2- Momentum AND Vector Notation PDF

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MOMENTUM AND VECTOR NOTATION One of the important things to remember:

So if we have a defined system, focused in on a single point, like above, then we really should evaluate the system from all possible directions.

It is extremely beneficial to review vector calculus as well before we move forward. Unit vectors:

Scalar: Vector:

Tensor:

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Now, before we can start considering how pressure impacts a moving fluid, we need to begin considering overall impacts from stresses (which combine with pressure in many cases for overall impact). To be able to fully evaluate pressure and stress together, though, it helps to have some grasp on vector notation. So let’s review. Partial time differentiation:

Total time derivative:

Substantial time derivative:

Gradient:

Gradient of a scalar:

Gradient of a vector:

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Divergence: We represent the divergence with: And use a dot product because: Divergence of a vector:

Divergence of a tensor:

Now, what are the different forces we are concerned with acting upon?

Don’t want to evaluate an entire system all together all at once!

So instead, we define:

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Now let’s return to our defined volume within the flow field. We’ve acknowledged that there may be viscous forces exerted on the fluid, contributing to the overall momentum transfer. In general, What other force is likely to exert on the fluid? This will always be perpendicular to the exposed surface! Because it is a scalar, we must multiply by the unit vector, essentially acting to direct the fluid. -

We can combine both of these forces to define overall molecular stresses:

Normal stresses: Shear stresses:

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Let’s summarize these components!

How else can momentum be transferred?

Again, let’s consider our cube-shaped defined volume. Considering that momentum per unit volume is:

and the volume rate of flow through the unit area is:

then the flux through the defined perpendicular plane:

So we can have multiple descriptions of the momentum flux for the flux across the areas perpendicular to each axis:

But we also have flux resulting from effects of two dimensions:

This is similar to how the viscous stresses are noted! So overall, this leaves us with the convective momentum-flux tensor: Which we can combine with the molecular momentum-flux tensor to achieve the combined momentum-flux tensor:

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Summarized:

convective momentum-flux tensor: viscous momentum-flux tensor: molecular momentum-flux tensor: combined momentum-flux tensor:

Usually, we focus on specific components of the tensors, not the overall tensors themselves. These can be broken down depending on which dimensions we are evaluating. For example: φxx =

φxy =

Let’s take the difference between these different contributions to momentum transport a step further. Molecular transport:

Convective transport:

If we wanted to compare these two components against each other:

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Each of these as we have just written, however, is a reduced form simplified based on common assumptions or conditions that apply to general systems. It is important to make sure that we have the full equations before proceeding. Because the stress, τ, is a tensor and dependent on two dimensions, we need to recognize that: So:

Assume:

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And if the fluid is isotropic:

Based on simplifications and experimental determination, we know:

where: Written in vector-tensor notation:

Note:

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We can thus go through and derive what the constitutive equations are for each viscous stress, depending on dimensions, and depending on coordinate systems. Cartesian:

Cylindrical:

Spherical:

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At this point, we have worked towards being able to define constitutive equations depending on the system structure. From there, we can utilize the operating and boundary conditions to more fully solve and define the system properties. In order to succeed in doing so for momentum, however, we need to be able to return to the general laws of conservation, specifically the conservation of mass. The law of local conservation of mass is also known as:

This can be defined for each set of coordinate systems. Rectangular:

Cylindrical:

Spherical:

Before moving on to applying balances for flow systems, we should stop and define compressibility vs. incompressibility.

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