navier stoke equation and reynolds transport theorem

CVQ301Seminar course

NAVIER STOKES AND REYNOLDS TRANSPORT THEOREM

Supervisor Khemchand gurjarProf. Rakesh khosa 2013CE10351

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Introduction

• Material region – The fluid matter itself is define as the material

region • Balance law’s have two properties – They can’t be proved – They are stated for matter

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Case of fluid

• Problem with fluids– Fluid matter is not easily identifiable.

• Need to recast law’s for fluid• Two main assertions – Continuum approximation : the fluid matter is

assumed to be continuous not going down to atom or molecules.

– Nonprorability of laws: same balance laws applied.

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Vector Operations

• Del Operator :

• Laplacian Operator :

• Gradient :

• Vector Gradient : • Divergence : • Directional Derivative :

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Navier stokes equation

• For a small fluid element rate of change of momentum (Newton’s second law)

• The various forces acting on this element are– Body forces– Surface forces

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Surface forces

yxyx y x z

y

xxxx x y z

x

xx y z

yx x z

Y

X

Z

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Formulating The Forces

• Net forces along X direction

(Normal stress) (Shear stress) (Body force)

So combining the force and acceleration terms we get :

For x direction

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Continuity Equation

• Continuity equation for incompressible flow.

– Here u is velocity vector– The differential form is

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Navier stokes equation

• Kinematic viscosity = υ • Density = ρ• Velocity vector = u• Pressure = p• Body force = f

Now in our previous equation putting the values of shear and normal stress we get:

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Understanding Terms

Advection term : Diffusion term : Pressure term :

Body force term :

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Navier-Stokes equations for Newtonian fluid

Continuity equation for incompressible flow:

Navier-Stokes equation for incompressible flow : In x- direction:

In y- direction

In z- direction:

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Application of Navier Stoke Equation

• It is used in pipe flow problems.• Parallel plate fluid flow is also solved using this

equation.• Boundary layer equations are derived using

navier-stokes.

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Parallel Plate Flow From continuity equation

Assuming h<<<<LAssuming 2D ,w=0

Assuming fully developed , V=Vwall=0

Y-momentum equation

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Parallel Plate Flow

• As V=0 We get P=P0–ρgy• Now for this is not a function of y.• X-momentum eqn:

The flow being fully developed,steady,2-D and V=0;gx =0 we get

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Parallel Plate Flow

• The LHS is not a function of y but u is only a function of y.

• For the equation to hold both have to be independently constant.

• Integrating the equation and setting the boundary condition v(0)=v(h)=0 , we get

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Example : Skiing

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Reynolds Transport Theorem• All fluid laws are applied to system and a

system has to be consisting of mass.• Reynolds transport theorem however helps us

to change to control volume approach from system approach.

• Let B is termed an extensive property, and b is an intensive property.

– So we get B=mb

BASICS

• The extensive property for a system is :

• The rate of change of B for a system is:

• Now for a control volume

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DERIVATION

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DERIVATION

• Initially assuming system coincides with the CV we get :

• The rate of change of B for the system is:

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DERIVATION

• Now simplifying terms we get:

• And the other term as

• So the final form of equation comes to be

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Applications

• The applications of RTT can be applied in various fields.

• Taking B our extensive property as mass we can get the mass balance equation.

• Setting B as p we get the momentum equation from which we can get force acting on our control volume system.

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Flow in Bucket From Tap• Let the velocity with which water flows from

the tap is v0 and the area of cross-section of tap is a and that of bucket is A.

• We need to find the rate at which height of water increases in the bucket or V.

• Taking a fixed CV and B=m. = 0 ; = ρ(A-a)V ; = - ρav0

Finally we get V= av0 / (A-a)

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Assumption of RTT

• The fluid is consider as continuum matter.• The dervation of RTT is for a fixed CV but can

be extend for a moving CV.

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Advantages of RTT

• We can solve problem by identifying a control volume which is easy enough then identifying a system.

• It is a general theorem and can be applied for any extensive property.

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Disadvantages of RTT

• That can not be use in non inertial frame.• The first term is at times difficult to calculate.• Identify a suitable control volume is a problem.

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Advantages of Navier Stoke Equation

• Equation is based on simple newton’s second law so therefore it is not a empirical equation .

• The terms in the equation are velocity ,pressure and body force which can be easily determine for a fluid so therefore the equation are fairly simple to use.

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Assumption of Navier Stoke Equation

• Continuum approximation • Viscosity is independent of shear rate • Stress tensor invariant at the interface of two

fluid. • Assumption on boundary condition – Determine the nature of solution.– Boundary condition are no slip condition u(0)=u(h)=0

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References

• Fox and McDonald's Introduction to Fluid Mechanics.

• Incompressible Flow, Ronald L. Panton.

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Thank you