Fluid Mechanics for MAP/Analytical solutions of internal and external flows

>back to Chapters of Fluid Mechanics for MAP

>back to Chapters of Microfluid Mechanics

Internal and External Incompressible Viscous Flows edit

Flows completely bounded by solid surfaces are called internal flows. External flows are flows over bodies immersed in an unbounded fluid[1].

Internal flows might be laminar or turbulent. The state of the flow regime is dependent on Re. There might be an analytical solution for laminar flows but not for turbulent flows.

Flows through pipes, ducts, nozzles, diffusers, valves and fittings are examples of internal flows.
Flow around an aircraft is classified under external flows.
Flow around a car is classified under external flows.

Laminar and Turbulent Channel and Pipe flows edit

At fully developed state the velocity profile becomes parabolic for laminar flow. The average velocity at any cross section is:


For the same flow value i.e.  , the fully developed turbulent pipe flow, would have higher velocity close to the wall and lower velocity at the center.

The reason is the turbulent eddies, which causes more momentum loss to the wall i.e. higher velocity gradients close to the wall. Note that such a direct comparison is only valid at the same  .

Velocity profiles for laminar (upper) and turbulent (lower) states at the same mass flow rate
Signal measured by a hot-wire at a center of a pipe during transition showing the developement of transitional structures along the pipe at different Reynolds numbers[2] [3] and the change of friction coefficient at laminar, transitional and turbulent regimes.
Development of velocity profile in a pipe with increasing Reynolds number

Concept of Fully Developed Flow edit

Flow between two plates

Consider the flow in a channel between two plates having a height of   and an infinite depth in   direction. Starting from the entrance, the boundary layers develop due to the no-slip condition on the wall. At a finite distance, the boundary layers merge and the inviscid core (field with no velocity gradient in   direction) vanishes. The flow becomes fully viscous. The velocity field in   direction adjusts slightly further until   and it no longer changes with   direction. This state of the flow is called fully-developed. At that state:


Because of the two-dimensional nature of the flow, no gradient of the velocity quantities in   direction is expected starting from the entrance.




The entrance lengths for laminar pipe and channel flows are, respectively[4]:



Fully Developed Laminar Flow Between Infinite Parallel Plates edit

Fully developed flow in a channel with an infinite depth

Consider the fully developed laminar flow between two infinite plates.

Consider the continuity equation and momentum equation in   direction for an incompressible steady flow between two infinite plates as shown.

Continuity Equation


Since   ,   because it is a fully-developed and two dimensional flow. Hence,   reads


As   is zero on the walls, it should be zero in the whole fully developed region, i.e.


Momentum Equation in j-direction


in   direction , 


Consider term A:


Consider term B:


hence  .

Thus the momentum equation in   direction reads:


This equation should be valid for all   and  . This requires that   = constant.

Remember   is the stress in   direction on a face normal to   direction.





Thus the momentum equation reads:


This equation can be obtained also by using the Reynold's transport equations for a differential volume.

Treatment of channel flow with the RTT approach

The momentum equation in   direction,


The flux term becomes zero since for fully-developed flow incoming flux is equal to the outgoing flux. Thus,


That is:





Finally, the governing equation of this kind of flow becomes:


with the following boundary conditions:


Integrating the equation once results in a linear function of  :


The second integration reads:


The integration constants is obtained by using the boundary conditions:



Finally, the velocity profile reads:


Note that the velocity profile is parabolic!

The shear stress becomes:


at the wall i.e. at   = 0 and  = D



Velocity and shear stress profiles in fully developed channel flow

Note that   is maximum near the wall, i.e. momentum loss is maximum near the wall. This is due to the maximum velocity gradient   near the wall!

The volume flow rate is,


where   is the depth of the channel.

Thus the volume flow rate per depth   is given by:



Note that   should be constant for the fully developed flow. Hence, for a channel with a finite length  :


Where   is the pressure drop along L.


or the pressure drop can be calculated from:


For the same flow rate, increasing the height of the channel would cause a drastic reduction in the pressure drop.

The average velocity   is:


The maximum velocity occurs when:


Hence, at  ,  


The velocity profile can be written as functions of bulk velocity   or maximum velocity   by replacing their value the velocity profile equation:


Same problem can be solved by using moving plates.

Example edit

Flow through the gap between a cylindrical piston and the surrounding wall

Consider the hydrolic control valve coprising a piston, fitted to a cylinder with a mean radial clearance of 0,005mm. Determine the leakage flow rate. The fluid is SAE low oil (  = 932  ,   =0.018   at 55ºC). The flow can be assumed to be laminar, steady, incompressible, fully-developed flow.  

