Momentum Transport

Momentum transport deals with the transport of momentum which is responsible for flow in fluids.
Momentum transport describes the science of fluid flow also called fluid dynamics. A few basic assumptions
are involved in fluid flow and these are discussed below.

No slip boundary condition

This is the first basic assumption used in momentum transport. It deals with the fluid flowing over a solid
surface, and states that whenever a fluid comes in contact with any solid boundary, the adjacent layer of the
fluid in contact with the solid surface has the same velocity as the solid surface. Hence, we assumed that
there is no slip between the solid surface and the fluid or the relative velocity is zero at the fluid–solid
interface. For example, consider a fluid flowing inside a stationary tube of radius R as shown in Fig 7.1. Since
the wall of the tube at r=R is stationary, according to the no-slip condition implies that the fluid velocity
at r=R is also zero.

Fig 7.1 Fluid flow in a circular tube of radius R

In the second example as shown in Fig. 7.2, there are two plates which are separated by a distance h, and
some fluid is present between these plates. If the lower plate is forced to move with a velocity V in x
direction and the upper plate is held stationary, no-slip boundary conditions may be written as follows
 Fig 7.2 Two parallel plates at stationary condition

Thus, every layer of fluid is moving at a different velocity. This leads to shear forces which are described in
the next section.

Newton’s Law of Viscosity

Newton’s law of viscosity may be used for solving problem for Newtonian fluids. For many fluids in chemical
engineering the assumption of Newtonian fluid is reasonably acceptable. To understand Newtonian fluid, let
us consider a hypothetical experiment, in which there are two infinitely large plates situated parallel to each
other, separated by a distance h. A fluid is present between these two plates and the contact area between
the fluid and the plates is A.
A constant force F1 is now applied on the lower plate while the upper plate is held stationary. After steady
state has reached, the velocity achieved by the lower plate is measured as V1. The force is then changed,
and the new velocity of the plate associated with this force is measured. The experiment is then repeated to
take sufficiently large readings as shown in the following table.
If the F/A is plotted against V/h, we may observe that they lie on a straight line passing through the origin.

Fig 7.4 Shear stress vs. shear stain

Thus, it may be said that F/A is proportional to v/h for a Newtonian fluid.
It may be noted that it is the velocity gradient which leads to the development of shear forces. The above
equation may be re-written as

In the limiting case, as h → 0, we have

where, µ is a constant of proportionality, and is called as the viscosity of the fluid. The
quantity F/A represents the shear forces/stress. It may be represented as , where the
subscript x indicates the direction of force and subscript y indicates the direction of outward normal of the

surface on which this force is acting. The quantity or the velocity gradient is also called the shear

rate. µ is a property of the fluid and is measured the resistance offered by the fluid to flow. Viscosity may be
constant for many Newtonian fluids and may change only with temperature.

Thus, the Newton’s law of viscosity, in its most basic form is given as
Here, both ‘+’ or ‘–’ sign are valid. The positive sign is used in many fluid mechanics books whereas the
negative sign may be found in transport phenomena books. If the positive sign is used then may be
called the shear force while if the negative sign is used may be referred to as the momentum flux
which flows from a higher value to a lower value.

The reason for having a negative sign for momentum flux in the transport phenomena is to have similarities
with Fourier's law of heat conduction in heat transport and Ficks law of diffusion in mass transport. For
example, in heat transport, heat flows from higher temperature to lower temperature indicating that heat
flux is positive when the temperature gradient is negative. Thus, a minus sign is required in the Fourier's law
of heat conduction. The interpretation of as the momentum flux is that x directed momentum flows
from higher value to lower value in y direction.

The dimensions of viscosity are as follows:

The SI unit of viscosity is kg/m.s or Pa.s. In CGS unit is g/cm.s and is commonly known as poise (P).
where 1 P = 0.1 kg/m.s. The unit poise is also used with the prefix centi-, which refers to one-hundredth of a
poise, i.e. 1 cP = 0.01 P. The viscosity of air at 25oC is 0.018 cP, water at 25oC is 1 cP and for many polymer
melts it may range from 1000 to 100,000 cP, thus showing a long range of viscosity.

