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Navier-Stokes Equation

Navier-Stokes Equation

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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Cylindrical Coordinates (r, 0, z): Continuity equation ap 10(prv,) 10(pve), a(pv₂) + + + at r ər ae дz Navier-Stokes Equation for constant p and u 10² vr (avr avr να δν, avr vo + p² მ02 r-dir: at ər r 20 дz r 1 ap (ave ave ve ave aveva) = + a (1a(rve) 1 0² ve + arr Ər 2 მ02 + + 1 - 0-dir: P at ər r 20 дz r rae avz ap əvz ve dv₂ д 1 0² vzd²v₂) r² 20² дz2 {+ - ( 1² / (+²) + r Əvz at + z-dir: Р + Vr + дz ər r 20 дz rər ər Stokes viscosity relation 1 dur dve, Tez-Tze=μ[+1 duz T₁z=z=μ(dur + dr Tre Ter= dz r de r de A Newtonian fluid flows inside an annular (¹) cylinder with height H as shown in Figure p3. The radius of the inner cylinder is kR; while the outer cylinder with radius R. The fluid is filled between the inner and outer cylinders. The inner cylinder rotates with a constant angular velocity Qi (Voi = Ni kR) and the outer cylinder rotates with a constant angular velocity No (V₂0= No R). Assume V₁ #0, dp/d 0 = 0, d V₁/dz = 0, and g₁ = 0. (a) Analyze the continuity equation. (b) Analyze the Navier-Stokes equations in the 3 dimensions. (c) Find the velocity distribution in the cylindrical flow system. (d) Determine the torques (E) on the outer and inner cylinders by the fluid flow. Si r-kR 20 outer cylinder P · + vr + Dr + + [Q² +2: + vz + Vz + Vz = 0 ap va (10(v₂)) +μ Ər arr ar r-Rr H 2 ave r2 80 + Pgz inner cylinder ² vr дz2 2 avr a²e + r2 де дz2 + + pgr +pge
Transcribed Image Text:Cylindrical Coordinates (r, 0, z): Continuity equation ap 10(prv,) 10(pve), a(pv₂) + + + at r ər ae дz Navier-Stokes Equation for constant p and u 10² vr (avr avr να δν, avr vo + p² მ02 r-dir: at ər r 20 дz r 1 ap (ave ave ve ave aveva) = + a (1a(rve) 1 0² ve + arr Ər 2 მ02 + + 1 - 0-dir: P at ər r 20 дz r rae avz ap əvz ve dv₂ д 1 0² vzd²v₂) r² 20² дz2 {+ - ( 1² / (+²) + r Əvz at + z-dir: Р + Vr + дz ər r 20 дz rər ər Stokes viscosity relation 1 dur dve, Tez-Tze=μ[+1 duz T₁z=z=μ(dur + dr Tre Ter= dz r de r de A Newtonian fluid flows inside an annular (¹) cylinder with height H as shown in Figure p3. The radius of the inner cylinder is kR; while the outer cylinder with radius R. The fluid is filled between the inner and outer cylinders. The inner cylinder rotates with a constant angular velocity Qi (Voi = Ni kR) and the outer cylinder rotates with a constant angular velocity No (V₂0= No R). Assume V₁ #0, dp/d 0 = 0, d V₁/dz = 0, and g₁ = 0. (a) Analyze the continuity equation. (b) Analyze the Navier-Stokes equations in the 3 dimensions. (c) Find the velocity distribution in the cylindrical flow system. (d) Determine the torques (E) on the outer and inner cylinders by the fluid flow. Si r-kR 20 outer cylinder P · + vr + Dr + + [Q² +2: + vz + Vz + Vz = 0 ap va (10(v₂)) +μ Ər arr ar r-Rr H 2 ave r2 80 + Pgz inner cylinder ² vr дz2 2 avr a²e + r2 де дz2 + + pgr +pge
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