{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Bernoulli's equation computationally\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/bernoullis-equation-computationally). 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": [ "Bernoulli's equation is a mechanical-energy relation for flowing fluid. It connects\n", "pressure, speed, and elevation under specific assumptions. It does not say that every\n", "fast-moving fluid has low pressure regardless of streamline, height, losses, or external\n", "energy input.\n", "\n", "## The ideal relation\n", "\n", "For steady, incompressible, inviscid flow along a streamline, with gravity as the only\n", "body force,\n", "\n", "\n", "$$\n", "p+\\frac12\\rho v^2+\\rho gz=\\text{constant along a streamline}\n", "$$\n", "\n", "\n", "The three terms are pressure, kinetic energy per volume, and gravitational potential\n", "energy per volume. Every term has units of pascals, equivalent to joules per cubic meter.\n", "\n", "Between two points,\n", "\n", "\n", "$$\n", "p_1+\\frac12\\rho v_1^2+\\rho gz_1=p_2+\\frac12\\rho v_2^2+\\rho gz_2\n", "$$\n", "\n", "\n", "The speed $v$ is the magnitude of\n", "velocity. In a one-dimensional pipe calculation it is usually a section-average speed,\n", "an approximation that may require a kinetic-energy correction for strongly nonuniform\n", "profiles.\n", "\n", "## Where the equation comes from\n", "\n", "Consider a small fluid volume moving from point 1 to point 2. Upstream pressure does work\n", "$p_1\\Delta V$ on it, while\n", "downstream pressure does opposing work\n", "$-p_2\\Delta V$.\n", "The net pressure work changes kinetic and gravitational potential energy:\n", "\n", "\n", "$$\n", "(p_1-p_2)\\Delta V=\\frac12\\rho\\Delta V(v_2^2-v_1^2)+\\rho\\Delta Vg(z_2-z_1)\n", "$$\n", "\n", "\n", "Divide by volume and rearrange to obtain Bernoulli's equation. This derivation makes the\n", "assumptions visible: density is constant, the flow is steady, viscous dissipation is\n", "neglected, and the two points follow one streamline with no intervening shaft work.\n", "\n", "## Three equivalent unit forms\n", "\n", "Dividing the pressure form by density gives energy per mass:\n", "\n", "\n", "$$\n", "\\frac{p}{\\rho}+\\frac{v^2}{2}+gz=\\text{constant}\n", "$$\n", "\n", "\n", "Dividing by $\\rho g$ gives head,\n", "which has units of length:\n", "\n", "\n", "$$\n", "\\frac{p}{\\rho g}+\\frac{v^2}{2g}+z=\\text{constant}\n", "$$\n", "\n", "\n", "Pressure head, velocity head, and elevation head must not be added across mixed unit\n", "forms. Choose one form and convert every source and loss term consistently." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "density, pressure, speed, elevation, gravity = 1000.0, 101325.0, 2.0, 1.0, 9.81\n", "bernoulli_value = pressure + 0.5 * density * speed**2 + density * gravity * elevation\n", "print(f\"Bernoulli sum: {bernoulli_value:.1f} Pa\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "## Continuity supplies compatible speeds\n", "\n", "For steady incompressible flow through a pipe,\n", "\n", "\n", "$$\n", "A_1\\bar v_1=A_2\\bar v_2=Q\n", "$$\n", "\n", "\n", "If a pipe narrows, its average speed rises at fixed volume flow rate. At equal elevation,\n", "ideal Bernoulli balance then requires a pressure decrease:\n", "\n", "\n", "$$\n", "p_1-p_2=\\frac12\\rho(v_2^2-v_1^2)\n", "$$\n", "\n", "\n", "Continuity alone does not predict pressure. Bernoulli alone does not determine both\n", "unknown speeds. Their assumptions must describe the same control volume and flow." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "flow, area1, area2 = 0.02, 0.01, 0.005\n", "speed1, speed2 = flow / area1, flow / area2\n", "pressure1_minus_pressure2 = 0.5 * density * (speed2**2 - speed1**2)\n", "print(f\"Pressure drop for a level pipe: {pressure1_minus_pressure2:.1f} Pa\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "This is the basis of an ideal Venturi calculation. Real meters require calibration and\n", "loss corrections.\n", "\n", "## Torricelli outflow\n", "\n", "For a large open tank draining through a small opening a vertical distance\n", "$h$ below its free surface,\n", "take both points at atmospheric pressure. If the tank surface speed is negligible,\n", "Bernoulli gives\n", "\n", "\n", "$$\n", "v=\\sqrt{2gh}\n", "$$\n", "\n", "\n", "This resembles free-fall speed through height\n", "$h$ because gravitational\n", "potential energy becomes kinetic energy. A finite tank area gives a nonzero surface\n", "speed, and a real opening has contraction and viscous losses, so measured discharge is\n", "lower than the simplest ideal estimate.