕𝑑

+𝜕𝜏𝑦𝑧𝜕𝑑

+𝜕𝜏𝑧𝑧𝜕𝑑

+ 𝜌𝑓𝑧

Presenter

Presentation Notes

Contains terms that include divergence of some quantity 𝛻∙ 𝜌𝑉 on the left hand side (also known as the divergence form). For incompressible flow the term 𝛻∙V = 0, therefore is eliminated from the equation

EXAMPLE

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navier stokes equations

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