Dimensional Analysis

Hello

I am looking for some help with the following question:

A sphere of radius r, is moving through a fluid of density ρ, at velocity v. The velocity is sufficiently high such that the effect of the fluids viscosity may be ignored. It is postulated that the resistive force acting on the sphere: F ∝ r v ρ

Use dimensional analysis to confirm the relationship and develop a final formula for the resistive force, F.

I have used force, density, velocity and length and equated the indices with the outcome of: F=K ρr^2v^2

I am unsure if this answers the question of confirming the relationship and whether this is how the final formula for F should look.

Any help is appreciated.

Reply 1

mqb2766

21

Original post by thomas0611

Hello

I am looking for some help with the following question:

A sphere of radius r, is moving through a fluid of density ρ, at velocity v. The velocity is sufficiently high such that the effect of the fluids viscosity may be ignored. It is postulated that the resistive force acting on the sphere: F ∝ r v ρ

Use dimensional analysis to confirm the relationship and develop a final formula for the resistive force, F.

I have used force, density, velocity and length and equated the indices with the outcome of: F=K ρr^2v^2

I am unsure if this answers the question of confirming the relationship and whether this is how the final formula for F should look.

Any help is appreciated.

Its a bit of a non trivial question (not hard, but not easy). You should be able to confirm the formula, so if youve got something different it would help to see what you did.

Reply 2

thomas0611

OP

8

Original post by mqb2766

Its a bit of a non trivial question (not hard, but not easy). You should be able to confirm the formula, so if youve got something different it would help to see what you did.

I assume by confirming the relationship it means both sides are the same:

F ∝ r v ρ

MLT^-2 ∝ L x LT^-1 x ML^-3

MLT^-2 ∝ MLT^-1

Have I done something wrong here?

Reply 3

thomas0611

OP

8

Original post by thomas0611

I assume by confirming the relationship it means both sides are the same:

F ∝ r v ρ

MLT^-2 ∝ L x LT^-1 x ML^-3

MLT^-2 ∝ MLT^-1

Have I done something wrong here?

To add to my previous reply, would this be the correct solution to confirm the relationship:

F=K ρr^2v^2

MLT^-2 = ML^-3 L^2 LT^-2

MLT^-2 = M L^-3 + L^2 + L^1 T^-2

MLT^-2 = MLT^-2

Thanks for the help

Reply 4

mqb2766

21

Original post by thomas0611

I assume by confirming the relationship it means both sides are the same:

F ∝ r v ρ

MLT^-2 ∝ L x LT^-1 x ML^-3

MLT^-2 ∝ MLT^-1

Have I done something wrong here?

I just skim read and thought it was viscosity on the right rather than density which is the problem with the given equation. As you say
F = MLT^-2
Right hand side = L x LT^-1 x ML^-3 = ML^-1T^-1
(note the final expression is slightly different from yours - typo). So you want an L^2T^-1 which is the viscosity / density "problem"

Reply 5

thomas0611

OP

8

Original post by mqb2766

I just skim read and thought it was viscosity on the right rather than density which is the problem with the given equation. As you say
F = MLT^-2
Right hand side = L x LT^-1 x ML^-3 = ML^-1T^-1
(note the final expression is slightly different from yours - typo). So you want an L^2T^-1 which is the viscosity / density "problem"

So should I be using viscosity not density to confirm the relationship?

Reply 6

DFranklin

18

Original post by thomas0611

So should I be using viscosity not density to confirm the relationship?

It's a bit unclear what's going on TBH (and some suspicion there's a mistake in the question).

There's a known formula for motion of a sphere through a viscous liquid (https://en.wikipedia.org/wiki/Stokes%27_law):

F \propto \mu r v

which looks very like what you've been asked to confirm, only here

\mu

is viscosity with units

ML^{-1}T^{-1}

There's also a known formula for rapid motion of a sphere through a non-viscous liquid of density

\rho

:

F \propto \rho r^2 v^2

What you've done is (correctly) deduced that if F depends on

\rho, r, v

, then by dimensional analysis,

F \propto \rho r^2 v^2

.

But, that isn't the relationship you were asked to confirm.

So we're left with the following possibilities:

1.

The question meant to use viscosity, at some point the incorrect word density was substituted but the formula remained that for viscosity.

2.

The question meant to use density, but incorrectly used the formula involving viscosity.

3.

The question meant to use density, and the intent was to give you an incomplete formula (where you needed to find the indices).

4.

Something else.

For cases 2 + 3, you've basically done the right thing - just not matched the formula they gave you (because it's incorrect/incomplete).
For case 1, you should be using viscosity.
For case 4, who knows...?

[And because of the ambiguity, it's hard for anyone to really know what to say at this point].

Dimensional Analysis

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