To determine The velocity at the interface. Explanation Write the expression of Navier-strokes equation in x direction. ρ ( ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z ) = − ∂ p ∂ x + μ ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 ) (I) Here, the density of the fluid is ρ , the dynamic viscosity of the fluid is μ , the x component of the velocity is x , the y component of the velocity is v , the z component of the velocity is w and the pressure is p . Here, the component of velocity is not given. Consider the flow is steady, laminar and the change in velocity with respect to time is 0 . ∂ u ∂ t = 0 Consider the flow is steady, laminar and the pressure gradient in x direction is 0 ∂ p ∂ x = 0 Consider the velocity along the y direction is 0 . v = 0 (II) Consider the velocity along the z direction is 0 . w = 0 (III) Write the expression of continuity Equation. ∂ u ∂ x + ∂ v ∂ y + ∂ w ∂ z = 0 (IV) Differentiate Equation (II) partially on both sides with respect to y . ∂ v ∂ y = 0 (V) Differentiate Equation (III) partially on both sides with respect to z ∂ w ∂ z = 0 (VI) Substitute 0 for ∂ w ∂ z and 0 for ∂ v ∂ y in Equation (IV). ∂ u ∂ x + 0 + 0 = 0 ∂ u ∂ x = 0 (VII) Differentiate Equation (VII) with respect to x on both sides. ∂ ∂ x ( ∂ u ∂ x ) = 0 ∂ 2 u ∂ x 2 = 0 (VIII) Differentiate Equation (VI) with respect to z on both sides. ∂ ∂ z ( ∂ w ∂ z ) = 0 ∂ 2 w ∂ z 2 = 0 (IX) Substitute 0 for ∂ u ∂ t , 0 for ∂ u ∂ x , 0 for ∂ u ∂ y , 0 for ∂ u ∂ z , 0 for ∂ p ∂ x 0 for ∂ 2 u ∂ x 2 , 0 for ∂ 2 v ∂ z 2 in Equation (I). ρ ( 0 + u × 0 + v × 0 + w × 0 ) = − ∂ p ∂ x + μ ( 0 + ∂ 2 u ∂ y 2 + w × 0 ) ∂ 2 u ∂ y 2 = 0 (X) Integrate Equation (X) on both sides. ∫ ∂ 2 u ∂ y 2 = 0 ∂ u ∂ y = C 1 ∂ u = ∂ y C 1 (XI) Here, the integral constant is C 1 . Integrate Equation (XI) on both sides. ∫ ∂ u = ∫ ∂ y C 1 u = y C 1 + C 2 (XII) Here, the integral constant is C 2 . For fluid 1. Substitute u 1 for u , C 1 a for C 1 and C 2 a for C 2 in Equation (XII). u 1 = y C 1 a + C 2 a (XIII) For fluid 2. Substitute u 2 for u , C 1 b for C 1 and C 2 b for C 2 in Equation (XII)

8th Edition

ISBN: 9781119571490

Author: GERHART

Publisher: WILEY

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Properties of Fluids

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Chapter 6.9, Problem 80P

To determine

The velocity at the interface.

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FUNDAMENTALS OF FLUID MECHANICS

Ch. 6.1 - Prob. 1P Ch. 6.1 - The velocity in a certain flow field is given by...Ch. 6.1 - The flow in the plane two-dimensional channel...Ch. 6.1 - The three components of velocity in a flow field...Ch. 6.1 - Determine an expression for the vorticity of the...Ch. 6.1 - According to Eq. 6.134, the x-velocity in fully...Ch. 6.1 - For a certain incompressible, two-dimensional flow...Ch. 6.1 - An incompressible viscous fluid is placed between...Ch. 6.1 - A viscous fluid is contained in the space between...Ch. 6.1 - ..Air is delivered through a constant-diameter...

