Fluids as continuous media

A fluid contains molecules, but tracking each molecule is unnecessary for most flows. Continuum mechanics replaces microscopic motion with smoothly varying fields such as density, pressure, velocity, and temperature. The approximation works when a sampling volume is small compared with the flow feature of interest yet contains many molecules.

The continuum approximation

At microscopic scale, density in an extremely tiny volume fluctuates as molecules enter and leave. At a larger scale, average mass divided by volume becomes smooth enough to treat as a field:

\[\rho(\mathbf x,t)=\lim_{\Delta V\to\Delta V_{\mathrm c}}\frac{\Delta m}{\Delta V}\]

The notation emphasizes a representative continuum volume \(\Delta V_{\mathrm c}\), not a literal limit to zero molecules. It must be much smaller than the macroscopic length scale and much larger than molecular spacing or mean free path.

Total mass in a region \(V\) is

\[M=\int_V\rho(\mathbf x,t)\,dV\]

Density is an intensive field; total mass is extensive and depends on region size.

Fields that describe fluid state

Common continuum variables are

\[\rho(\mathbf x,t),\qquad p(\mathbf x,t),\qquad\mathbf v(\mathbf x,t),\qquad T(\mathbf x,t)\]

Density, pressure, and temperature are scalar fields. Velocity is a vector field. A complete model may also need composition, internal energy, viscosity, and thermal conductivity. An equation of state connects thermodynamic fields; they cannot all be chosen independently.

Pressure has units of force per area and acts normally on a surface in a fluid at rest. Velocity gives the local mean motion at a point. It is not the rapidly varying molecular velocity of one particle.

import numpy as np

cell = {"density_kg_m3": 1000.0, "volume_m3": 0.002, "pressure_pa": 101325.0}
cell_mass = cell["density_kg_m3"] * cell["volume_m3"]
print(f"Cell mass: {cell_mass:.3f} kg")

Each field value must be associated with either a cell center, face, node, or particle sample. Mixing those locations without interpolation rules creates inconsistency.

Eulerian and Lagrangian descriptions

An Eulerian description asks what flows through fixed locations in space. The field \(\mathbf v(\mathbf x,t)\) is Eulerian. A Lagrangian description follows identified material parcels as they move.

A parcel path \(\mathbf X(t)\) satisfies

\[\frac{d\mathbf X}{dt}=\mathbf v(\mathbf X(t),t)\]

Streamlines are curves tangent to the velocity field at one instant. Pathlines are actual parcel trajectories through time. In steady flow they coincide; in unsteady flow they need not.

position = np.array([1.0, 0.0])
dt = 0.001
for _ in range(1000):
    velocity = np.array([-position[1], position[0]])
    position += dt * velocity
print("Advected parcel position (m):", position)

This calculation traces a parcel through a prescribed field. It does not solve how the field itself evolves.

Change following a moving parcel

A scalar field can change because time passes at a fixed point and because a parcel moves into a region with a different value. The material derivative combines both effects:

\[\frac{Dq}{Dt}=\frac{\partial q}{\partial t}+\mathbf v\cdot\boldsymbol\nabla q\]

For velocity, parcel acceleration is

\[\frac{D\mathbf v}{Dt}=\frac{\partial\mathbf v}{\partial t}+(\mathbf v\cdot\boldsymbol\nabla)\mathbf v\]

A steady velocity field has zero explicit time derivative, but a parcel can still accelerate as it moves into regions where velocity differs. This convective acceleration explains how steady flow through a narrowing pipe can speed up.

Conservation of mass

Mass cannot appear or disappear in an ordinary fluid model. The local continuity equation is

\[\frac{\partial\rho}{\partial t}+\boldsymbol\nabla\cdot(\rho\mathbf v)=0\]

The first term is local density change. The divergence term is net mass flux leaving a small volume. For constant-density incompressible flow, this reduces to

\[\boldsymbol\nabla\cdot\mathbf v=0\]

Incompressible does not mean motionless. It means a material parcel does not change volume, and in the common constant-density setting the velocity field has zero divergence.

For a fixed control volume, mass changes according to flux through its boundary. A numerical finite-volume method mirrors this statement by using one shared face flux with opposite signs for neighboring cells.

density, area, dt = 1000.0, 0.01, 0.1
outward_normal_speeds = np.array([-0.02, 0.02])
fluxes_kg_s = density * area * outward_normal_speeds
new_mass = cell_mass - dt * fluxes_kg_s.sum()
assert np.isclose(new_mass, cell_mass)
print("Balanced cell mass:", new_mass)

If each neighboring cell invents its own unrelated face flux, global conservation can fail even when the pictures look smooth.