Since   = 5000 the flow in the clearance can be accepted to be 2-D, with the depth  , thus:




Check the Reynolds number to ensure that laminar flow assumption is correct.



Re  , i. e. the flow is laminar.

Layered Channel Flow edit


This channel flow contains two different and non miscible fluids. Fluids A and B flow at the same time through a channel, which is bounded two flat plates. They both occupy the half height of the channel. The fluid A has a viscosity  , a density   and the mass flow  . Fluid B, which is located above fluid A, has a viscosity  , a density   and the mass flow  . The following differential equations correspond to the molecular momentum   for each Fluid.

  and  .

With   yields the velocity field:


After integration of both equations we obtain:


As boundary condition we consider that shear stress on the interface between A and B is the same. Therefore we obtain:




After the integration for the velocity field:




The second boundary condition turns out to be on the interface:

 , i.e.  


 . The integration constants can be calculated with the following boundary conditions:

At  :


At  :


Therefore we obtain for the velocity distribution in the fluids A and B:




For the distribution of the shear stress we get:


If we choose  ,



The solution gives that of the channel flow. In other words, velocity has a parabolic profile with the peak in the middle of the channel and a linear shear stress distribution  , where   at the channel's centerline.

If  , the position where the maximal velocity occurs can be calculated by introducing   on the velocity profile equation:


The shear stress on the upper plate is:


and the shear stress on the lower plate reads:


The average velocities of the fluids A and B are:




Hence the respectively mass flow rates are:




Navier Stokes Equation in Cyclindrical Coordinates edit

A change of variables on the Cartesian equations will yield[5] the following equations of momentum in r,  , and z directions:




The continuity equation is:


Fully Developed Pipe Flow edit

Treatment of pipe flow with the RTT approach(with infinitesimal cylinder)

It is possible to use the same mathematical treatment like before to find and understand the velocity profile for fully developed flow inside a pipe with diameter D and infinite length. To show the flexibility, the same solution for this problem will be approached via 3 different ways.

(i) Infinitesimal Cylinder at the center of the pipe

Applying the RTT to the infinitesimal cylindrical CV along the symmetry axis of horizontal pipe, in which the flow is fully developed, the conservation of mass and the transport side of the conservation of momentum equation drops. Only remaining term governing this kind of flow is the balance of the forces on the CV in   direction.



(ii) Infinitesimal thin hollow cylinder at the center

Treatment of pipe flow with the RTT approach(with infinitesimal hollow cylinder)

This time the pressure and viscous force is considered for a concentric hollow cylinder with radius of R and infinitesimally small thickness dr and length dx(as shown in the image besides) along x-direction. Considering pressure term on the cross-section of cylinder


considering viscous shear stress on the surface of the cylinder


combining both term,we get the balance equation,


it could be rewritten as


dividing both side with  ,we get




integrating both side ,






However,in a laminar flow!




The boundary condition is:


Thus   can be calculated from the boundary condition.




(iii) Using NS in cylindrical co-ordinates:








Now, integrating both side with respect to  , we get




dividing by r in both side , we get then


integrating again with respect to   gives ,


Consequently, when   then   and  , as a result C=0.

On the other hand, when   then   , so




By putting D to the primitive equation, we get,


Knowing the velocity profile we can evaluate relevant quantities. The shear stress profile will look like:


at r = 0  

at r = R  

Shear stress distribution in the fully developed laminar pipe flow.

The volume flow rate would read




When we approximate  



Increase radius to create drastic reduction in the pressure drop.

The mean velocity is:


The location where maximum velocity occurs can be found be setting:


at r = 0   U =  .


Note that in a channel was  .

  can be written as a function of   i.e.



Again, the velocity profile becomes parabolic.

References edit

  1. Fox, R.W. and McDonald, A.T., “Introduction to Fluid Mechanics”, John Willey and Sons.
  2. M. Nishi. PhD Thesis Friedrich-Alexander-Universität Erlangen-Nürnberg, 2009.
  3. M. Nishi, B. Ünsal, F. Durst, and G. Biswas. J. Fluid Mech., 614:425–446, 2008.
  4. Durst, F., Ray, S., Unsal, B., and Bayoumi, O. A., 2005, “The Development Lengths of Laminar Pipe and Channel Flows,” J. Fluids Eng., 127, pp. 1154– 1160.
  5. Acheson, D.J.: Elementary fluid dynamics, Clarendon Press, 1990.