Fluid mechanics

Preet Kumar

M.Tech 1st Year

Enrollment No.- 14551009

Centre of Nanotechnology

IIT Roorkee

Types of FlowLaminar Flow

Turbulent Flow

Transition Flow

motion

flow in laminar

6

highly viscous fluids such as oils flow

flow in laminarturbulent flow

flows in a pipe.candle smoke.

8–2 ■ LAMINAR AND Laminar flow is encountered when

TURBULENT FLOWS in small pipes or narrow passages.

Laminar: Smooth

streamlines and highly

ordered motion.

Turbulent: Velocity

fluctuations and highly

disordered motion.

Transition: The flow

fluctuates between

laminar and turbulent

flows.

Most flows encountered

in practice are turbulent.

The behavior of

colored fluid

Laminar and injected into the

regimes of and turbulent

Principles of Fluid Flow in PipesIn laminar flow , the fluid travels as parallel layers (known as streamlines) that do not mix as they move in the direction of the flow.

If the flow is turbulent, the fluid does not travel in parallel layers, but moves in a haphazard manner with only the average motion of the fluid being parallel to the axis of the pipe.

If the flow is transitional , then both types may be present at different points along the pipeline or the flow may switch between the two.

In 1883, Osborne Reynolds performed a classic set of experiments that showed that the flow characteristic can be predicted using a dimensionless number, now known as the Reynolds number.

Principles of Fluid Flow in Pipes

The Reynolds number Re is the ratio of the inertia forces in the flow to the viscous forces in the flow and can be calculated using:

• If Re < 2000, the flow will be laminar.

• If Re > 4000, the flow will be turbulent.

• If 2000<Re<4000, the flow is transitional

• The Reynolds number is a good guide to the type of flow



The Bernoulli equation defines the relationship between fluid velocity (v), fluid pressure (p), and height (h) above some fixed point for a fluid flowing through a pipe of varying cross-section, and is the starting point for understanding the principle of the differential pressure flowmeter.

Bernoulli’s equation states that:

Bernoulli’s equation can be used to measure flow rate.

Consider the pipe section shown in figure below. Since the pipe is horizontal, h 1 = h 2,

and the equation reduces to:

Principles of Fluid Flow in PipesThe conservation of mass principle requires that:

Compressible or Incompressible

Fluid FlowMost liquids are nearly incompressible; that is, the density of a liquid remains almost constant as the pressure changes.

To a good approximation, then, liquids flow in an incompressible manner.

In contrast, gases are highly compressible. However, there are situations in which the density of a flowing gas remains constant enough that the flow can be considered incompressible.

Recalling vector operationsDel Operator:

Laplacian Operator:

Gradient:

Vector Gradient:

Divergence:

Directional Derivative:

Momentum Conservation

below.shown as zyxelement small aConsider

leration)mass)(acce(Force:law second sNewton' From

x

y

z

The element experiences an acceleration

DVm ( )

Dt

as it is under the action of various forces:

normal stresses, shear stresses, and gravitational force.

V V V Vx y z u v w

t x y z

r r r r r

xxxx x y z

x

xx y z

yx

yx y x zy

yx x z

Momentum Balance (cont.)

yxxx zx

Net force acting along the x-direction:

x x xxx y z x y z x y z g x y z

Normal stress Shear stresses (note: zx: shear stress acting on surfaces perpendicular to the z-axis, not shown in previous slide)

Body force

yxxx zx

The differential momentum equation along the x-direction is

x x x

similar equations can be derived along the y & z directions

x

u u u ug u v w

t x y z

Euler’s Equations

xx yy zz

For an inviscid flow, the shear stresses are zero and the normal stresses

are simply the pressure: 0 for all shear stresses,

x

Similar equations for

x

P

P u u u ug u v w

t x y z

y & z directions can be derived

y

z

y

z

P v v v vg u v w

t x y z

P w w w wg u v w

t x y z

Note: Integration of the Euler’s equations along a streamline will give rise to the Bernoulli’s equation.

