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Fluid Mechanics (Analysis of Fluid Flow motion) Essay Example
 Category:Engineering and Construction
 Document type:Math Problem
 Level:Undergraduate
 Page:1
 Words:628
Mathematical modeling of viscous incompressible flow through a diffuser
a) Design flow
In this design we consider oil of viscosity 0.048 kg/ms flowing at a mean velocity of 0.5m/s before the diffuser assuming that it flows in pipe of 0.1.5 m diameter before the diffuser. We assume the oil density is 900 kg/m^{3}. The entire pipe is to be taken horizontal. The conical nose base diameter is 0.1m.the conical nose rotates at 1200rpm and the external tunnel is fixed
Calculating the Reynolds number we have.
b) Governing equation

Continuity equation
We use cylindrical coordinate to as shown below;
Using the mass conservation in the control volume;

Momentum equation
The summation of force includes the surface forces plus the body forces;

The body forces includes the gravitational forces that is

The surface force is due to shearing stresses on all the surfaces of the control volume.
Therefore the momentum equation simplifies to;
The momentum equation in r direction
The momentum equation in
direction
The momentum equation in z direction
The Lapsian operator
C) Assumptions

The tunnel before the diffuser is infinitely long such that the flow is fully developed,

The flow is steady that is the flow properties do not change with time therefore all the time derivatives becomes zero,

The fluid is incompressible and Newtonian with constant properties such as density and viscosity,

A constant pressure drop is applied along the axial direction therefore the pressure gradient is constant all over,

The flow is entirely parallel and there is no movement along the radial direction,

The flow before the conical the nose do not tend to swirl and therefore all derivative with respect to
are equal to zero, 
In this analysis we can assume the gravitational forces.
Therefore the Continuity differential equation reduce to
Which means that
is not a function of z. again considering assumption ii the flow is steady therefore
is not a function of time t hence can only be a function of r. simplifying the momentum equation in the z direction (axial direction) we have
In the diffuser region the continuity would require that
Considering our
therefore
Pressure actually increases
When the fluid particle slides over the nose it will have two velocity components that is the tangential component and the axial component. To determine the axial component we must assume the nose does not rotate.
And therefore we apply the boundary condition for the equation,
At r=D/2 the axial velocity is equal zero and therefore
At r=R the axial velocity is equal to zero
At r=D/2+(RD/2)/2 velocity is maximum and therefore
Evaluating for
and substituting
The axial velocity will be equal to
To find the average velocity
In our design flow
To determine the tangential component we assume there is no end effect and therefore axial velocity is equal to zero, since the nose rotates at constant velocity therefore there is no variation of angular velocity with the change in the angle. The continuity equation reduce to,
We also assume there is no motion along the r and therefore radial velocity is equal to zero and
the particles do not accelerate tangentially due to constant angular velocity. The pressure does
not also vary tangentially. Therefore the momentum equation reduce to
Therefore
Integrating
Applying the boundary conditions
At r=D/2
therefore
At r=R
0= solving this simultaneously the velocity profile is defined as
The motion of fluid components moves over the nose with this two velocity components.
References
WHITE, F. M. (2001). Fluid mechanics. Boston, Mass, WCB/McGrawHill.
ÇENGEL, Y. A., & CIMBALA, J. M. (2006). Fluid mechanics: fundamentals and applications. Boston, McGrawHillHigher Education.
SHAUGHNESSY, E. J., KATZ, I. M., & SCHAFFER, J. P. (2005). Introduction to fluid mechanics. New York, Oxford University Press.
WEAST, R. C., & SELBY, S. M. (1995). Handbook of tables for mathematics. Cleveland, CRC Press.