\n", "\n", "## Stagnation pressure\n", "\n", "Along a horizontal ideal streamline, bring flow to rest at a stagnation point. Bernoulli\n", "then gives\n", "\n", "\n", "$$\n", "p_0=p+\\frac12\\rho v^2\n", "$$\n", "\n", "\n", "Here $p_0$ is stagnation\n", "pressure and $p$ is static pressure.\n", "Their difference is dynamic pressure. A Pitot-static tube uses this relation to infer\n", "speed under appropriate calibration and flow assumptions.\n", "\n", "Dynamic pressure is not an extra pressure that always adds at every location. It is the\n", "kinetic term in this energy relation.\n", "\n", "## Pumps, turbines, and losses\n", "\n", "Real steady pipe systems may gain shaft energy from a pump, remove it through a turbine,\n", "and lose mechanical energy through viscosity, fittings, separation, and turbulence. A\n", "common head-form ledger is\n", "\n", "\n", "$$\n", "\\frac{p_1}{\\rho g}+\\frac{v_1^2}{2g}+z_1+h_{\\mathrm p}-h_{\\mathrm t}-h_{\\mathrm L}=\\frac{p_2}{\\rho g}+\\frac{v_2^2}{2g}+z_2\n", "$$\n", "\n", "\n", "Pump head $h_{\\mathrm p}$ is energy added\n", "per weight, turbine head\n", "$h_{\\mathrm t}$ is useful energy removed,\n", "and loss head\n", "$h_{\\mathrm L}$ is nonnegative mechanical\n", "energy dissipated per weight under this convention." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "upstream_head, pump_head, turbine_head, loss_head = 10.0, 5.0, 0.0, 2.0\n", "predicted_downstream_head = upstream_head + pump_head - turbine_head - loss_head\n", "measured_downstream_head = 12.8\n", "print(f\"Head residual: {predicted_downstream_head-measured_downstream_head:.2f} m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "A nonzero residual may indicate measurement error, an omitted device, heat or unsteady\n", "storage, an invalid incompressible assumption, or an incorrect loss model.\n", "\n", "## Along one streamline or across the field\n", "\n", "For steady inviscid flow, Bernoulli's constant is generally constant along each\n", "streamline but can differ between streamlines. If the flow is also irrotational, the same\n", "constant applies throughout a connected region. Applying two points from unrelated\n", "streamlines without this extra condition can be wrong.\n", "\n", "Across curved streamlines, a pressure gradient can provide centripetal acceleration.\n", "Thus a high-speed region can have a pressure pattern that cannot be inferred from speed\n", "alone. The slogan “faster means lower pressure” is safe only after holding the other\n", "Bernoulli terms and constant appropriately fixed.\n", "\n", "## When ideal Bernoulli fails\n", "\n", "Viscosity converts mechanical energy into internal energy. Boundary layers and flow\n", "separation make loss depend on geometry and Reynolds number. Pumps and fans add energy.\n", "Turbines remove it. Rapidly changing flow has unsteady terms. Compressible flow couples\n", "density, pressure, temperature, and possibly shocks, so the incompressible formula cannot\n", "be used unchanged at high Mach number.\n", "\n", "Bernoulli also does not replace momentum balance. Forces on bends, jets, and nozzles are\n", "usually found with a control-volume momentum equation even if Bernoulli helps determine\n", "the speeds and pressures.\n", "\n", "## A computational residual\n", "\n", "At two supposedly ideal comparable points, calculate\n", "\n", "\n", "$$\n", "R_{\\mathrm B}=\\left(p_2+\\frac12\\rho v_2^2+\\rho gz_2\\right)-\\left(p_1+\\frac12\\rho v_1^2+\\rho gz_1\\right)\n", "$$\n", "\n", "\n", "The ideal model predicts zero. Scale the residual by a meaningful pressure such as the\n", "largest term or measured pressure difference, but always retain the dimensional residual\n", "too. Large terms can nearly cancel, so input precision matters.\n", "\n", "Check the elevation datum: changing it adds the same constant to both ideal sides and\n", "leaves the residual unchanged. Check gauge versus absolute pressure: a common atmospheric\n", "reference cancels, but inconsistent references do not. Use cross-section average velocity\n", "when pairing with pipe continuity.