Ch. 6.2 - For a certain incompressible flow field it is...Ch. 6.2 - Prob. 12P Ch. 6.2 - Prob. 14P Ch. 6.2 - For each of the following stream functions, with...Ch. 6.2 - The stream function for an incompressible,...Ch. 6.2 - Prob. 17P Ch. 6.2 - Prob. 18P Ch. 6.2 - In a two-dimensional, incompressible flow field,...Ch. 6.2 - The stream function for an incompressible flow...Ch. 6.2 - The stream function for an incompressible,...Ch. 6.2 - Consider the incompressible, two-dimensional flow...Ch. 6.3 - A fluid with a density of 2000 kg/m3 flows...Ch. 6.3 - Prob. 24P Ch. 6.3 - Prob. 25P Ch. 6.4 - The stream function for a given two-dimensional...Ch. 6.4 - Prob. 27P Ch. 6.4 - Prob. 28P Ch. 6.4 - Prob. 29P Ch. 6.4 - The velocity potential for a certain inviscid flow...Ch. 6.4 - Prob. 31P Ch. 6.4 - Prob. 32P Ch. 6.4 - Prob. 33P Ch. 6.4 - Prob. 34P Ch. 6.4 - Prob. 35P Ch. 6.4 - Prob. 36P Ch. 6.4 - Prob. 37P Ch. 6.5 - Prob. 38P Ch. 6.5 - Prob. 39P Ch. 6.5 - Water flows through a two-dimensional diffuser...Ch. 6.5 - Prob. 41P Ch. 6.5 - Prob. 42P Ch. 6.5 - Prob. 43P Ch. 6.5 - Prob. 44P Ch. 6.5 - Prob. 45P Ch. 6.5 - Prob. 46P Ch. 6.5 - Consider the flow of a liquid of viscosity μ and...Ch. 6.5 - Prob. 48P Ch. 6.5 - Show that the circulation of a free vortex for any...Ch. 6.5 - Prob. 50P Ch. 6.6 - Potential flow against a flat plate (Fig. P6.51a)...Ch. 6.6 - Prob. 52P Ch. 6.6 - Prob. 53P Ch. 6.6 - Prob. 54P Ch. 6.6 - Prob. 55P Ch. 6.6 - Prob. 56P Ch. 6.6 - A 15-mph wind flows over a Quonset hut having a...Ch. 6.6 - Prob. 58P Ch. 6.6 - Prob. 59P Ch. 6.6 - Prob. 60P Ch. 6.6 - Prob. 61P Ch. 6.6 - Prob. 62P Ch. 6.6 - The velocity potential for a cylinder (Fig. P6.63)...Ch. 6.6 - (See The Wide World of Fluids article titled “A...Ch. 6.6 - Prob. 65P Ch. 6.6 - Air at 25 °C flows normal to the axis of an...Ch. 6.8 - Determine the shearing stress for an...Ch. 6.8 - Prob. 68P Ch. 6.8 - The velocity of a fluid particle moving along a...Ch. 6.8 - “Stokes’s first problem” involves the...Ch. 6.9 - Oil (SAE 30) at 15.6 °C flows steadily between...Ch. 6.9 - Prob. 72P Ch. 6.9 - Prob. 73P Ch. 6.9 - We will see in Chapter 8 that the pressure drop in...Ch. 6.9 - (See The Wide World of Fluids article titled “10...Ch. 6.9 - The bearing shown in Fig. P6.76 consists of two...Ch. 6.9 - Prob. 77P Ch. 6.9 - Prob. 78P Ch. 6.9 - An incompressible, viscous fluid is placed between...Ch. 6.9 - Two immiscible, incompressible, viscous fluids...Ch. 6.9 - Prob. 81P Ch. 6.9 - A viscous fluid (specific weight = 80 lb/ft3;...Ch. 6.9 - A flat block is pulled along a horizontal flat...Ch. 6.9 - A viscosity motor/pump is shown in Fig. P6.84. The...Ch. 6.9 - A vertical shaft passes through a bearing and is...Ch. 6.9 - A viscous fluid is contained between two long...Ch. 6.9 - Verify that the momentum correction factor β for...Ch. 6.9 - Verify that the kinetic energy correction factor α...Ch. 6.9 - A simple flow system to be used for steady-flow...Ch. 6.9 - (a) Show that for Poiseuille flow in a tube of...Ch. 6.9 - An infinitely long, solid, vertical cylinder of...Ch. 6.9 - We will see in Chapter 8 that the pressure drop in...Ch. 6.9 - A liquid (viscosity = 0.002 N · s/m2; density =...Ch. 6.9 - Fluid with kinematic viscosity ν flows down an...Ch. 6.9 - Blood flows at volume rate Q in a circular tube of...Ch. 6.9 - An incompressible Newtonian fluid flows steadily...Ch. 6.9 - Prob. 97P Ch. 6.9 - Prob. 98P Ch. 6.9 - Prob. 99P Ch. 6.10 - Prob. 101P Ch. 6.10 - Prob. 102P Ch. 6.11 - Prob. 1LLP Ch. 6.11 - Prob. 2LLP Ch. 6.11 - Prob. 3LLP

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