Continuity equation for incompressible (constant

density) flow

where u is the velocity vector

u, v, w are velocities in x, y, and z directions

- derived from conservation of mass

ρυ

Navier-Stokes equation for incompressible flow of

Newtonian (constant viscosity) fluid

- derived from conservation of momentum

kinematic

viscosity

(constant)density

(constant)pressure

external force

(such as

gravity)




ρυ

ρυ




ρυ

Acceleration term:

change of velocity

with time




ρυ

Advection term:

force exerted on a

particle of fluid by the

other particles of fluid

surrounding it



viscosity (constant) controlled

velocity diffusion term:

(this term describes how fluid motion is

damped)

Highly viscous fluids stick together (honey)

Low-viscosity fluids flow freely (air)


ρυ




ρυ

Pressure term:

Fluid flows in the

direction of

largest change

in pressure




ρυ

Body force term:

external forces that

act on the fluid

(such as gravity,

electromagnetic,

etc.)




ρυ

change

in

velocity

with time

advection diffusion pressurebody

force= + + +

Continuity and Navier-Stokes equationsfor incompressible flow of Newtonian fluid

ρυ

Continuity and Navier-Stokes equationsfor incompressible flow of Newtonian fluidin Cartesian coordinates

Continuity:

Navier-Stokes:

x - component:

y - component:

z - component:

Steady, incompressible flow of Newtonian fluid in an infinite

channel with stationery plates

- fully developed plane Poiseuille flow

Fixed plate

Fixed plate

Fluid flow direction h

x

y

Steady, incompressible flow of Newtonian fluid in an

infinite channel with one plate moving at uniform velocity

- fully developed plane Couette flow

Fixed plate

Moving plate

h

x

yFluid flow direction

Continuity and Navier-Stokes equationsfor incompressible flow of Newtonian fluidin cylindrical coordinates

Continuity:

Navier-Stokes:

Radial component:

Tangential component:

Axial component:

Steady, incompressible flow of Newtonian fluid in a pipe

- fully developed pipe Poisuille flow

Fixed pipe

z

r

Fluid flow direction 2a 2a

φ

Steady, incompressible flow of Newtonian fluid between a

stationary outer cylinder and a rotating inner cylinder

- fully developed pipe Couette flow

aΩ

a

br

13developed laminar flow.

8–4 ■ LAMINAR FLOW IN PIPESWe consider steady, laminar, incompressible flow of a fluid with constant

properties in the fully developed region of a straight circular pipe.

In fully developed laminar flow, each fluid particle moves at a constant axial

velocity along a streamline and the velocity profile u(r) remains unchanged in the

flow direction. There is no motion in the radial direction, and thus the velocity

component in the direction normal to the pipe axis is everywhere zero. There is

no acceleration since the flow is steady and fully developed.

Free-body diagram of a ring-shaped

differential fluid element of radius r,

thickness dr, and length dx oriented

coaxially with a horizontal pipe in fully

t t li

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Boundaryconditions

Average velocity

Velocityprofile

Maximim velocity

Free-body diagram of a fluid disk element at centerline

of radius R and length dx in fully developed

laminar flow in a horizontal pipe.

l

only and is independent of the roughness of the pipe

types of fully developedlaminar

frictionpressure loss

15raised by a pump in order to overcome the frictional losses in the pipe.

Pressure Drop and Head Loss

A pressure drop due to viscous effects represents an irreversible pressure

loss, and it is called pressure loss ∆PL.

pressure loss for all Circular pipe,

internal flows

dynamic Darcy Head

factor

In laminar flow, the friction factor is a function of the Reynolds number

only and is independent of the roughness of the pipe surface.

The head loss represents the additional height that the fluid needs to be

Horizontalpipe

Poiseuille’slaw

The pumping power requirement for a laminarcircular or noncircular pipes, and

of 16 by doubling the pipe diameter.

For a specified flow rate, the pressure drop and

thus the required pumping power is proportional

to the length of the pipe and the viscosity of the

fluid, but it is inversely proportional to the fourth

power of the diameter of the pipe.

The relation for pressure loss (and

head loss) is one of the most general

relations in fluid mechanics, and it is

valid for laminar or turbulent flows,

pipes with smooth or rough surfaces. flow piping system can be reduced by a f1a6ctor

The pressure drop ∆P equals the pressure loss ∆PL in the case of a

horizontal pipe, but this is not the case for inclined pipes or pipes with

variable cross-sectional area.

This can be demonstrated by writing the energy equation for steady,

incompressible one-dimensional flow in terms of heads as

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Technology

Fluid mechanics