{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Fluids as continuous media\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/fluids-continuous-media). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "A fluid contains molecules, but tracking each molecule is unnecessary for most flows.\n", "Continuum mechanics replaces microscopic motion with smoothly varying fields such as\n", "density, pressure, velocity, and temperature. The approximation works when a sampling\n", "volume is small compared with the flow feature of interest yet contains many molecules.\n", "\n", "## The continuum approximation\n", "\n", "At microscopic scale, density in an extremely tiny volume fluctuates as molecules enter\n", "and leave. At a larger scale, average mass divided by volume becomes smooth enough to\n", "treat as a field:\n", "\n", "\n", "$$\n", "\\rho(\\mathbf x,t)=\\lim_{\\Delta V\\to\\Delta V_{\\mathrm c}}\\frac{\\Delta m}{\\Delta V}\n", "$$\n", "\n", "\n", "The notation emphasizes a representative continuum volume\n", "$\\Delta V_{\\mathrm c}$,\n", "not a literal limit to zero molecules. It must be much smaller than the macroscopic\n", "length scale and much larger than molecular spacing or mean free path.\n", "\n", "Total mass in a region\n", "$V$ is\n", "\n", "\n", "$$\n", "M=\\int_V\\rho(\\mathbf x,t)\\,dV\n", "$$\n", "\n", "\n", "Density is an intensive field; total mass is extensive and depends on region size.\n", "\n", "## Fields that describe fluid state\n", "\n", "Common continuum variables are\n", "\n", "\n", "$$\n", "\\rho(\\mathbf x,t),\\qquad p(\\mathbf x,t),\\qquad\\mathbf v(\\mathbf x,t),\\qquad T(\\mathbf x,t)\n", "$$\n", "\n", "\n", "Density, pressure, and temperature are scalar fields. Velocity is a vector field. A\n", "complete model may also need composition, internal energy, viscosity, and thermal\n", "conductivity. An equation of state connects thermodynamic fields; they cannot all be\n", "chosen independently.\n", "\n", "Pressure has units of force per area and acts normally on a surface in a fluid at rest.\n", "Velocity gives the local mean motion at a point. It is not the rapidly varying molecular\n", "velocity of one particle." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "cell = {\"density_kg_m3\": 1000.0, \"volume_m3\": 0.002, \"pressure_pa\": 101325.0}\n", "cell_mass = cell[\"density_kg_m3\"] * cell[\"volume_m3\"]\n", "print(f\"Cell mass: {cell_mass:.3f} kg\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Each field value must be associated with either a cell center, face, node, or particle\n", "sample. Mixing those locations without interpolation rules creates inconsistency.\n", "\n", "## Eulerian and Lagrangian descriptions\n", "\n", "An Eulerian description asks what flows through fixed locations in space. The field\n", "$\\mathbf v(\\mathbf x,t)$ is\n", "Eulerian. A Lagrangian description follows identified material parcels as they move.\n", "\n", "A parcel path\n", "$\\mathbf X(t)$ satisfies\n", "\n", "\n", "$$\n", "\\frac{d\\mathbf X}{dt}=\\mathbf v(\\mathbf X(t),t)\n", "$$\n", "\n", "\n", "Streamlines are curves tangent to the velocity field at one instant. Pathlines are actual\n", "parcel trajectories through time. In steady flow they coincide; in unsteady flow they\n", "need not." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "position = np.array([1.0, 0.0])\n", "dt = 0.001\n", "for _ in range(1000):\n", " velocity = np.array([-position[1], position[0]])\n", " position += dt * velocity\n", "print(\"Advected parcel position (m):\", position)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "This calculation traces a parcel through a prescribed field. It does not solve how the\n", "field itself evolves.\n", "\n", "## Change following a moving parcel\n", "\n", "A scalar field can change because time passes at a fixed point and because a parcel moves\n", "into a region with a different value. The material derivative combines both effects:\n", "\n", "\n", "$$\n", "\\frac{Dq}{Dt}=\\frac{\\partial q}{\\partial t}+\\mathbf v\\cdot\\boldsymbol\\nabla q\n", "$$\n", "\n", "\n", "For velocity, parcel acceleration is\n", "\n", "\n", "$$\n", "\\frac{D\\mathbf v}{Dt}=\\frac{\\partial\\mathbf v}{\\partial t}+(\\mathbf v\\cdot\\boldsymbol\\nabla)\\mathbf v\n", "$$\n", "\n", "\n", "A steady velocity field has zero explicit time derivative, but a parcel can still\n", "accelerate as it moves into regions where velocity differs. This convective acceleration\n", "explains how steady flow through a narrowing pipe can speed up.