Reference Reference Book isBook is

2. The flow is steady.2. The flow is steady. In steady (laminar) flow, the velocity of the fluid In steady (laminar) flow, the velocity of the fluid at each point remains constant.at each point remains constant.

Fluid DYNAMICSFluid DYNAMICSBecause the motion of real fluids is very complex and not fully understood, we make some simplifying assumptions in our approach. In our model of an ideal fluid, we make the following four assumptions:1.1. The fluid is nonviscous. The fluid is nonviscous. In a nonviscous fluid, internal friction is neglected. In a nonviscous fluid, internal friction is neglected.

Viscous force, Viscous force, Is the resistance that two adjacent Is the resistance that two adjacent

layers of fluid have to moving layers of fluid have to moving relative relative to each otherto each other

3. The fluid is incompressible. 3. The fluid is incompressible. The density of an incompressible fluid is constant.The density of an incompressible fluid is constant.4. The flow is irrotational. 4. The flow is irrotational. In irrotational flow, the fluid has no angular In irrotational flow, the fluid has no angular momentum about any point. momentum about any point.

STREAMLINES AND THE EQUATION OF CONTINUITYSTREAMLINES AND THE EQUATION OF CONTINUITY

Consider an ideal fluid flowing Consider an ideal fluid flowing through a pipe of nonuniform size, through a pipe of nonuniform size, as illustrated in Figureas illustrated in Figure

Streamline:Streamline: is the path taken by a fluid particle under is the path taken by a fluid particle under steady flowsteady flow

because mass is conserved and because the flow is steady, the mass that crosses A1 in a time t must equal the mass that crosses A2 in the time t. That is, m1 = m2 or ρ A1v1t = ρ A2v2t ; this means that

This expression is called the equation of continuity. It states thatThe product of the area and the fluid speed at The product of the area and the fluid speed at

all points along the pipe is a constant for an all points along the pipe is a constant for an incompressible fluid.incompressible fluid.

BERNOULLI’S EQUATIONBERNOULLI’S EQUATIONThe relationship between The relationship between fluid speed, pressure, and fluid speed, pressure, and elevation was first derived in elevation was first derived in 1738 by the Swiss physicist 1738 by the Swiss physicist Daniel Bernoulli. Consider the Daniel Bernoulli. Consider the flow of an ideal fluid through flow of an ideal fluid through a nonuniform pipe in a time a nonuniform pipe in a time t, t, as illustrated in Figureas illustrated in Figure. Thus, . Thus, the net work done by Fthe net work done by F1 1 and and FF2 2 forces forces are are WW11 = F = F11xx11 = P = P11AA11xx11 = P= P11VV. And . And WW22 = F = F22 x x22 = P = P22AA22xx22 = P= P22V.V.

If If m is the mass that enters one m is the mass that enters one end and leaves the other end and leaves the other in a time in a time t, then the change in the kinetic energy of this t, then the change in the kinetic energy of this mass ismass is

The change in gravitational The change in gravitational potential energy ispotential energy is

We can apply EquationWe can apply Equation

If we divide each term by If we divide each term by V and recall V and recall Rearranging terms, we obtainRearranging terms, we obtain

When the fluid is at rest When the fluid is at rest vv11 and and vv22 are zero so are zero so the Equation becomesthe Equation becomes

This is in agreement with pervious Equation This is in agreement with pervious Equation

This is This is Bernoulli’s equation Bernoulli’s equation as applied to an ideal as applied to an ideal fluid. It is often expressed asfluid. It is often expressed as

The The liftlift on an aircraft wing can be explained by on an aircraft wing can be explained by the Bernoulli effect. Airplane wings are designed the Bernoulli effect. Airplane wings are designed so that the air speed above the wing is greater so that the air speed above the wing is greater than that below the wing. As a result, the air than that below the wing. As a result, the air pressure above the wing is less than the pressure above the wing is less than the pressure below, and a net upward force on the pressure below, and a net upward force on the wing, wing, called called liftlift, results., results.

APPLICATIONS OF BERNOULLI’S EQUATIONAPPLICATIONS OF BERNOULLI’S EQUATION

APPLICATIONS OF BERNOULLI’S EQUATIONAPPLICATIONS OF BERNOULLI’S EQUATION

A stream of air passing over one end of an open tube, A stream of air passing over one end of an open tube, the other end of which is immersed in a liquid, the other end of which is immersed in a liquid, reduces the pressure above the tube. This reduction reduces the pressure above the tube. This reduction in pressure causes the liquid to rise into the air in pressure causes the liquid to rise into the air stream. The liquid is then dispersed into a fine spray stream. The liquid is then dispersed into a fine spray of droplets. This so-called atomizer is used in of droplets. This so-called atomizer is used in perfume bottles and paint sprayersperfume bottles and paint sprayers