Understanding Pressure as a Scalar Quantity: Exploring Tensors and Rank

[ovais]

Hello everybody, yesterday I stand to teach vectors and scalars to 12th standard students in a coaching.While giving examples of scalars I named mass , work , pressure etc.Then a student argued me that pressure should be a vector quantity since when you apply a push on wall that is force then the pressure would also acts in the direction where you are applying the force.So according to him pressure is a vector quantity.While in books I always read pressure a scalar.Actually I my self used to wonder how is pressure a scalar quantity while it seems associated with direction just as a force do, there I answer that since we do talk of pressure(while on solid surfaces) as pressure on a surface, which always acts normal to the surface no matter be the surface plane or curved thus it is immaterial to say the direction of the pressure since it is always calculated on the surface that is perpendicular to it or in opposite direction of area vector( just as we take are in guss law, which however has nothing to do with pressure but is helpful here to explain about area vector).Though I explain this, neither me nor he was fully satisfied, since still it has a direction though fixed.I search the net and get many useful points but still it is not getting clear as to how pressure a scalar quantity.Then I come across with tensors, I want to know-1: what is the tensor rank of pressure? 2:If its rank is zero (that is if is scalar) then please explain what physical quantities can be taken with non-zero rank. Thanks in advance

 

[DrDu]

I am also not a specialist on this, but, at least in a solid, pressure is the trace of the stress tensor (or minus one third of it, to be precise) or, the isotropic part of it. Hence it transforms as a scalar. Liquids are special in so far as all other components of strain are zero.

 

[Phrak]

If I may restate the question

Force is a vector. Force over an area is a pressure. How can pressure be a scalar?

I might add that area is properly expressed as a 2-form rather than a tensor of two upper indices such that the area is oriented; there is a unique choice for which direction the surface normal points.

 

[DrDu]

Well, so to speak, pressure is the scalar product of normal surface vector (of unit area) times force averaged over all orientations of the surface.

 

[ovais]

thanks for your replies DrDu, I will say I am not clear when you say pressure is the trace of the stress tensor or the isotropic part of it. And how a part of something(stress tensor) non -scalar could become scalar. Thankx again

 

[DrDu]

Well, the trace of a tensor T_ij is the sum over all T_ii, which is clearly a scalar.
E.g., I can define the tensor of the tensor product of the vectors r with itself $$T_{ij}=r_i r_j$$. It is important e.g. in the definition of the moment of inertia or of the electric multipole moments. Now the trace of this tensor $$tr(T)=\sum_i r_i r_i$$ is simply the squared length of the vector r.
What does that mean in the case of pressure:
Consider a cube made out of an elastic material in a bench vice. The force on one of the two forces in the vice is F and the area of the surface is A. The force acting on the other surfaces is 0. Then the pressure is 1/3 (F/A+0+0).

 

[ovais]

ok the trace of tensor T_ij is the sum of all t_ii, can you please state the physical relationship between T_ij and T_ii using words other than trace?

 

[DrDu]

T_ii are the diagonal elements of the tensor T_ij, i.e., the elements T_ij for which i=j.
Hence, if you think of the tensor as a 3x3 matrix, the trace (or spur) is the sum over all diagonal elements of the matrix.

 

[ovais]

ah i got it u explained very well.Now I will like to know what is that tensor whose trace you defined as pressure, i know it is stress tensor as you said i want to know how i visulaize the stress tensor?where can i feel its effect?I am familiar with pressure but stress tensor is new to me?

 

[Q_Goest]

When pressure is referred to as a scalar quantity, what is meant is that pressure at any point within a fluid has no intrinsic direction to it. Consider the pressure of a fluid NOT acting against a surface. For example, consider what pressure water has at a depth of 10 feet in the middle of the ocean. If the water isn't acting against anything, there is no directional force, so we can't call it a vector. But there is certainly pressure at that depth in the ocean, just as there is pressure in a fluid at all locations within the fluid. So the pressure in a fluid is a scaler quantity. We call pressure a scalar quantity because without considering what surface the pressure is acting against, there is no force and no vector. (Note that "fluid" here means any liquid or gas.)

