{"id":9,"date":"2026-10-08T16:58:45","date_gmt":"2026-10-08T16:58:45","guid":{"rendered":"https:\/\/marouane.site\/index.php\/lessons\/explore-laminar-pipe-flow\/"},"modified":"2026-10-08T16:58:45","modified_gmt":"2026-10-08T16:58:45","slug":"explore-laminar-pipe-flow","status":"publish","type":"ms_lesson","link":"https:\/\/marouane.site\/index.php\/lessons\/explore-laminar-pipe-flow\/","title":{"rendered":"03 \u00b7 Why a parabola appears"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Under a specific set of assumptions, the velocity of a viscous fluid in a straight circular pipe has a parabolic profile. The result comes from balancing the pressure gradient with viscous resistance.<\/p><h2 class=\"wp-block-heading\">Assumptions first<\/h2><ul class=\"wp-block-list\"><li>Steady, incompressible, laminar flow of a Newtonian fluid.<\/li><li>A straight circular pipe with constant radius and stationary walls.<\/li><li>A fully developed axial velocity profile and a positive inlet-to-outlet pressure drop.<\/li><li>Constant dynamic viscosity; no entrance, exit, or additional body-force effects in the model.<\/li><\/ul>    <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\">Understand the result<\/h2><p class=\"wp-block-paragraph\">The maximum velocity is twice the mean velocity for this profile. At fixed pressure drop, viscosity, and pipe length, the volumetric flow rate is proportional to the fourth power of radius. Small radius changes can therefore have a large effect.<\/p><h2 class=\"wp-block-heading\">Checkpoint<\/h2><details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>At fixed pressure drop, viscosity, and length, what happens to flow rate when the radius doubles?<\/summary><p class=\"wp-block-paragraph\">The analytical flow rate increases by a factor of sixteen. Check whether the resulting Reynolds number still supports the laminar assumption before treating that prediction as physically applicable.<\/p><\/details><h2 class=\"wp-block-heading\">What this example can and cannot establish<\/h2><p class=\"wp-block-paragraph\">This is an analytical teaching calculation, not a numerical flow solver or an experimental validation. Agreement with it can help verify a numerical model under matching conditions; it cannot establish accuracy for a different physical regime.<\/p>","protected":false},"excerpt":{"rendered":"<p>Adjust pressure drop, viscosity, radius, and length, then inspect the calculated velocity and flow rate.<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"_ms_course_id":6,"_ms_order":30,"_ms_minutes":9,"_ms_level":"Foundation","_ms_classification":"Analytical solution","_ms_evidence":"","_ms_accent":"orange","footnotes":""},"ms_topic":[2],"class_list":["post-9","ms_lesson","type-ms_lesson","status-publish","hentry","ms_topic-fluid-mechanics"],"_links":{"self":[{"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/ms_lesson\/9","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/ms_lesson"}],"about":[{"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/types\/ms_lesson"}],"version-history":[{"count":0,"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/ms_lesson\/9\/revisions"}],"wp:attachment":[{"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/media?parent=9"}],"wp:term":[{"taxonomy":"ms_topic","embeddable":true,"href":"https:\/\/marouane.site\/index.php\/wp-json\/wp\/v2\/ms_topic?post=9"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}