Subject Name: FLUID MECHANICS Subject Code:10ME36B Prepared By: PUNITH R Department: AE Date: 03-09-2014

Reynolds Experiment Reynold’s experiments involved injecting a dye streak into fluid moving at constant velocity through a transparent tube. Fluid type, tube diameter and the velocity of the flow through the tube were varied.

Reynolds’ Results Dye followed a straight path. Dye followed a wavy path with streak intact. Dye rapidly mixed through the fluid in the tube

Reynolds classified the flow type according to the motion of the fluid. Laminar Flow: every fluid molecule followed a straight path that was parallel to the boundaries of the tube. Transitional Flow: every fluid molecule followed wavy but parallel path that was not parallel to the boundaries of the tube. Turbulent Flow: every fluid molecule followed very complex path that led to a mixing of the dye.

Reynolds found that conditions for each of the flow types depended on: 1. The velocity of the flow (U) 2. The diameter of the tube (D) 3. The density of the fluid (r). 4. The fluid’s dynamic viscosity (m). He combined these variables into a dimensionless combination now known as the Flow Reynolds’ Number (R) where:

Flow Reynolds’ number is often expressed in terms of the fluid’s kinematic viscosity (n; lower case Greek letter nu), where: (units are m2/s) Rearranging: Substituting into R: R

The value of R determined the type of flow in the experimental tubes: < 1000 1000 - 2000 > 2000

Laminar Transitional Turbulent

Fluid flow between two parallel plates The bottom plate is fixed and the top plate is accelerated by applying some force that acts from left to right. The upper plate will be accelerated to some terminal velocity and the fluid between the plates will be set into motion. Terminal velocity is achieved when the applied force is balanced by a resisting force (shown as an equal but opposite force applied by the stationary bottom plate).

As the upper plate begins to accelerate the velocity of the fluid molecules in contact with the plate is equal to the velocity of the plate (a no slip condition exists between the plate and the fluid). Fluid molecules in contact with those against the plate will be accelerated due to the viscous attraction between them... and so on through the column of fluid. The viscosity of the fluid (m, the attraction between fluid molecules) results in layers of fluid that are increasingly further from the moving plate being set into motion.

The bottom plate and water molecules attached to it are stationary (zero velocity; no slip between molecules of fluid and the plate) so that eventually the velocity will vary from zero at the bottom to Uterm at the top which is equal to the terminal velocity of the upper plate. The velocity gradient (the rate of change in velocity between plates; du/dy) will be constant and the velocity will increase linearly from zero at the bottom plate to Uterm at the top plate. Terminal velocity is achieved when the resisting force (the force shown applied by the bottom plate) is equal but opposite to the force applied to the top plate (forces are equal so that there is no change in velocity with time).

What is this resisting force What is this resisting force? It is “fluid resistance” (m) rather than a force applied by the lower plate (viscosity is often called “fluid friction”). Note that as the velocity increases upwards through the column of fluid, there must be slippage across any plane that is parallel to the plates within the fluid. At the same time there must be resistance to the slippage or the upper plate would accelerate infinitely. This same resistance results in the initial acceleration of every layer of fluid to its own terminal velocity (that decreases downwards).

Fluid viscosity is the cause of fluid resistance and the total viscous resistance through the column of fluid equals the applied force when the terminal velocity is achieved. The viscous resistance results in the transfer of the force applied to the top plate through the column of fluid. Within the fluid this force is applied as a shear stress (t, the lower case Greek letter tau; a force per unit area) across an infinite number of planes between fluid molecules from the top plate down to the bottom plate.

The shear stress transfers momentum (mass times velocity) through the fluid to maintain the linear velocity profile. The magnitude of the shear stress is equal to the force that is applied to the top plate. The relationship between the shear stress, the fluid viscosity and the velocity gradient is given by:

From this relationship we can determine the velocity at any point within the column of fluid. Rearranging the terms: We can solve for u at any height y above the bottom plate by integrating with respect to y. Where c (the constant of integration) is the velocity at y=0 (where u=0) such that:

From this relationship we can see the following: 1. That the velocity varies in a linear fashion from 0 at the bottom plate (y=0) to some maximum at the highest position (i.e., at the top plate). 2. That as the applied force (equal to t) increases so does the velocity at every point above the lower plate. 3. That as the viscosity increases the velocity at any point above the lower plate decreases.

Flow between plates Driving force is only the force applied to the upper, moving plate. Shear stress (force per unit area) within the fluid is equal to the force that is applied to the upper plate. Fluid momentum is transferred through the fluid due to viscosity.

Laminar Flow through Circular Tubes Different geometry, same equation development Apply equation of motion to cylindrical sleeve (use cylindrical coordinates)

Laminar Flow through Circular Tubes: Equations R is radius of the tube Max velocity when r = 0 Velocity distribution is paraboloid of revolution therefore _____________ _____________ average velocity (V) is 1/2 vmax Q = VA = VpR2

Laminar Flow through Circular Tubes: Diagram Shear (wall on fluid) Velocity Laminar flow Next slide! Shear at the wall True for Laminar or Turbulent flow Remember the approximations of no shear, no head loss?

Relationship between head loss and pressure gradient for pipes cv energy equation Constant cross section In the energy equation the z axis is tangent to g x is tangent to V z x l is distance between control surfaces (length of the pipe)

The Hagen-Poiseuille Equation Relationship between head loss and pressure gradient Hagen-Poiseuille Laminar pipe flow equations From Navier-Stokes What happens if you double the pressure gradient in a horizontal tube? Ans: flow doubles V is average velocity

Thank you