Presentation on theme: "Problems 6.8 An incompressible viscous fluid is placed between two large parallel plates. The bottom plate is fixed and the top moves with the velocity."— Presentation transcript:
1Problems6.8 An incompressible viscous fluid is placed between two large parallel plates. The bottom plate is fixed and the top moves with the velocity U. Determine:volumetric dilation rate;rotation vector;vorticity;rate of angular deformation.6.74 Oil SAE30 at 15.6C steadily flows between fixed horizontal parallel plates. The pressure drop per unit length is 20kPa/m and the distance between the plates is 4mm, the flow is laminar. Determine the volume rate of flow per unit width; magnitude and direction of the shearing stress on the bottom plate; velocity along the centerline of the channel
2Problems7.19. One type of viscosimeter is designed as shown in the figure. The reservoir is filled with liquid and the time required for the liquid to drop from Hi to Hf determined. Obtain relationship between viscosity m and draining time t. Assume the variables involved Hi,Hf, D and specific weight g.7.50 The drag D=f(d,D,V,r). What dimensional parameters will be used? If in experiment d=5mm, D=12.5mm and V=0.6m/s, the drag is 6.7x10-3N. Estimate drag on a sphere in 0.6 m diam tube where water flowing with a velocity of 1.8 m/s and the diameter of sphere is such that the similarity is maintained
4General characteristic of Pipe flow pipe is completely filled with watermain driving force is usually a pressure gradient along the pipe, though gravity might be important as wellopen-channel flowPipe flow
5Laminar or Turbulent flow well defined streakline, one velocity componentAs was first found by British scientist Osborne Reynolds flow in pipes can be laminar at low velocity or turbulent at high velocity.If we want to correctly describe what is low and what is high we have to assign a dimensionless number.velocity along the pipe is unsteady and accompanied by random component normal to pipe axis
6Laminar or Turbulent flow In this experiment water flows through a clear pipe with increasing speed. Dye is injected through a small diameter tube at the left portion of the screen. Initially, at low speed (Re <2100) the flow is laminar and the dye stream is stationary. As the speed (Re) increases, the transitional regime occurs and the dye stream becomes wavy (unsteady, oscillatory laminar flow). At still higher speeds (Re>4000) the flow becomes turbulent and the dye stream is dispersed randomly throughout the flow.
7Entrance region and fully developed flow fluid typically enters pipe with nearly uniform velocitythe length of entrance region depends on the Reynolds numberdimensionless entrance length
8Pressure and shear stress no acceleration, viscous forces balanced by pressurepressure balanced by viscous forces and acceleration
9Fully developed laminar flow we will derive equation for fully developed laminar flow in pipe using 3 approaches:from 2nd Newton law directly appliedfrom Navier-Stokes equationfrom dimensional analysis
112nd Newton’s law directly applied doesn’t depend on radius
122nd Newton’s law directly applied for Newtonian liquid:Flow rate:
132nd Newton’s law directly applied if gravity is present, it can be added to the pressure:
14Navier-Stokes equation applied in cylindrical coordinates:The assumptions and the result are exactly the same as Navier-Stokes equation is drawn from 2nd Newton law
15Dimensional analysis applied assuming pressure drop proportional to the length:
16Turbulent flowin turbulent flow the axial component of velocity fluctuates randomly, components perpendicular to the flow axis appearheat and mass transfer are enhanced in turbulent flowin many cases reasonable results on turbulent flow can be obtained using Bernoulli equation (Re=inf).
17Fluctuation in turbulent flow All parameters fluctuate in turbulent flow (velocity, pressure, shear stress, temperature etc.) behave chaoticallyflow parameters can be described as an average value + fluctuations (random vortices)can be characterized by turbulence intensity and time scale of fluctuationturbulence intensity
18Shear stress in turbulent flow Turbulent flow can often be thought of as a series of random, 3-dimensional eddy motions (swirls) ranging from large eddies down through very small eddiesVortices transfer momentum, so the shear force is higher compared with laminar flow:
19Shear stress in turbulent flow The turbulent nature of the flow of soup being stirred in a bowl is made visible by use of small reflective flakes that align with the motion. The initial stirring causes considerable small and large scale turbulence. As time goes by, the smaller eddies dissipate, leaving the larger scale eddies. Eventually, all of the motion dies out. The irregular, random nature of turbulent flow is apparent.
20Shear stress in turbulent flow Shear stress is a sum of laminar portion and a turbulent portionpositiveshear stress is larger in turbulent flowAlternatively:h – eddy viscosityPrandtl suggested that turbulent flow is characterized by random transfer over certain distance lm:
22Turbulent velocity profile in the viscous sublayerwhere, y=R-r, u – time averaged x component, u*=(t/r)½ friction velocityvalid near smooth wall:function of Reynolds numberin the turbulent layer:
23Turbulent velocity profile An approximation to the velocity profile in a pipe is obtained by observing the motion of a dye streak placed across the pipe. With a viscous oil at Reynolds number of about 1, viscous effects dominate and it is easy to inject a relatively straight dye streak. The resulting laminar flow profile is parabolic. With water at Reynolds number of about 10,000, inertial effects dominate and it is difficult to inject a straight dye streak. It is clear, however, that the turbulent velocity profile is not parabolic, but is more nearly uniform than for laminar flow.
24Dimensional analysis of pipe flow major loss in pipes: due to viscous flow in the straight elementsminor loss: due to other pipe components (junctions etc.)Major loss:roughnessthose 7 variables represent complete set of parameters for the problemas pressure drop is proportional to length of the tube:
25Dimensional analysis of pipe flow friction factorfor fully developed laminar flowfor fully developed steady incompressible flow (from Bernoulli eq.):
26Moody chartFriction factor as a function of Reynolds number and relative roughness for round pipesColebrook formula
27Non-circular ducts Reynolds number based on hydraulic diameter: cross-sectionwetted perimeterFriction factor for noncircular ducts:for fully developed laminar flow:
28Equivalent circuit theory flow:electricity:channels connected in series
29Equivalent circuit theory channels connected in parallel
30Compliancecompliance (hydraulic capacitance): Q – volume V/time I – charge/timeflow:electricity:
33ProblemsEthanol solution of a dye (h=1.197 mPa·s) is used to feed a fluidic lab-on-chip laser. Dimension of the channel are L=122mm, width w=300um, height h=10 um. Calculate pressure required to achieve flow rate of Q=10ul/h.8.7 A soft drink with properties of 10 ºC water is sucked through a 4mm diameter 0.25m long straw at a rate of 4 cm3/s. Is the flow at outlet laminar? Is it fully developed?Calculate total resistance of a microfluidic circuit shown. Assume that the pressure on all channels is the same and equal Dp.