Problems 6 8 An incompressible viscous fluid

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 U. Determine: – – volumetric dilation rate; rotation vector; vorticity; rate of angular deformation. • 6. 74 Oil SAE 30 at 15. 6 C steadily flows between fixed horizontal parallel plates. The pressure drop per unit length is 20 k. Pa/m and the distance between the plates is 4 mm, 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

Problems • 7. 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=5 mm, D=12. 5 mm and V=0. 6 m/s, the drag is 6. 7 x 10 -3 N. 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

Viscous flow in pipes

General characteristic of Pipe flow • pipe is completely filled with water • main driving force is usually a pressure gradient along the pipe, though gravity might be important as well Pipe flow open-channel flow

Laminar or Turbulent flow well defined streakline, one velocity component velocity along the pipe is unsteady and accompanied by random component normal to pipe axis

Laminar 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.

Entrance region and fully developed flow • fluid typically enters pipe with nearly uniform velocity • the length of entrance region depends on the Reynolds number dimensionless entrance length

Pressure and shear stress no acceleration, viscous forces balanced by pressure balanced by viscous forces and acceleration

Fully developed laminar flow • we will derive equation for fully developed laminar flow in pipe using 3 approaches: – from 2 nd Newton law directly applied – from Navier-Stokes equation – from dimensional analysis

2 nd Newton’s law directly applied

2 nd Newton’s law directly applied doesn’t depend on radius

2 nd Newton’s law directly applied for Newtonian liquid: Flow rate:

2 nd Newton’s law directly applied • if gravity is present, it can be added to the pressure:

Navier-Stokes equation applied in cylindrical coordinates: • The assumptions and the result are exactly the same as Navier-Stokes equation is drawn from 2 nd Newton law

Dimensional analysis applied assuming pressure drop proportional to the length:

Turbulent flow • in turbulent flow the axial component of velocity fluctuates randomly, components perpendicular to the flow axis appear • heat and mass transfer are enhanced in turbulent flow • in many cases reasonable results on turbulent flow can be obtained using Bernoulli equation (Re=inf).

Fluctuation in turbulent flow • • • All parameters fluctuate in turbulent flow (velocity, pressure, shear stress, temperature etc. ) behave chaotically flow parameters can be described as an average value + fluctuations (random vortices) can be characterized by turbulence intensity and time scale of fluctuation turbulence intensity

Shear 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 eddies • Vortices transfer momentum, so the shear force is higher compared with laminar flow:

Shear 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.

Shear stress in turbulent flow • Shear stress is a sum of laminar portion and a turbulent portion positive shear stress is larger in turbulent flow • Alternatively: h – eddy viscosity Prandtl suggested that turbulent flow is characterized by random transfer over certain distance lm:

Turbulent velocity profile

Turbulent velocity profile • in the viscous sublayer where, y=R-r, u – time averaged x component, u*=(t/r)½ friction velocity valid near smooth wall: • in the turbulent layer: function of Reynolds number

Turbulent 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.

Dimensional analysis of pipe flow • major loss in pipes: due to viscous flow in the straight elements • minor loss: due to other pipe components (junctions etc. ) Major loss: roughness • those 7 variables represent complete set of parameters for the problem as pressure drop is proportional to length of the tube:

Dimensional analysis of pipe flow friction factor • for fully developed laminar flow • for fully developed steady incompressible flow (from Bernoulli eq. ):

Moody chart Friction factor as a function of Reynolds number and relative roughness for round pipes Colebrook formula

Non-circular ducts • Reynolds number based on hydraulic diameter: cross-section wetted perimeter • Friction factor for noncircular ducts: for fully developed laminar flow:

Equivalent circuit theory flow: electricity: • channels connected in series

Equivalent circuit theory • channels connected in parallel

Compliance • compliance (hydraulic capacitance): Q – volume V/time flow: I – charge/time electricity:

Equivalent circuits

Pipe networks • Serial connection • Parallel connection

Problems • Ethanol solution of a dye (h=1. 197 m. Pa·s) is used to feed a fluidic lab-onchip laser. Dimension of the channel are L=122 mm, width w=300 um, height h=10 um. Calculate pressure required to achieve flow rate of Q=10 ul/h. • 8. 7 A soft drink with properties of 10 ºC water is sucked through a 4 mm diameter 0. 25 m long straw at a rate of 4 cm 3/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.