so what is the net effect of shear
as it applies to a cubic fluid particle
well we've seen before that when we try
to break up shear
as it applies to for example a cubic
particle like this
and we break this up into x y and z
components
when we look at only one direction the x
direction
we have to add up the contributions from
three pairs
of faces one is this pair front and back
one is this pair here left and right and
one is this pair here
top and bottom and so we wrote this
equation here that says
the shear in the x direction so the x
component of shear
as it applies to this particle here is
the volume of the particle
multiplied by the sum of three vectors
and every time those vectors
are the change in the direction
of the shear pointing in x on the
surface perpendicular to that direction
and so we said um there must be a better
way of writing this and the good news is
there is
and so for this we need to go back a
little bit in time
to the previous operator that we saw
before when we
looked at pressure we use the operator
gradient gradient doesn't mean anything
it's waiting to be applied to something
and the gradient is applied to scalar
fields so if you have a scalar field
a field of pressure a field of
temperature you apply a gradient to it
and it will tell you
it will give you a vector in space and
every time the local vector
is in the x direction it is the change
x of that thing, in the y direction
it is a change in y of that thing and so
on and so forth
so that if you apply it to a field a for
example
then it tells you in which direction a
is getting bigger so every time you have
a vector [that] points
into which direction and by which
magnitude
the value a is getting larger in space
okay this is the gradient and now we
introduce
a cool new toy and this toy is the
divergence
the divergent is similar to the gradient
but it's a little bit different
the divergent like the gradient is
a tool it doesn't mean anything by
itself it's
an operator that's waiting to be applied
when you apply it to a vector field you
will take
the change in x of the dot product
of the i vector and your vector, the
change in y
of the [dot product of the] j vector and your vector
so that every time you have this dot
product of a unit vector in the
direction and your vector
which means you're going to have the
component
in this direction of your vector so
let me give you an example you have a
vector field A
like this you take the sum here
of three terms and every time it is the
change in x
of the x component of a the change in y
of the y component of a and the change
in z
of the z component of a
so if you write it perhaps in a more
compact way
partial partial x of the x component of
a
partial partial y of the y component of
a and so on and so forth for z
this is a scalar it's a number
at any point in space you're going to
have a scalar value
that gives you by how much the x
component of a is changing in a
plus by how much the y component of y of
a is changing in y
plus by how much the z component of a is
changing in z
so what happens if we apply this to
now a tensor field well the same process
happens
if we apply this to a tensor field the
number of dimensions reduces
by one step
and we can come back now to this net
force, or the
 x component of the net force
due to shear
which we said was a change in z of
something on planes perpendicular to z
plus
the change in y of something on planes
perpendicular to y and so on and so
forth
this thing here we can write it like
so we can say it is the divergence
of the component of the shear tensor
that is in the x direction
okay this is super cool because now when
we sum up
not just x but x y and z as it applies
to this
approximately cubic infinitely small
particle
we have bam net force due to shear
is made of three components one in x one
in y one z
and every time each component is the
divergence
of the component of the shear tensor in
this direction
okay so we can write it even in a more
compact way
by saying wait a second the divergent of
the x component divergent of the y
component
and divergent of the z components as a
vector this is
just the divergence of the shear tensor
and so we get this equation here which
is kind of cool which is
the net force due to shear per unit
volume
on a certain small box like this it is
the divergence
of the shear tensor shear tensor has
here
18 components and when we take the
divergence of this we reduce this
into three components and this will give
us
in newtons per meter cube the amount of
force
that the shear is exerting
on this particle around here so
the net force per volume is the
divergence of the shear tensor
this will be extremely useful once we
start adding
the three kinds of forces that apply to
a fluid pressure
gravity and shear together and say the
sum of those forces is equal to
the mass times acceleration of the
particle
but this we keep for the next chapter
in the wait for that chapter
let's just recap what we saw before
pressure has only one component pressure
is a scalar
field it means at every point in space
there is only
one local value of pressure doesn't have
a direction
when we have pressure on a certain
volume
we need to take into account six
components one for each of the six
faces of the cube so it's a matrix if
you want
and once we take into account all of
those values
to calculate the net effect of all those
pressure effects then we take the
gradient of pressure
and the gradient of pressure has three
components it's a vector field
so we have everywhere in space one
vector with three components
pointing as to
the direction and the magnitude with
which pressure is pushing the particle
this is for pressure now let's take a
look at shear
here at a point it has three components
shear is an
arrow it points in some specific
direction with some magnitude
so it's a vector field when we apply
shear to a volume there are six faces on
this cube
on each of those six faces we're going
to have one three component vector
so there are 18 components in total this
is a vector
made out of vectors which we call a
tensor
and then the net force due to shear we
don't want
all 18 values we want the net effect of
those 18 values
we take the divergence of shear and the
divergence of this tensor
has three components and it is a vector
a vector field um so here we are
let's just take a look at a
few pictures
to conclude this is not exactly a
fluid
it's not exactly a solid either it is
ice flowing down a glacier and i like
ice flows on the glaciers because
they display pretty well the effect
of shear because they fracture and they
fracture in ways that are very elegant
and so i like to see i like to visualize
in those photos the direction
and the net effect of shear on the ice
here for example accelerating the ice
but you also have shear which is
downwards as the ice progresses
horizontally
and you also have shear sideways as the
as the eyes flows
downwards this is actually this one
here is my favorite picture
because we can see pretty well the
sideways
shear effect on the ice as it moves
down
the valley so here you are this is
how we quantify
shear in fluid mechanics.