In contrast, pressure acting on a surface becomes a vector quantity because the interaction of the fluid against the solid surface creates a normal force. That normal force is just the pressure times the area (integrated of course) so that FORCE is a vector quantity.

 

[ovais]

did you mean the concept of pressure applies only to fluid?and the thing which acts on solid surfaces is pressure force(a vector) not pressure.Is that you mean?

 

[Q_Goest]

When referring to pressure as a scalar quantity, the underlying assumption is that we're talking about a fluid. For a fluid acting on a solid, there is a force produced which is a vector quantity, but the concept of pressure in a fluid shouldn't be confused with the concept of that pressure acting at a surface.

 

[ovais]

so pressure in fluid is scalar and pressure acting on solid surface has direction normal to it?and in the bulk fluids have pressure in scalar form?

 

[dulrich]

In contrast, pressure acting on a surface becomes a vector quantity because the interaction of the fluid against the solid surface creates a normal force. That normal force is just the pressure times the area (integrated of course) so that FORCE is a vector quantity.

Here is a quote from the http://en.wikipedia.org/wiki/Pressure#Definition" article:

Pressure ... relates the vector surface element (a vector normal to the surface) with the normal force acting on it. The pressure is the scalar proportionality constant that relates the two normal vectors

This way of saying it works for both fluids and solids. The difference is that in a fluid the pressure is the same in all directions.

 

[ovais]

the statement that pressure is the scalar proportionality constant that relates two normal vectors.please tell me what are these two vectors and how is pressure(as a constant) relates these two vectors. thanks a ton

 

[Studiot]

Interesting discussion, but won't all of this pass over the head of your 12th graders?

If you want a description to convince them, forget tensors and try this.

How do you combine (add) vectors?
How do you combine (add) scalars?
How do you combine (add) pressures?

Think of a sealed vessel half full of water. At any point in the water there is a pressure.
Now pump up the air pressure in the other half.

How much does the pressure increase at any point in the water?

This is scalar addition.

Another thing to bear in mind.

Only scalars can affect every point in this way. Vectors can only affect things in their line of action.

 

[dulrich]

the statement that pressure is the scalar proportionality constant that relates two normal vectors.please tell me what are these two vectors and how is pressure(as a constant) relates these two vectors. thanks a ton

(1) The vector normal to the surface, and (2) the force that creates the pressure. One way to think of it is that P = F/A implies F = PA. But we can write the second using both scalars and vectors so that it applies in both cases.

 

[Studiot]

I have to observe that pressure can never be a vector, simply because it does not combine according to the laws of vector addition.

 

[ovais]

dear studiot your explanation works well when dealing with fluid in containers but what should I call the quantity F/A.Suppose i apply a force of 10 N normal to a plate .1 m2 in area.then we say we are applying a pressure of 100 N/m2 on the wall.Now if we say our pressure has nothing to do with direction, dosn't it seem wrong?

 

[ovais]

(1) The vector normal to the surface, and (2) the force that creates the pressure. One way to think of it is that P = F/A implies F = PA. But we can write the second using both scalars and vectors so that it applies in both cases.

The vector normal to the surface-it that you write A and the force vector F?If this the case then P=F/A should be meaningless, since vectors do not follow division.we can give a meaning to it by your second relation F=PA.

 

[Studiot]

No, the pressure does not have a direction.

The wall has a direction.

Take the wall away, what is then the direction of the pressure? Does it remain the same?

 

[ovais]

so if i imagine that just mass relates velocity and momentum as mv=p.where mass is thought as a scalar constant relating two vector quantities-velocity and momentum,scalar pressure relates area and force as two vectors as PA=F.Is this how one think?

 

[Studiot]

Being in the UK I am not sure what standard 12 grade is but I guess it is some sort of middle school? So I don't know if your students have done any calculus.

You need to be very careful with pressure = force/area.

When you shrink your control area to zero and take the limit, what do you mean by area?

You need also to be careful with force = pressure x area

What force? Individual forces posess a line of action, they are not distributed over an area or in space.