\n", "\n", "Repeat measurements or calculations under spatial and temporal refinement. A beautiful\n", "streamline plot cannot establish energy balance. A trustworthy use of Bernoulli names the\n", "points, units, streamline, assumptions, source and loss terms, and the residual that tests\n", "them.\n", "\n", "## A worked Python notebook\n", "\n", "Start with water at 101325 pascals and speed 2 metres per second in a level pipe. The question for this calculation is precise: how does **downstream pressure**\n", "change when we change **downstream flow speed**, 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", "Bernoulli's ideal form excludes viscous loss, pumps, and changes of elevation. 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", "1 to 6 m/s. 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 downstream flow speed, expressed in m/s. The output is\n", "downstream pressure, expressed in Pa. 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", "$$\n", "f(q) = 103325 - 500 q^{2}\n", "$$\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 4 m/s, the expression gives 95325 Pa. 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(\"101325+500*(4-q**2)\", locals={\"q\": q})\n", "reference_input = 4.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m/s\")\n", "print(\"Predicted downstream pressure:\", reference_output, \"Pa\")\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", "102825 Pa. The midpoint, at 3.5 m/s, gives\n", "97200 Pa. The upper endpoint gives 85325 Pa.\n", "Read these together with the assumptions: they describe different choices of downstream flow speed\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([1.0, 3.5, 6.0])\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/s -> {output_value:.6g} Pa\")\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 downstream flow speed; the vertical axis shows downstream pressure. 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(1.0, 6.0, 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(\"Downstream flow speed (m/s)\")\n", "ax.set_ylabel(\"Downstream pressure (Pa)\")\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", "downstream pressure change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 4 m/s is\n", "-4000 in those ratio units. Locally, the output decreases 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", "$$\n", "f'(q) = - 1000 q\n", "$$\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 = 4.025\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, \"Pa\")\n", "print(\"Full expression:\", full_prediction, \"Pa\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"Pa\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 4.025 m/s. The tangent estimate is\n", "95225 Pa, while the full expression gives\n", "95224.7 Pa. 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 = 5.0\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", "downstream flow speed inside the stated interval, predict whether downstream pressure 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: Bernoulli's ideal form excludes viscous loss, pumps, and changes of elevation. 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 Pa:\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(6.0) - evaluate(1.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(1.0, 6.0, 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 14.5, \"Fluid Dynamics\"](https://openstax.org/books/university-physics-volume-1/pages/14-5-fluid-dynamics).\n", "- [OpenStax, *University Physics Volume 1*, Section 14.6, \"Bernoulli's Equation\"](https://openstax.org/books/university-physics-volume-1/pages/14-6-bernoullis-equation).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 28, Sections 28.4-28.5](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter28/).\n", "- [Benjamin Crowell, *Light and Matter*, fluid energy discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Bernoulli's Equation\"](https://physics.info/bernoulli/).\n", "- [Physics LibreTexts, \"Bernoulli's Equation\"](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/14.07%3A_Bernoullis_Equation)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "bernoullis-equation-computationally.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/bernoullis-equation-computationally", "source_sha256": "f8936747d73ea1994c6cff15098f62bbd971347eecc3b24ba569a21022c953bb" } }, "nbformat": 4, "nbformat_minor": 5 }