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Mechanical Engineering Module 8

By Martin Holmes,2014-05-07 19:27
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Mechanical Engineering Module 8

    ME 6601: Introduction to Fluid Mechanics

    Module 8

    Table of Contents

    ? Slide 1 Balance of Linear Momentum I

    ? Slide 2 Balance of Linear Momentum

    ? Slide 3 - Forces

    ? Slide 4 Forces (revisited)

    ? Slide 5 Balance of Linear Momentum mathematical statement

    ? Slide 6 Balance of Linear Momentum mathematical statements (continued)

    ? Slide 7 Local Equilibrium

    ? Slide 8 Local Equilibrium - proof

    ? Slide 9 Stress Tensor

    ? Slide 10 Stress Tensor (continued)

    ? Slide 11 - Stress Tensor (continued)

    ? Slide 12 - Stress Tensor (continued)

    ? Slide 13 Balance of Linear Momentum I - Summary

Slide 1 Balance of Linear Momentum I

Welcome back. You will recall that in a previous module we applied the

Reynolds Transport Theorem to the principle of the conservation of mass in

order to derive a local form of that principle that is going to work for us in

spatial coordinates. In this module, what we would like to do is start the

same process for the balance of linear momentum.

That is ultimately going to lead us to a set of equations of motion for viscous

fluids as equations of motion which are known as Navier-Stokes equations,

and along the way we shall derive the set of equations governing the motion

of an inlet viscid fluid as well as a viscous fluid.

Again, the balance of linear momentum is nothing more than a statement of

Newton’s Second Law of Motion,

and so we will be applying the Reynolds Transport Theorem to that,

expressed in material coordinates,

to give us the spatial coordinate form.

Slide 2 Balance of Linear Momentum

Now, Newton’s Second Law of Motion applied to a particular piece of the

continuum tells us the following in words. It says that the time rate of

change, or increase, of the linear momentum within a fluid material volume

that we will indicate by capital V, just as we did with conservation of mass,

is equal to the resultant external force acting on the fluid in the volume,

capital V. So that is a very simple statement of Newton’s Second Law.

We need to examine both sides of this equation. We need to look at the time

rate of increase of linear momentum within the fluid material volume (it

turns out that is the easy part),

and we also need to look at the types of forces that are going to be acting on

the fluid in that volume. We will start with the forces to begin our process.

Slide 3 - Forces

There are two types of forces that we are going to be dealing with. The first

    type of force is called a body force. Body forces are very long-range forces,

which permeate the matter.

The examples, the easy examples to think of, are gravity and

    electromagnetic forces. In addition, body forces depend upon the mass of the

body on which they are acting.

    Gravitational force, my weight, depends upon my mass, not just my volume

or my surface area. So body forces depend upon that mass.

    The other type of force is a surface force. Surface forces are forces that are

    exerted by material on one side of a surface element, an imaginary surface

element,

    on the material, on the opposite side of that surface element. The length

    scale over which these surface forces act, unlike the body forces,

    which are long-range forces, is intermolecular distance. So surface forces act

     the material on this side of this surface is exerting a force on the material

    on the other side of the surface. Now, this concept of surface forces seems

like a simple one to us,

    but it comes from something called the Cauchy Stress Principle, which we

will examine a statement of right now.

    The Cauchy Stress Principle dates back to about 1827. The statement that we

are going to be looking at is due to Clifford Truesdell,

    and the statement is the following. Upon any imagined closed surface S,

closed surface, there exists a distribution of stress vectors

    (we are going to call those stress vectors t. We will be using t for something

else later, but for right now it is the stress vector),

    whose resultant and moment are equivalent to the actual forces of material

continuity exerted by the material outside the surface S

    (it is a closed surface) upon the material inside the surface capital S.

    Now this seems like a very simple concept, but as Truesdell points out, a

    whole generation of mathematicians and hydrodynamicists were trying to

    come up with an expression for the forces of material continuity.

    That means the forces that keep the material from tearing itself apart. Now, a

    whole generation of people had tried to do this, in mostly an ad hoc fashion,

    without any success, until Cauchy postulated this very simple statement.

    Okay, so this is how we are going to be modeling these intermolecular

    forces that exist between pieces of the fluid continuum. That is just a

distribution of the stress vectors acting on surfaces.

    Slide 4 Forces (revisited)

    Now we are going to be designating the two types of forces differently.

Body forces we will designate with vector lowercase f.

    That is going to indicate the body force per unit mass, so that if we want the

total body force acting on a particular body of mass,

    we are going to need to multiply f by m to get the total force. Surface forces,

on the other hand, are just a force per unit area,

    and we are going to indicate surface forces by the vector t. As I said, later

we will be using t to indicate something else,

    but for right now just think of that vector t as the surface force exerted on

    some element. That force does not have to be normal to the element,

    and it does not have to be tangential to the element. In fact, in general it will

contain both components.

Slide 5 Balance of Linear Momentum mathematical statement

    Now, what we are going to do is create a mathematical statement of

Newton’s Second Law of motion,

    as applied to an arbitrary fluid material volume, capital V of lowercase t,

bounded by a surface capital S of lowercase t.

    And we can do that in a very simple fashion, now that we know how to

represent the forces.