\n", "\n", "## Conservation of mass\n", "\n", "Mass cannot appear or disappear in an ordinary fluid model. The local continuity equation\n", "is\n", "\n", "\n", "$$\n", "\\frac{\\partial\\rho}{\\partial t}+\\boldsymbol\\nabla\\cdot(\\rho\\mathbf v)=0\n", "$$\n", "\n", "\n", "The first term is local density change. The divergence term is net mass flux leaving a\n", "small volume. For constant-density incompressible flow, this reduces to\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla\\cdot\\mathbf v=0\n", "$$\n", "\n", "\n", "Incompressible does not mean motionless. It means a material parcel does not change\n", "volume, and in the common constant-density setting the velocity field has zero divergence.\n", "\n", "For a fixed control volume, mass changes according to flux through its boundary. A\n", "numerical finite-volume method mirrors this statement by using one shared face flux with\n", "opposite signs for neighboring cells." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "density, area, dt = 1000.0, 0.01, 0.1\n", "outward_normal_speeds = np.array([-0.02, 0.02])\n", "fluxes_kg_s = density * area * outward_normal_speeds\n", "new_mass = cell_mass - dt * fluxes_kg_s.sum()\n", "assert np.isclose(new_mass, cell_mass)\n", "print(\"Balanced cell mass:\", new_mass)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "If each neighboring cell invents its own unrelated face flux, global conservation can\n", "fail even when the pictures look smooth.\n", "\n", "## Momentum and pressure\n", "\n", "For an inviscid fluid under body acceleration\n", "$\\mathbf b$, Euler's equation is\n", "\n", "\n", "$$\n", "\\rho\\frac{D\\mathbf v}{Dt}=-\\boldsymbol\\nabla p+\\rho\\mathbf b\n", "$$\n", "\n", "\n", "Pressure gradients accelerate fluid from higher toward lower pressure, while body forces\n", "such as gravity act throughout the volume. A viscous Newtonian fluid adds stress terms;\n", "for constant viscosity in a simple incompressible setting, the momentum equation contains\n", "$\\eta\\nabla^2\\mathbf v$.\n", "\n", "The ideal-fluid model used for elementary Bernoulli reasoning assumes zero viscosity and\n", "usually incompressible flow. Real fluids have viscosity, boundary layers, and losses.\n", "\n", "## Discretizing a continuum\n", "\n", "A grid represents averages or samples, not infinitely detailed fields. In three\n", "dimensions, halving spacing in all directions increases cell count by roughly eight for\n", "the same domain. It also often requires a smaller time step and more memory.\n", "\n", "A particle method represents moving samples and reconstructs fields through a kernel or\n", "neighborhood. A finite-difference method approximates derivatives at points. A\n", "finite-volume method updates cell-integrated conserved quantities through face fluxes.\n", "These approaches can approximate the same continuum laws but have different conservation,\n", "boundary, and stability properties.\n", "\n", "Resolution must be compared with physical scales. A vortex or interface narrower than a\n", "few cells is not resolved. Numerical diffusion can smear features; numerical dispersion\n", "can create oscillations. Refining a grid without checking convergence does not certify the\n", "answer.\n", "\n", "## Boundaries and closure\n", "\n", "Equations need initial and boundary conditions. A solid wall may impose no penetration;\n", "a viscous model commonly also uses no slip. Inlets need specified flow data, outlets need\n", "consistent pressure or flux treatment, and free surfaces require interface conditions.\n", "\n", "The fields also need closure. An incompressible model may treat density as constant and\n", "solve pressure as the constraint enforcing divergence-free velocity. A compressible gas\n", "needs an equation of state and an energy equation. Choosing pressure and density updates\n", "independently can violate thermodynamics.\n", "\n", "## When the continuum model fails\n", "\n", "The Knudsen number compares molecular mean free path\n", "$\\ell$ with characteristic\n", "length $L$:\n", "\n", "\n", "$$\n", "\\mathrm{Kn}=\\frac{\\ell}{L}\n", "$$\n", "\n", "\n", "When this ratio is not small, such as in rarefied gas or micro-scale flow, ordinary\n", "continuum equations and no-slip boundaries may fail. Molecular or kinetic models may be\n", "needed. Continuum fields can also need extra physics at shocks, phase boundaries,\n", "reactive fronts, and turbulent scales smaller than the grid.