 

[ovais]

oh 12th stanard here is equavalent to UK's High school and yes they have studied calculas but not advanced advanced calculus.and how can anyone talk of pressure without considering surface so my friend if i say my students to think of pressure(on a wall)without considering the direction of wall they will suspect me.and i think i am getting understand what pressure actually mean.i believe it must be scalar as you said.do u agree with my above posts?regards

 

[dulrich]

so if i imagine that just mass relates velocity and momentum as mv=p.where mass is thought as a scalar constant relating two vector quantities-velocity and momentum,scalar pressure relates area and force as two vectors as PA=F.Is this how one think?

This is what I was getting at. Using some better symbols: $\vec{F} = P \vec{A}$

 

[Studiot]

Introducing momentum is not a good idea.

You need to be careful not to say or introduce anything which would actually be incorrect at higher level.

Unfortunately the curriculum in the UK no longer deals with basic concepts such as centre of pressure, and also what we can and cannot apply pressure to etc.

Remember that when you discuss 'force' in you mean the normal force. Pressure is not the tangential force/area for instance.

You also need to consider how the pressure is to be applied. In my experiment what would be the difference if the bottom half of the vessel was filled with ice not water?

 

[Dale]

Now I will like to know what is that tensor whose trace you defined as pressure, i know it is stress tensor as you said i want to know how i visulaize the stress tensor?where can i feel its effect?I am familiar with pressure but stress tensor is new to me?

The Wikipedia page seems reasonably comprehensive, but I did not go into detail:
http://en.wikipedia.org/wiki/Stress_(mechanics)

If you have a material in static equilibrium then you can imagine cutting a small cube out of it. We label the faces of the cube x,y,z where the x face is normal to the x-axis and so forth. On each face there is some force which we can further break up into x,y,z components. So the x,x component of the stress tensor would be the component of the force on the x face of the cube in the x direction, and the x,y component of the stress tensor would be the component of the force on the x face of the cube in the y direction. So the diagonal elements (x,x y,y z,z) are pressure and the off-diagonal elements (x,y x,z y,z) are shear stresses. Now, a fluid cannot support shear stresses without deforming, so in a static fluid all of the off-diagonal terms are 0 and all of the diagonal terms are equal. So in a static fluid the stress tensor can be reduced to a single number. That number is the pressure.

 

[DrDu]

Nearly right, only that the diagonal elements are not called "pressures" but Nnormal stresses".
Only their mean is called pressure.

 

[Studiot]

component of the force on the x face of the cube in the x direction

component of force

or

component of stress ?

Ovais,

I think you have the basis of a good classroom discussion or three

What is the difference between Force and Pressure?

What is the difference between Pressure and Stress?

Can you always add two vectors?

Can you always add two scalars?

What Force is represented by the expression Force = Pressure x Area?

 

[Dale]

Sure, it is normal stress and it is not the force on a given face in a given direction but the differential force divided by the differential area of the face. I was trying to use terminology that ovasis would recognize. For an accurate treatment he should get a good book on statics.

 

[ovais]

Nearly right, only that the diagonal elements are not called "pressures" but Nnormal stresses".
Only their mean is called pressure.

Dale spam has given a very good explanation.And he talks of(xx yy zz)as pressure.But as you extend that this diagonal elements are not pressures but infact normal stresses.now please let me know how we find the pressure using these normal stresses.also tell if the pressure found will be uniform on every side of the cube?

 

[ovais]

ok x,x (the normal stress) is the component of the stresses on face xx, and similarly y,y &z,z are component of stresses on yy and zz faces of the cube,.now how to find pressure on the cube?

 

[DrDu]

I already explained this in post #6 and gave an example.

 

[ovais]

As x,x y,y z,z are normal stresses along the three faces. So you mean we have sum up them?I will be thankfull to know if these three NORMAL STRESSES scalar or vector?Sorry if I seem too argumentive.Thanks!

 

[DrDu]

You have to sum them and divide by -3. So pressure is minus the arithmetical average of the three normal stresses.
The normal stresses are neither a vector nor a scalar. They are simply three components of a second rank tensor.
Btw., it might be interesting for you to look up the meanings of "contraction" of a tensor and "irreducible tensors".