couette flow
TRANSCRIPT
Couette Flow
BY
VIRENDRA KUMAR
PHD PURSUING (IIT DELHI)
IntroductionIn fluid dynamics, Couette flow is the laminar flow of a viscous
fluid in the space between two parallel plates, one of which is moving relative to the other.
The flow is driven by virtue of viscous drag force acting on the fluid and the applied pressure gradient parallel to the plates.
This kind of flow has application in hydro-static lubrication, viscosity pumps and turbine.
The present analysis can be applied to journal bearings, which are widely used in mechanical systems.
When the bearing is subjected to a small load, such that the rotating shaft and bearing remain concentric, the flow characteristic of the lubricant can be modeled as flow between parallel plates where the top plate moves at a constant velocity.
Journal bearing
Navier-Stokes Equation: Cartesian Coordinates
Continuity equation for 3-D flow
X-momentum
Y-momentum
Z-momentum
ππππ‘ +
ππ π₯ ( Οπ’)+ ππ π¦ (Οπ£ )+ ππ π§ ( Οπ€)=0
π (ππ’ππ‘ +π’ ππ’π π₯ +π£ ππ’π π¦ +π€ ππ’π π§ )=β πππ π₯ +πππ₯+π (π2π’ππ₯2 + π
2π’π π¦2 +
π2π’π π§ 2 )+π3 π
π π₯ (ππ’ππ₯ + ππ£π π¦ + ππ€π π§ )
Ο+
Ο+
ππ’ππ₯ +
ππ£π π¦ +
ππ€π π§ =0 Continuity equation for study incompressible flow
Analytical solution oF Couette flow
means
Now Steady Navier-Stroke equation can be reduce to
Invoking 0 0
Ο X-momentum
πππ π₯=π( π
2π’π π¦ 2 )
We choose to be the direction along which all fluid particles travel, and assume the plates are infinitely large in z-direction, so the z-dependence is not there.
0 0 0 0 00 0
πππ π¦=
πππ π§=0 meansπ=π (π₯ )ππππ¦
β’ The governing equation is :
π’= 12πππππ₯ π¦
2+πΆ1 π¦+πΆ2
The boundary conditions are:
After invoking boundary conditions:
Where P is non-dimensional Pressure gradient.
sπ ,π’= π¦hπ β h
2
2π βπππ π₯ β
π¦h (1β π¦h )
π’π=
π¦hβ h2
2ππ βπππ π₯ β
π¦h (1β π¦h ) Let P
The velocity profile in non-dimensional form
β’ when the equation reduced to:
(simple couette flow )
β’ It can be produced by sliding a parallel plate at constant speed relative to a stationary wall.
Fig. Simple couette flow
β’ For simple shear flow, there is no pressure gradient in the direction of the flow.
The velocity profiles for various P β’ For P < 0, the fluid motion created
by the top plate is not strong enough to overcome the adverse pressure gradient, hence backflow (i.e., u/U is negative) occurs at the lower-half region.
β’ For P>0, the fluid motion created by top plate is enough strong to overcome the adverse pressure gradient, hence u/U is +ve over the whole gap.
Velocity Profiles
Maximum and minimum velocity and itβs locationβ’ For maximum velocity :
β’ It is interesting to note that maximum velocity for P=1 occurs at y/h =1 and equals to U. For P>1, the maximum velocity occurs at a location y/h<1.
β’ This means that with P>1, the fluid particles attain a velocity higher than that of the moving plate at a location somewhere below the moving plate.
β’ For P=-1 the minimum velocity occurs, at y/h=0. For P<-1, the minimum velocity occurs at allocation y/h>1, means occurrence of back flow near the fixed plate.
The Max. velocity : For P β₯ 1The Min. velocity : For P β€ 1
Volume flow rate and average velocity
β’ The volume flow rate per unit width is:
π’ππ£π=( 12+ π
6 )π
β’ The Average velocity:
β’ For P=-3, volume flow rate (Q) and average velocity uavg=0
Shear stress distribution
β’ By invoking Newtonβs law of viscosity:
β’ In the dimensionless form, the shear stress distribution becomes
hπππ=1+π (1β 2 π¦
h )β’ Shear stress varies linearly with the distance from the
boundary.
β’ For P=0, Shear stress remains constant across the flow passage:
β’ At y=h/2, i.e., at the center of the flow passage, shear stress is independent of pressure gradient (P).
Force, Torque and Power
ΒΏππ π·π
60 π‘ βππ·πΏ=ππ 2π·2ππΏ
60 π‘
πππππ’πππππ’πππππ‘πππ£ππππππ hπ‘ ππ£ππ πππ’π ππππππ‘=π£ππ πππ’π πππ ππ π‘ππππΓ π·2
π=ππ 2π·3ππΏ
120 π‘
πππ€ππ πππ πππππππππ£ππππππππ hπ‘ ππ£ππ πππ’π πππ ππ ππ‘ππππ=π βπ
πππ€ππ=ππ3π·3π2 πΏ360 0 π‘ watts
Thanks