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
๐๐๐๐ก +
๐๐ ๐ฅ ( ฯ๐ข)+ ๐๐ ๐ฆ (ฯ๐ฃ )+ ๐๐ ๐ง ( ฯ๐ค)=0
๐ (๐๐ข๐๐ก +๐ข ๐๐ข๐ ๐ฅ +๐ฃ ๐๐ข๐ ๐ฆ +๐ค ๐๐ข๐ ๐ง )=โ ๐๐๐ ๐ฅ +๐๐๐ฅ+๐ (๐2๐ข๐๐ฅ2 + ๐
2๐ข๐ ๐ฆ2 +
๐2๐ข๐ ๐ง 2 )+๐3 ๐
๐ ๐ฅ (๐๐ข๐๐ฅ + ๐๐ฃ๐ ๐ฆ + ๐๐ค๐ ๐ง )
ฯ+
ฯ+
๐๐ข๐๐ฅ +
๐๐ฃ๐ ๐ฆ +
๐๐ค๐ ๐ง =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
ฯ X-momentum
๐๐๐ ๐ฅ=๐( ๐
2๐ข๐ ๐ฆ 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
๐๐๐ ๐ฆ=
๐๐๐ ๐ง=0 means๐=๐ (๐ฅ )๐๐๐๐ฆ
โข The governing equation is :
๐ข= 12๐๐๐๐๐ฅ ๐ฆ
2+๐ถ1 ๐ฆ+๐ถ2
The boundary conditions are:
After invoking boundary conditions:
Where P is non-dimensional Pressure gradient.
s๐ ,๐ข= ๐ฆh๐ โ h
2
2๐ โ๐๐๐ ๐ฅ โ
๐ฆh (1โ ๐ฆh )
๐ข๐=
๐ฆhโ h2
2๐๐ โ๐๐๐ ๐ฅ โ
๐ฆh (1โ ๐ฆh ) Let P
The velocity profile in non-dimensional form
โข when the equation reduced to:
(simple couette flow )
โข It can be produced by sliding a parallel plate at constant speed relative to a stationary wall.
Fig. Simple couette flow
โข For simple shear flow, there is no pressure gradient in the direction of the flow.
The velocity profiles for various P โข 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.
โข 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 itโs locationโข For maximum velocity :
โข 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.
โข 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.
โข 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 โฅ 1The Min. velocity : For P โค 1
Volume flow rate and average velocity
โข The volume flow rate per unit width is:
๐ข๐๐ฃ๐=( 12+ ๐
6 )๐
โข The Average velocity:
โข For P=-3, volume flow rate (Q) and average velocity uavg=0
Shear stress distribution
โข By invoking Newtonโs law of viscosity:
โข In the dimensionless form, the shear stress distribution becomes
h๐๐๐=1+๐ (1โ 2 ๐ฆ
h )โข Shear stress varies linearly with the distance from the
boundary.
โข For P=0, Shear stress remains constant across the flow passage:
โข 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
ยฟ๐๐ ๐ท๐
60 ๐ก โ๐๐ท๐ฟ=๐๐ 2๐ท2๐๐ฟ
60 ๐ก
๐๐๐๐๐ข๐๐๐๐๐ข๐๐๐๐๐ก๐๐๐ฃ๐๐๐๐๐๐ h๐ก ๐๐ฃ๐๐ ๐๐๐ข๐ ๐๐๐๐๐๐ก=๐ฃ๐๐ ๐๐๐ข๐ ๐๐๐ ๐๐ ๐ก๐๐๐๐ร ๐ท2
๐=๐๐ 2๐ท3๐๐ฟ
120 ๐ก
๐๐๐ค๐๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐ฃ๐๐๐๐๐๐๐๐ h๐ก ๐๐ฃ๐๐ ๐๐๐ข๐ ๐๐๐ ๐๐ ๐๐ก๐๐๐๐=๐ โ๐
๐๐๐ค๐๐=๐๐3๐ท3๐2 ๐ฟ360 0 ๐ก watts
Thanks