\n", "\n", "## Verification foundations\n", "\n", "A uniform fluid at rest under no unbalanced force should remain uniform. Sum cell mass\n", "and confirm that changes equal declared boundary flux and sources. Check density\n", "positivity, dimensions, boundary orientation, and shared face-flux cancellation. Refine\n", "space and time separately while comparing pressure, velocity, integrated flux, and\n", "conservation residuals at the same physical locations and times.\n", "\n", "The continuum viewpoint is powerful because it turns an enormous molecular system into\n", "fields governed by conservation laws. Its predictions remain credible only when the\n", "averaging scale, material closure, boundaries, and numerical resolution fit the physical\n", "question.\n", "\n", "## A worked Python notebook\n", "\n", "Represent water at fixed density by cells of different volumes. The question for this calculation is precise: how does **cell mass**\n", "change when we change **cell volume**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "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\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "0.001 to 0.01 m^3. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for cell volume, expressed in m^3. The output is\n", "cell mass, expressed in kg. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = 1000 q\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 0.002 m^3, the expression gives 2 kg. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"1000*q\", locals={\"q\": q})\n", "reference_input = 0.002\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m^3\")\n", "print(\"Predicted cell mass:\", reference_output, \"kg\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "1 kg. The midpoint, at 0.0055 m^3, gives\n", "5.5 kg. The upper endpoint gives 10 kg.\n", "Read these together with the assumptions: they describe different choices of cell volume\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.001, 0.0055, 0.01])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} m^3 -> {output_value:.6g} kg\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows cell volume; the vertical axis shows cell mass. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "input_grid = np.linspace(0.001, 0.01, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Cell volume (m^3)\")\n", "ax.set_ylabel(\"Cell mass (kg)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "cell mass change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 0.002 m^3 is\n", "1000 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = 1000\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 0.002045\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"kg\")\n", "print(\"Full expression:\", full_prediction, \"kg\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"kg\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 0.002045 m^3. The tangent estimate is\n", "2.045 kg, while the full expression gives\n", "2.045 kg. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "input_span = 0.009000000000000001\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "cell volume inside the stated interval, predict whether cell mass will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "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\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of kg:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-019", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(0.01) - evaluate(0.001))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.001, 0.01, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 1*, Section 14.1, \"Fluids, Density, and Pressure\"](https://openstax.org/books/university-physics-volume-1/pages/14-1-fluids-density-and-pressure).\n", "- [OpenStax, *University Physics Volume 1*, Section 14.5, \"Fluid Dynamics\"](https://openstax.org/books/university-physics-volume-1/pages/14-5-fluid-dynamics).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 28, Sections 28.1-28.3](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter28/).\n", "- [Benjamin Crowell, *Light and Matter*, fluid mechanics discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Density\"](https://physics.info/density/).\n", "- [Physics LibreTexts, \"Fluid Density and Pressure\"](https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/14%3A_Fluid_Mechanics/14.02%3A_Fluids_Density_and_Pressure)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "fluids-continuous-media.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/fluids-continuous-media", "source_sha256": "f0682d7c8fb77835fa9a5c57f2ea3040b999e870b1f57e9fd1b5760d8aa80ae8" } }, "nbformat": 4, "nbformat_minor": 5 }