navier stoke equation and reynolds transport theorem
TRANSCRIPT
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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
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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