SAE 30 oil at 20\u00b0C (density p-918 kg\/m3, viscosity -0.440 Pa- s) flows between two horizontal, infinite, parallel plates separated by 40 mm. The bottom plate is fixed, and the upper plate moves parallel to the bottom plate with a velocity 5 cm\/s. The pressure gradient in the direction of flow is 500 Pa\/m. (a) (5 points) Determine the velocity distribution between the plates. (b) (5 points) Determine the flowrate passing between the plates (per a unit width).<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n To solve this problem, we use the Navier-Stokes equation for laminar flow between two infinite parallel plates, known as Couette-Poiseuille flow.<\/p>\n\n\n\n For flow between two parallel plates, with one plate moving and a pressure gradient present, the velocity profile is given by:<\/p>\n\n\n\n [ Since the bottom plate is fixed (( u(0) = 0 )), we determine ( C_1 = 0 ). The velocity profile simplifies to:<\/p>\n\n\n\n [ [ [ The volumetric flow rate per unit width is:<\/p>\n\n\n\n [ [ Evaluating the integral:<\/p>\n\n\n\n [ [ [ [ Thus, the flow rate per unit width is ( 0.01113 ) m\u00b2\/s<\/strong>.<\/p>\n\n\n\n This problem involves both Couette flow (due to the moving upper plate) and Poiseuille flow (due to the pressure gradient). The velocity profile is parabolic due to pressure-driven flow but is skewed due to the moving plate. The negative sign in the flow rate indicates that the pressure-driven flow opposes the upper plate movement.<\/p>\n","protected":false},"excerpt":{"rendered":" SAE 30 oil at 20\u00b0C (density p-918 kg\/m3, viscosity -0.440 Pa- s) flows between two horizontal, infinite, parallel plates separated by 40 mm. The bottom plate is fixed, and the upper plate moves parallel to the bottom plate with a velocity 5 cm\/s. The pressure gradient in the direction of flow is 500 Pa\/m. (a) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-203974","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203974","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=203974"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203974\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=203974"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=203974"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=203974"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Given Data:<\/strong><\/h3>\n\n\n\n
\n
(a) Velocity Distribution:<\/strong><\/h3>\n\n\n\n
u(y) = -\\frac{1}{2\\mu} \\frac{dP}{dx} y^2 + \\frac{U}{h} y + C_1
]<\/p>\n\n\n\n
u(y) = -\\frac{1}{2(0.440)} (500) y^2 + \\frac{0.05}{0.04} y
]<\/p>\n\n\n\n
u(y) = -\\frac{500}{0.88} y^2 + 1.25y
]<\/p>\n\n\n\n
u(y) = -568.18 y^2 + 1.25 y
]<\/p>\n\n\n\n(b) Flow Rate Per Unit Width:<\/strong><\/h3>\n\n\n\n
Q = \\int_0^h u(y) dy
]<\/p>\n\n\n\n
Q = \\int_0^{0.04} (-568.18 y^2 + 1.25 y) dy
]<\/p>\n\n\n\n
Q = \\left[ -\\frac{568.18}{3} y^3 + \\frac{1.25}{2} y^2 \\right]_{0}^{0.04}
]<\/p>\n\n\n\n
Q = \\left[ -189.39 (0.04)^3 + 0.625 (0.04)^2 \\right]
]<\/p>\n\n\n\n
Q = \\left[ -189.39 (0.000064) + 0.625 (0.0016) \\right]
]<\/p>\n\n\n\n
Q = (-0.01213 + 0.001) = -0.01113 \\text{ m}^2\\text{\/s}
]<\/p>\n\n\n\nExplanation:<\/strong><\/h3>\n\n\n\n