{"id":13,"date":"2026-10-08T16:58:45","date_gmt":"2026-10-08T16:58:45","guid":{"rendered":"https:\/\/marouane.site\/index.php\/simulations\/laminar-pipe-flow-lab\/"},"modified":"2026-10-08T16:58:45","modified_gmt":"2026-10-08T16:58:45","slug":"laminar-pipe-flow-lab","status":"publish","type":"ms_simulation","link":"https:\/\/marouane.site\/index.php\/simulations\/laminar-pipe-flow-lab\/","title":{"rendered":"The laminar pipe-flow lab"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Explore the analytical velocity profile of fully developed laminar flow. Change the inputs, inspect the profile table, and watch how the computed flow rate responds.<\/p>    <section class=\"ms-flow-demo\" data-flow-demo aria-label=\"Laminar pipe flow analytical calculator\">\n      <div class=\"ms-flow-heading\"><span class=\"ms-eyebrow\">VISUAL LAB<\/span><span class=\"ms-badge\">ANALYTICAL RESULT<\/span><\/div>\n      <h3>Change the conditions. See the flow.<\/h3>\n      <p>Steady, incompressible, fully developed Newtonian flow in a straight circular pipe. Water-like density is fixed at 998 kg\/m\u00b3.<\/p>\n      <svg viewBox=\"0 0 620 300\" role=\"img\" aria-label=\"Pipe cross-section: axial velocity is zero at the stationary walls and greatest on the centreline.\">\n        <defs><marker id=\"ms-flow-1arrow\" markerWidth=\"6\" markerHeight=\"6\" refX=\"5\" refY=\"3\" orient=\"auto\"><path d=\"M0 0L6 3L0 6z\" fill=\"currentColor\"\/><\/marker><\/defs>\n        <rect x=\"55\" y=\"45\" width=\"465\" height=\"205\" rx=\"3\" fill=\"none\" stroke=\"#91a4b7\" stroke-width=\"6\"\/>\n        <path d=\"M55 147.5H540\" stroke=\"#91a4b7\" stroke-dasharray=\"5 6\"\/>\n                <path d=\"M90 65.5H219.6\" stroke=\"currentColor\" stroke-width=\"2\" marker-end=\"url(#ms-flow-1arrow)\"\/>\n                <path d=\"M90 86H320.4\" stroke=\"currentColor\" stroke-width=\"2\" marker-end=\"url(#ms-flow-1arrow)\"\/>\n                <path d=\"M90 106.5H392.4\" stroke=\"currentColor\" stroke-width=\"2\" marker-end=\"url(#ms-flow-1arrow)\"\/>\n                <path d=\"M90 127H435.6\" stroke=\"currentColor\" stroke-width=\"2\" marker-end=\"url(#ms-flow-1arrow)\"\/>\n                <path d=\"M90 147.5H450\" stroke=\"currentColor\" stroke-width=\"2\" marker-end=\"url(#ms-flow-1arrow)\"\/>\n                <path d=\"M90 168H435.6\" stroke=\"currentColor\" stroke-width=\"2\" marker-end=\"url(#ms-flow-1arrow)\"\/>\n                <path d=\"M90 188.5H392.4\" stroke=\"currentColor\" stroke-width=\"2\" marker-end=\"url(#ms-flow-1arrow)\"\/>\n                <path d=\"M90 209H320.4\" stroke=\"currentColor\" stroke-width=\"2\" marker-end=\"url(#ms-flow-1arrow)\"\/>\n                <path d=\"M90 229.5H219.6\" stroke=\"currentColor\" stroke-width=\"2\" marker-end=\"url(#ms-flow-1arrow)\"\/>\n                <path d=\"M90 45 Q810 147.5 90 250\" stroke=\"currentColor\" stroke-width=\"3\" fill=\"none\"\/>\n        <text x=\"55\" y=\"28\">u = 0 \u00b7 stationary wall<\/text><text x=\"465\" y=\"138\">u max<\/text><text x=\"55\" y=\"285\">Normalized shape u\/u max \u00b7 r\/R<\/text>\n      <\/svg>\n      <div class=\"ms-flow-controls\">\n              <label>Pressure drop \u0394p <span>(Pa)<\/span><input data-flow-input=\"dp\" type=\"number\" min=\"0.1\" max=\"100\" step=\"0.1\" value=\"2\"><\/label>\n              <label>Dynamic viscosity \u03bc <span>(Pa\u00b7s)<\/span><input data-flow-input=\"mu\" type=\"number\" min=\"0.0001\" max=\"0.1\" step=\"0.0001\" value=\"0.001\"><\/label>\n              <label>Pipe radius R <span>(m)<\/span><input data-flow-input=\"radius\" type=\"number\" min=\"0.001\" max=\"0.02\" step=\"0.001\" value=\"0.005\"><\/label>\n              <label>Pipe length L <span>(m)<\/span><input data-flow-input=\"length\" type=\"number\" min=\"0.1\" max=\"10\" step=\"0.1\" value=\"1\"><\/label>\n            <\/div>\n      <p class=\"ms-equation\">u(r) = \u0394p (R\u00b2 \u2212 r\u00b2) \/ (4 \u03bc L)<\/p>\n      <div class=\"ms-flow-results\" aria-live=\"polite\"><p>Maximum velocity <strong data-result=\"umax\">0.0125 m\/s<\/strong><\/p><p>Mean velocity <strong data-result=\"mean\">0.00625 m\/s<\/strong><\/p><p>Flow rate <strong data-result=\"flow\">4.909 \u00d7 10\u207b\u2077 m\u00b3\/s<\/strong><\/p><p>Reynolds number <strong data-result=\"re\">62.4<\/strong><\/p><\/div>\n      <p class=\"ms-flow-status\" role=\"status\">Laminar-flow estimate. No entrance effects or experimental validation are included.<\/p>\n      <details><summary>Read the profile as a table<\/summary><table><thead><tr><th>r\/R<\/th><th>u (m\/s)<\/th><\/tr><\/thead><tbody data-flow-table><tr><td>0<\/td><td>0.0125<\/td><\/tr><tr><td>0.5<\/td><td>0.009375<\/td><\/tr><tr><td>1<\/td><td>0<\/td><\/tr><\/tbody><\/table><\/details>\n      <p class=\"ms-muted\">The normalized parabola keeps its shape; calculated speeds and the profile table change with the inputs. \u0394p = p in \u2212 p out &gt; 0. Reynolds number uses diameter 2R and mean velocity.<\/p>\n      <noscript><p>Enable JavaScript to change the inputs. The equation, diagram and default results remain available above.<\/p><\/noscript>\n    <\/section>\n    <h2 class=\"wp-block-heading\">Use it with a clear model<\/h2><p class=\"wp-block-paragraph\">The calculation assumes a straight circular pipe, stationary walls, steady incompressible Newtonian flow, and a fully developed laminar profile. Density is fixed at 998 kilograms per cubic metre. The normalized diagram is intentionally unchanged as the dimensional velocities change.<\/p><h2 class=\"wp-block-heading\">Check the solution<\/h2><p class=\"wp-block-paragraph\">At r = R the velocity is zero. At r = 0 it is \u0394p R\u00b2 \/ (4 \u03bc L). Integrating over the circular section gives Q = \u03c0 \u0394p R\u2074 \/ (8 \u03bc L), so the mean speed is half the maximum. A warning appears when the calculated Reynolds number reaches 2,300; this threshold is a guide, not a guarantee of the flow regime.<\/p><div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\"><div class=\"wp-block-button\"><a class=\"wp-block-button__link wp-element-button\" href=\"https:\/\/marouane.site\/index.php\/learn\/fluid-mechanics-first-principles\/\">Take the introductory course<\/a><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>An interactive analytical example with transparent assumptions, calculated results, and a readable data table.<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"_ms_course_id":0,"_ms_order":0,"_ms_minutes":6,"_ms_level":"","_ms_classification":"Analytical solution","_ms_evidence":"","_ms_accent":"orange","footnotes":""},"ms_topic":[2],"class_list":["post-13","ms_simulation","type-ms_simulation","status-publish","hentry","ms_topic-fluid-mechanics"],"_links":{"self":[{"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/ms_simulation\/13","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/ms_simulation"}],"about":[{"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/types\/ms_simulation"}],"version-history":[{"count":0,"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/ms_simulation\/13\/revisions"}],"wp:attachment":[{"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/media?parent=13"}],"wp:term":[{"taxonomy":"ms_topic","embeddable":true,"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/ms_topic?post=13"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}