Laminar and turbulent flow

Fluid flow can broadly be categorized into two kinds: laminar and turbulent. In laminar flow, the fluid layers
do not inter-mix, and flow separately. This is the flow encountered when a tap is just opened and water is
allowed to flow very slowly. As the flow increases, it becomes much more irregular and the different fluid
layers start mixing with each other leading to turbulent flow. Osborne Reynolds tried to distinguish between
the two kinds of flow using an ingenious experiment and known as the Reynolds’s experiment. The basic idea
behind this experiment is described below.

Reynolds’s experiment

Fig 7.5 Reynolds’s experiments

The experiment setup used for performing the Reynolds's experiment is shown in Fig. 7.5. The average
velocity of fluid flow through the pipe diameter can be varied. Also, there is an arrangement to inject a
colored dye at the center of the pipe. The profile of the dye is observed along the length of the pipe for
different velocities for different fluids. If this experiment is performed, it may be seen that for certain cases
the dye shows a regular thread type profile, which is seen at low fluid velocity and flow is called laminar flow.
when the fluid velocity is increased the dye starts to mixed with the fluid and for larger velocities simply
disappears. At this point fluid flow becomes turbulent.
For the variables average velocity of fluid vz avg, pipe diameter D, fluid density ρ, and the fluid viscosity µ,
Reynolds found a dimensionless group which could be used to characterize the type of fluid flow in the tube.
This dimensionless quantity is known as the Reynolds number. From the experiment, It was observed that
if Re >2100, the dye simply disappeared and the flow has changed to laminar to turbulent flow.
Thus, for Re <2100, we have laminar flow, i.e., no mixing in the radial direction leading to a thread like flow
and for Re >2100, we have the turbulent flow, i.e., mixing in the radial direction between layers of fluid.

In laminar flow, the fluid flows as a stream line flow with no mixing between layers. In turbulent flow, the
fluid is unstable and mixes rapidly due to fluctuations and disturbances in the flow. The disturbance might be
present due to pumps, friction of the solid surface or any type of noise present in the system. This makes
solving fluid flow problem much more difficult. To understand the difference in the velocity profile in two
kinds of fluid flows, we consider a fluid flowing to a horizontal tube in z direction under steady state
condition. Then, we can intuitively see the velocity profile may be shown below

For laminar flow, it is observed that fluid flows as smooth stream line and all other components of velocity
are zero. Thus

For turbulent flow, if we observe the fluid flows at a local point. It is observed that fluid flows in very random
manner in all directions where these local velocities may be the function of any dimensions.

Thus, we see that for laminar flow there is only one component of velocity present and it depends only on
one coordinate whereas the solution of turbulent flow may be vary complex.
For turbulent flow, one can ask the question that if the fluid is flowing in the z direction then why are the
velocity components in r and θ direction non-zero? The mathematical answer for this question can be
deciphered from the equation of motion. The equation of motion is a non-linear partial differential equation.
This non-linear nature of the equation causes instability in the system which produces flow in other
directions. The instability in the system may occur due to any disturbances or noise present in the
environment. On the other hand, if the velocity of fluid is very low the deviation due to disturbances may
decay with time, and becomes negligible after that. Thus the flow remains in laminar region. Consider a
practical example in which some cars are moving on the highway in the same direction but in the different
lanes at different speeds. If suddenly, some obstacle comes on the road, then if the car's speed is sufficiently
low, it can move on to other lane smoothly and come back to its original lane after the obstacle is crossed.
This is the regular laminar case. On the other hand, if the car is moving at a high speed and suddenly
encounters an obstacle, then the driver may lose control, and this car may move haphazardly and hit other
cars and after that traffic may never return to normal traffic conditions. This is the turbulent case.