Consider 2-D, steady, laminar flow of an incompressible Newtonian fluid down an inclined plate. The inclination...

Transcribed Image Text:

Consider 2-D, steady, laminar flow of an incompressible Newtonian fluid down an inclined plate. The inclination angle is with respect to horizontal, gravity acts downward relative to horizontal, and the flow is in a film of thickness h normal to the plate, as shown in the figure below. The fluid density is p, and viscosity is µ. You may neglect all pressure gradients. The velocity profile for this fluid is given as u(y) = ky(2h — y) - (x-component) (y-component) a) Provide a list of assumptions that can be used to simplify the Navier-Stokes equations given below. Identify which assumptions are used to simplify each term. (z-component) Р əu Ət Ət + u 60 du + u +v + w w 3₂2) = du əx əw p(3. P at Əv Əx +u. 0 aw əx du Əy Əv + v. +w Əy + v Əv √³₂) = Əw dy == Əwy -+w) = 3 Әр əx ap [a² u 2²u a²u] Ləx² Əy² дz2 ! +pgy +μ[3/²/2/² Əy +p9x +μ + + 2²v 2²] + + მy2 əz² ap [a²w 2²w 2²w + Pg₂ + μ + + дz Ləx² მyz дz2 b) Find k in terms of p, g, µ, and 0. c) Find the mass flow rate of fluid in the film (per unit depth normal to the page). d) Find a velocity potential for this flow if one exists. Otherwise demonstrate one cannot exist. Consider 2-D, steady, laminar flow of an incompressible Newtonian fluid down an inclined plate. The inclination angle is with respect to horizontal, gravity acts downward relative to horizontal, and the flow is in a film of thickness h normal to the plate, as shown in the figure below. The fluid density is p, and viscosity is µ. You may neglect all pressure gradients. The velocity profile for this fluid is given as u(y) = ky(2h — y) - (x-component) (y-component) a) Provide a list of assumptions that can be used to simplify the Navier-Stokes equations given below. Identify which assumptions are used to simplify each term. (z-component) Р əu Ət Ət + u 60 du + u +v + w w 3₂2) = du əx əw p(3. P at Əv Əx +u. 0 aw əx du Əy Əv + v. +w Əy + v Əv √³₂) = Əw dy == Əwy -+w) = 3 Әр əx ap [a² u 2²u a²u] Ləx² Əy² дz2 ! +pgy +μ[3/²/2/² Əy +p9x +μ + + 2²v 2²] + + მy2 əz² ap [a²w 2²w 2²w + Pg₂ + μ + + дz Ləx² მyz дz2 b) Find k in terms of p, g, µ, and 0. c) Find the mass flow rate of fluid in the film (per unit depth normal to the page). d) Find a velocity potential for this flow if one exists. Otherwise demonstrate one cannot exist. Consider 2-D, steady, laminar flow of an incompressible Newtonian fluid down an inclined plate. The inclination angle is with respect to horizontal, gravity acts downward relative to horizontal, and the flow is in a film of thickness h normal to the plate, as shown in the figure below. The fluid density is p, and viscosity is µ. You may neglect all pressure gradients. The velocity profile for this fluid is given as u(y) = ky(2h — y) - (x-component) (y-component) a) Provide a list of assumptions that can be used to simplify the Navier-Stokes equations given below. Identify which assumptions are used to simplify each term. (z-component) Р əu Ət Ət + u 60 du + u +v + w w 3₂2) = du əx əw p(3. P at Əv Əx +u. 0 aw əx du Əy Əv + v. +w Əy + v Əv √³₂) = Əw dy == Əwy -+w) = 3 Әр əx ap [a² u 2²u a²u] Ləx² Əy² дz2 ! +pgy +μ[3/²/2/² Əy +p9x +μ + + 2²v 2²] + + მy2 əz² ap [a²w 2²w 2²w + Pg₂ + μ + + дz Ləx² მyz дz2 b) Find k in terms of p, g, µ, and 0. c) Find the mass flow rate of fluid in the film (per unit depth normal to the page). d) Find a velocity potential for this flow if one exists. Otherwise demonstrate one cannot exist.

Answer rating: 100% (QA)

a Assumptions and terms simplified The NavierStokes equations are xcomponent ut uux vuy wuz px gx ... View the full answer