fluid kinematics flow of the fluid

9. CONSERVATION OF MOMENTUM
• momentum is the product of
the mass and velocity of an
object. It is a vector quantity,
possessing a magnitude and a
direction in three-dimensional
space. If m is an object's mass
and v is the velocity (also a
vector), then the momentum is

10. CONSERVATION OF MOMENTUM
• Linear momentum is defined as
• for a volume of interest. In writing Newton’s law, we had to consider
the summation of the forces that act on the fluid. Recall from our
earlier discussion that the summation of the forces that act on a fluid
element must include all body forces (denoted as ~Fb) and all surface
forces (denoted as ~Fs), thus in considering linear momentum we
may need to quantify all of the forces that act on the volume/surfaces
of interest. Physically, linear momentum is a force of motion, which is
conserved (and thus remains constant) unless other forces are
applied to the system.

11. CONSERVATION OF MOMENTUM
• we can get the formulation for conservation of linear momentum
using the Reynolds Transport Theorem (RTT) :
• Using Newton’s relationship for momentum:

12. CONSERVATION OF MOMENTUM
• We make use of forces in instead of the time rate of change of
momentum, because it is typically easier to quantify forces acting on
a fluid of interest. Last equation states that the summation of all
forces acting on a volume of interest is equal to the time rate of
change of momentum within the control volume and the summation
of momentum flux through the surface of interest. To solve
conservation of momentum problems, the first step will be to define
the volume of interest and surfaces of interest and label all of the
forces that are acting on this system (note though that some of the
forces can cancel out and thus they may not be included in the
formulation).

13. CONSERVATION OF MOMENTUM
• CONSERVATION OF MOMENTUM in 3 dimension.
• Cartesian coordinate system
• Unlike the conservation of mass formula, the formulas derived for the
conservation of linear momentum are vector equations

14. CONSERVATION OF MOMENTUM
• where u, v, and w are the velocity components in the x-, y-, and z-
directions, respectively. As before, the product of ρv and dA is a
scalar whose sign depends on the directions of the normal area
vector and the velocity vector (the sign of u, v, and w are defined by
the velocity components).

15. THE NAVIERSTOKES EQUATIONS
• The Navier-Stokes equations are time-dependent and consist of a
continuity equation for conservation of mass, three conservation of
momentum equations and a conservation of energy equation. There are
four independent variables in the equation - the x, y, and z spatial
coordinates, and the time t; six dependent variables - the pressure p,
density , temperature T, and three components of the velocity vector u.
Together with the equation of state such as the ideal gas law - p V = n R
T, the six equations are just enough to determine the six dependent
variables. In general, all of the dependent variables are functions of all
four independent variables. Usually, the Navier-Stokes equations are too
complicated to be solved in a closed form

17. THE NAVIERSTOKES EQUATIONS
• where v is the kinematic viscosity, u is the velocity of the
fluid parcel, p is the pressure, and ρ is the fluid density.
• equation that describes the motion of the fluid at any time or location
within the flow field
• The NavierStokes equations are the solutions of Newton’s second law
of motion applied to fluid flow. For incompressible flows with a
constant viscosity, these equations simplify to

18. Bernoulli equation
• The Bernoulli equation is a useful formula that relates the
hydrostatic pressure, the fluid height, and the speed of a fluid
element. However, there are a few important assumptions that
are made to derive this formula, which makes this powerful
equation not necessarily useful in many biofluid mechanics
applications.
• Although as a back-of-the-envelope calculation, the Bernoulli
equation can approximate real flow scenarios with reasonable
accuracy.
• To derive this equation, the conservation of mass and
conservation of momentum equations are simplified by making
the assumptions that the flow is steady, incompressible , and
inviscid (has no viscosity).

21. • Viscoelasticity is the property of materials that exhibit
both viscous and elastic characteristics when undergoing deformation.
Viscous materials, like water, resist shear flow and strain linearly with time
when a stress is applied. Elastic materials strain when stretched and
immediately return to their original state once the stress is removed.
• For example Ligaments and tendons.
• disks in the human spine

22. Viscoelasticity
• Viscoelastic materials are those for which the
relationship between stress and strain
depends on time.. Strain-time curves for
various constant stresses are shown in the
following figure for a linear viscoelastic
material.

23. Elasticity is the ability of a body to resist a distorting influence and to
return to its original size and shape when that influence or force is
removed. Solid objects will deform when adequate forces are applied
to them. If the material is elastic, the object will return to its initial
shape and size when these forces are removed.
• For example rubber.

24. • Viscoelastic material :the loading
and unloading curves do not
coincide, Fig a, but form a
hysteresis loop.

25. • Elastic material: After load is
removed, there are long range
residual stresses that get relaxed
over time • Returns to original
state after a sufficient time.

26. • The effect of rate of stretching shows that the viscoelastic material
depends on time. This contrasts with the elastic material, whose
constitutive equation is independent of time, for example it makes no
difference whether an elastic material is loaded to some given stress
level for one second or one day, or loaded slowly or quickly; the
resulting strain will be the same.

27. Testing of Viscoelastic Materials
• The creep-recovery test involves loading a material at constant stress,
holding that stress for some length of time and then removing the
load.
• First there is an instantaneous elastic straining, followed by an ever-
increasing strain over time known as creep strain. The creep strain
usually increases with an ever decreasing strain rate so that
eventually a more-or-less constant-strain steady state is reached, but
many materials often do not reach such a noticeable steady-state,
even after a very long time.

28. The creep-recovery test
• When unloaded, the elastic strain is recovered immediately.
• There is then anelastic recovery – strain recovered over time; this
anelastic strain is usually very small for metals, but may be significant
in polymeric materials.
• A permanent strain may then be left in the material1 . A test which
focuses on the loading phase only is simply called the creep test.

29. The creep-recovery test

30. Stress Relaxation Test
• The stress relaxation test involves
straining a material at constant strain
and then holding that strain. The
stress required to hold the
viscoelastic material at the constant
strain will be found to decrease over
time. This phenomenon is called
stress relaxation; it is due to a re-
arrangement of the material on the
molecular or micro-scale.

31. The Cyclic Test
• The cyclic test involves a repeating
pattern of loading-unloading‫ز‬
• The results of a cyclic test can be quite
complex, due to the creep, stress-
relaxation and permanent
deformations.