PHY.1:5 - Archetypal Grounding

The following constructed cases derive conditional comparisons. The physical laws and idealizations used in each case are stated; no apparatus measurements are reported.

PHY.1:5.1 - Find conflicting requirements before building a smaller flow experiment

A team proposes a quarter-size free-surface water experiment to investigate a flow in which inertia, gravity and viscosity may affect the result. The proposed geometry is similar, gravitational acceleration is unchanged, and the same liquid is initially intended. Use an incompressible Newtonian-fluid account. A relevant surface-tension, compressibility or other omitted effect would add a condition to this account.

Let L be characteristic length, U speed, g gravitational acceleration and nu kinematic viscosity. Comparing the acceleration contributions from :4.3 gives the Froude and Reynolds numbers:

Fr = U/sqrt(g*L)
Re = U*L/nu.

Write L'=lambda*L for the model length. To preserve Fr at the same g, solve:

U'/sqrt(g*lambda*L) = U/sqrt(g*L)
U' = sqrt(lambda)*U.

The same liquid then gives Re'/Re=lambda^(3/2). For lambda=1/4, the required speed is U/2 and Re’=Re/8. A corresponding transit time L’/U’ is one-half of L/U. Those settings preserve the inertia/gravity ratio while changing the viscosity/inertia ratio.

If the desired output needs both ratios reproduced, changing speed alone cannot repair the comparison. At fixed g, solving both conditions requires nu'=lambda^(3/2)*nu, which is nu/8 in this case. Selecting a liquid with that viscosity remains a physical-material question, including its other properties. Alternatively, with the same liquid and variable effective gravity, the conditions give U'=U/lambda and g'=g/lambda^3. At quarter scale those values are 4*U and 64*g; the transit time is divided by sixteen. A rotating apparatus proposed to create that acceleration also introduces rotation and spatial variation that may affect the comparison.

The first result is a decision about the experiment. If viscosity is negligible for the intended output throughout the relevant regime, the Froude-scaled experiment may suffice under that approximation. If viscosity changes separation or another needed behavior, retain that dependence and change the arrangement or the method of obtaining the answer. Matching initial and boundary conditions remains part of either construction.

Now change the question from a large-scale surface response to a local viscous effect near a wall. The previous gravity-dominated approximation leaves the new output unsupported. The method returns to the relative contributions and near-wall scale; the old choice of speed remains useful only for the question it answered.

PHY.1:5.2 - Reproduce one deformation without claiming the whole field

Consider a straight uniform bar, fixed at the top, with an axial force F pulling down at the lower end. Let L be length, A cross-sectional area, Y Young’s modulus, rho density and g gravity. Use small-strain linear elasticity and uniform material properties. Let x measure height from the bottom. The part below that point contributes weight rho*A*g*x, so the tensile force there is F+rho*A*g*x. Hooke’s law gives local strain:

strain(x) = F/(Y*A) + rho*g*x/Y.

Integrating from 0 to L gives total extension delta and mean strain:

delta = F*L/(Y*A) + rho*g*L^2/(2*Y)
delta/L = F/(Y*A) + rho*g*L/(2*Y).

A geometrically similar bar made of the same material has L'=lambda*L and A'=lambda^2*A. To reproduce the applied-force contribution to strain, use F'=lambda^2*F. Its own weight instead changes by lambda^3. At unchanged g, the self-weight contribution to strain changes by lambda. Merely using a smaller copy with the same material does not reproduce the two load contributions together.

Suppose the original bar hangs under its own weight, with F=0, and the question concerns only total extension relative to length. An added lower-end force on the smaller bar can match that output. Solve the mean-strain equation for the new force:

F' = A'*rho*g*L*(1-lambda)/2.

At quarter scale, F’ is 3/128 of the original bar’s weight. Substitution into the smaller bar’s mean-strain equation gives rho*g*L/(2*Y), the original value. The comparison therefore reproduces the requested normalized extension within this model.

Now ask for the strain at the lower end. The original unloaded bar has zero strain there; the smaller bar with the added force has F'/(Y*A') greater than zero. The output-specific construction cannot answer this new local question. To reproduce the complete strain profile, revisit the distributed loading or the combination rho*g*L/Y. The successful first comparison is retained for total extension.

PHY.1:5.3 - Preserve competing physical time scales

A substance diffuses along an interval and is consumed by a first-order reaction. A smaller comparison is proposed using the same diffusivity D and reaction rate k. Let c(x,t) be concentration, with the idealized equation:

c_t = D*c_xx - k*c,   0 < x < L.

Both ends absorb the substance, so c(0,t)=c(L,t)=0. For a simple worked preparation take c(x,0)=c0*sin(pi*x/L). Substitution gives:

c(x,t) = c0*sin(pi*x/L)*exp(-(pi^2*D/L^2 + k)*t).

Use x=L*xi and the diffusion time T=L^2/D. At scaled time tau=t/T, the equation becomes partial c/partial tau = partial^2 c/partial xi^2 - Da*c on 0<xi<1, where Da=k*L^2/D compares reaction with diffusion. The midpoint concentration relative to c0 is exp(-(pi^2+Da)*tau).

After L’=L/4 at unchanged D and k, the diffusion time is T/16 but Da becomes Da/16. If the original Da is 1, then at tau=1 the smaller experiment has a midpoint concentration exp(15/16), approximately 2.55, times the original normalized value. At that corresponding time, the smaller experiment has lost less substance to reaction.

To reproduce the dimensionless evolution with the same D, the smaller model would require k'=16*k, together with the corresponding initial and boundary conditions. A physical change intended to obtain that rate may change D too; solve using the resulting pair of properties. If the question instead concerns diffusion during an interval when consumption has a negligible effect, derive and use that shorter-time approximation. The reaction can re-enter when the requested duration changes.