Momentum and pressure

For an inviscid fluid under body acceleration \(\mathbf b\), Euler's equation is

\[\rho\frac{D\mathbf v}{Dt}=-\boldsymbol\nabla p+\rho\mathbf b\]

Pressure gradients accelerate fluid from higher toward lower pressure, while body forces such as gravity act throughout the volume. A viscous Newtonian fluid adds stress terms; for constant viscosity in a simple incompressible setting, the momentum equation contains \(\eta\nabla^2\mathbf v\).

The ideal-fluid model used for elementary Bernoulli reasoning assumes zero viscosity and usually incompressible flow. Real fluids have viscosity, boundary layers, and losses.

Discretizing a continuum

A grid represents averages or samples, not infinitely detailed fields. In three dimensions, halving spacing in all directions increases cell count by roughly eight for the same domain. It also often requires a smaller time step and more memory.

A particle method represents moving samples and reconstructs fields through a kernel or neighborhood. A finite-difference method approximates derivatives at points. A finite-volume method updates cell-integrated conserved quantities through face fluxes. These approaches can approximate the same continuum laws but have different conservation, boundary, and stability properties.

Resolution must be compared with physical scales. A vortex or interface narrower than a few cells is not resolved. Numerical diffusion can smear features; numerical dispersion can create oscillations. Refining a grid without checking convergence does not certify the answer.

Boundaries and closure

Equations need initial and boundary conditions. A solid wall may impose no penetration; a viscous model commonly also uses no slip. Inlets need specified flow data, outlets need consistent pressure or flux treatment, and free surfaces require interface conditions.

The fields also need closure. An incompressible model may treat density as constant and solve pressure as the constraint enforcing divergence-free velocity. A compressible gas needs an equation of state and an energy equation. Choosing pressure and density updates independently can violate thermodynamics.

When the continuum model fails

The Knudsen number compares molecular mean free path \(\ell\) with characteristic length \(L\):

\[\mathrm{Kn}=\frac{\ell}{L}\]

When this ratio is not small, such as in rarefied gas or micro-scale flow, ordinary continuum equations and no-slip boundaries may fail. Molecular or kinetic models may be needed. Continuum fields can also need extra physics at shocks, phase boundaries, reactive fronts, and turbulent scales smaller than the grid.

Verification foundations

A uniform fluid at rest under no unbalanced force should remain uniform. Sum cell mass and confirm that changes equal declared boundary flux and sources. Check density positivity, dimensions, boundary orientation, and shared face-flux cancellation. Refine space and time separately while comparing pressure, velocity, integrated flux, and conservation residuals at the same physical locations and times.

The continuum viewpoint is powerful because it turns an enormous molecular system into fields governed by conservation laws. Its predictions remain credible only when the averaging scale, material closure, boundaries, and numerical resolution fit the physical question.

A worked Python notebook

Represent water at fixed density by cells of different volumes. The question for this calculation is precise: how does cell mass change when we change cell volume, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The continuum approximation assumes each cell contains many molecules and resolves the relevant macroscopic variation. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0.001 to 0.01 m^3. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for cell volume, expressed in m^3. The output is cell mass, expressed in kg. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 1000 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.002 m^3, the expression gives 2 kg. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("1000*q", locals={"q": q})
reference_input = 0.002
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m^3")
print("Predicted cell mass:", reference_output, "kg")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 1 kg. The midpoint, at 0.0055 m^3, gives 5.5 kg. The upper endpoint gives 10 kg. Read these together with the assumptions: they describe different choices of cell volume within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.001, 0.0055, 0.01])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m^3 -> {output_value:.6g} kg")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows cell volume; the vertical axis shows cell mass. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.001, 0.01, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Cell volume (m^3)")
ax.set_ylabel("Cell mass (kg)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does cell mass change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.002 m^3 is 1000 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 1000\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.002045
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "kg")
print("Full expression:", full_prediction, "kg")
print("Approximation difference:", full_prediction - linear_prediction, "kg")

Here the changed input is 0.002045 m^3. The tangent estimate is 2.045 kg, while the full expression gives 2.045 kg. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 0.009000000000000001
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of cell volume inside the stated interval, predict whether cell mass will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The continuum approximation assumes each cell contains many molecules and resolves the relevant macroscopic variation. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of kg: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(0.01) - evaluate(0.001))
for points in [101, 201, 401]:
    grid = np.linspace(0.001, 0.01, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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