Question

Question #2984a

Answer 1

Velocity profile is an expression, that can be used to find the velocity at any depth of fluid by changing the variable.

Explanation:

We know that all fluid particles do not travel at the same velocity within a pipe. The velocity profile across any given section of the pipe, which describes the shape of the velocity curve, depends upon whether the flow is laminar or turbulent.

As shown in the figure below, fluid enters a circular tube and develops a velocity profile due to viscous forces propagating from the interior wall of the tube. Hydrodynamic region is characterized by non-uniform flow.

The fluid enters the tube at a uniform velocity. In the irrotational flow region viscous effects and velocity changes are negligible

Fluid particles of the layer which is contact with the surface of the tube come to a complete stop.

Viscous forces within the fluid, the layer in contact with the tube surface resist the motion of adjacent layers and slows the adjacent layers of fluid gradually down the path. Thus forming a velocity profile.

Due to the Law of Conservation of mass, the velocity of layers of the fluid in the center of the pipe increases to compensate for the reduced velocities of the layers of fluid as these near the surface of tube. Thus, a velocity gradient across the cross section of the tube gets developed.

In case of laminar flow, the velocity profile in the fully developed region is parabolic. The profile remains unchanged in this region.

It can be shown that velocity `v(r)` of fluid layer located at a distance `r` from center of tube of radius `R` can be expressed as

`v(r)=v_max[1-r^2/R^2]` ........(1)

Equation (1) represents velocity profile for Laminar flow.

Here velocity of the layer which is in contact with the tube surface `v(R)=0`, and maximum velocity of the layer at the center of tube

`v_max=1/(4eta)(DeltaP)/(Deltax)R^2` .......(2)
where `eta` is the viscosity of fluid and `(DeltaP)/(Deltax)` is pressure gradient.