    The left-hand side just represents the time rate of change following the

    motion (we are using the material derivative notation capital D by capital D

    lowercase t) of the total linear momentum of the fluid in the material volume

capital V of lowercase t.

    So lowercase v is the velocity vector, rho is the density function, as I used

    with the continuity equation derivation, lowercase d capital V is the

    differential volume element, so rho times lowercase d capital V represents

the differential mass multiplied by the velocity,

    integrated over the entire volume. That is the time rate of change of linear

    momentum following the motion, is equal to the sum of the forces.

    And for the body forces, remember f is the body force per unit mass, rho

    times lowercase d capital V is the mass of a differentially small element,

    so if we integrate that over the entire volume, we get the total body force.

    Lowercase t, on the other hand, is the surface force per unit area.

    We multiply that by the differentially small area lowercase d capital S, and

    integrate over the entire area S of t, which bounds the material volume

capital V of lowercase t. So that is our global statement now of Newton’s

Second Law of Motion, in terms of material formulation.

    And again, we are just going through the various changes, so the red terms

    indicate the time rate of change of linear momentum in capital V.

    The blue terms, the middle terms indicate the total body force acting on the

fluid material in V of t.

    And finally, the magenta terms are the total surface forces acting on the fluid

material in V of t. So that is our integral form of Newton’s Second Law.

Slide 6 Balance of Linear Momentum mathematical statements

    (continued)

Using the corollary to the Reynolds Transport Theorem you will remember

that corollary stated if conservation of mass held,

    then we had capital D by capital D lowercase t of the integral of rho times

    something. We could write that just as the integral over capital V of

    lowercase t of rho times the material derivative of what is left. So therefore,

    we have the integral over capital V of lowercase t of rho times the material

    derivative of the velocity field, integrated over the volume. This is not a

working form, however.

    Slide 7 Local Equilibrium

    We want a local form of this equation. We want form of this equation valid

    at every point in the fluid, and in order to pursue that local form of the

equation,

    we are going to have to prove a little theorem called local equilibrium. The

    theory of local equilibrium tells us that the stress forces are in local

equilibrium at each point

    (that means that balance themselves at each point) in the flow field.

    So in order to prove that, it is a pretty simple proof, the first thing we are

    going to have to do is begin by recognizing that volumes are of the order of

a length scale cubed. ‘l’ is a length scale representative of our volume capital

V of lowercase t. So capital V is of order lowercase l cubed.

    This big-O notation indicates the way that capital V behaves as lowercase l

    changes. In particular, we are going to be concerned with how it behaves as l

goes to zero.

    So the volume goes to zero very quickly as the length scale cubed, whereas

    the surface, on the other hand, goes to zero as the length squared.

    Alright, so S is of order l squared. We can then rewrite our global form that

we have in the following manner.

    Slide 8 Local Equilibrium - proof

That is our form, now we are going to rewrite it.

    We are going to divide by lowercase l squared on both sides of the equation,

    and we are going to lump the two volume integral terms together on the left-

hand side of the equation.

    So all we have done is taken the body force integral to the left hand side of

    the equation, and combined it with the time rate of change of the linear,

    and we have divided by lowercase l squared. And now, we take the limit.

    If we take the limit as lowercase l goes to zero, we recognize that nothing in

here varies with length, but the volume goes as l cubed,

    and so something that goes as l cubed divided by something that goes as

    lowercase l squared in the limit as lowercase l goes to zero, will vanish.

    That means, therefore, that the limit of the right-hand side as lowercase l

goes to zero must also vanish.

    And, as we shrink down to a point, that is going to tell us that our

    distribution of stress vectors is going to be at equilibrium at that particular

point.

    Slide 9 Stress Tensor

    Okay, so that is our principle of local equilibrium. And now what we wish to

    do is to apply that principle of local equilibrium to a very particular fluid

volume.

    And in undergraduate courses, this is usually done to a rectangular piped.

    Here, we are going to be a little bit more sophisticated with it,

    and apply it to a trapezoidal fluid volume. I am sorry, not trapezoidal, but

tetrahedral. Pardon the mistake.

    So this is the tetrahedron bound by the three coordinate axes x subscript one,

x subscript two, and x subscript three.

    And I see my figure has overlapped there, which is a bit of a mistake. We

are going to indicate the unit outward vector at each face.

    So lowercase n here, indicated in red, represents the unit outward normal, on

the slanted face which is colored blue,

    and it has components lowercase n subscript one times lowercase i, where

    lowercase i is the unit vector along the lowercase x subscript one axis,

    plus lowercase n subscript two times lowercase j, where lowercase j is the

unit vector along the lowercase x subscript two axis,

    plus n subscript three times k, and n subscript one, n subscript two, and n

subscript three are the three direction cosines.

    The unit vector on this back face, marked by the x subscript one/x subscript

two plane, is minus lowercase k, it’s the unit outward normal,

    and the area of that back face is n subscript three times sigma, the area of the

    slanted face, where n subscript three is again, that direction cosine that

appears right here.

    The unit vector here is minus i. The associated area is n subscript one times

    sigma, and the unit vector here is minus j, with an area n subscript two times

sigma.

    I will clean this up before you see the printed version of it. Okay, so we have

    indicated now the unit vector and the area of each one of the faces of that

stress tetrahedron.

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