To determine The volumetric strain rate. The vorticity. The rate of angular deformation. The shear stress at the plate surface. Explanation Write the expression of the x component of the velocity. u = 1 2 μ ( ∂ p ∂ x ) ( y 2 − h 2 ) (I) Here, the dynamic viscosity is μ , the displacement in y direction is y , the height of the plate is h and the rate of change of pressure with respect to change in displacement in x direction s ∂ p ∂ x . Write the expression of volumetric strain rate. ∇ ⋅ V = ∂ u ∂ x + ∂ v ∂ y + ∂ w ∂ z (II) Here, the x component of change in velocity with respect to change in displacement is ∂ u ∂ x , the y component of change in velocity with respect to change in displacement is ∂ v ∂ y and the z component of change in velocity with respect to change in displacement is ∂ w ∂ z . Here, the y and z component of the velocity is not given only x component is given. ∇ ⋅ V = ∂ u ∂ x (III) Write the expression of vorticity. ζ = 2 ω (IV) Here, the rotational speed is ω . Write the expression of rotational speed. ω = 1 2 ∇ × V (V) Here, the cross product of volumetric strain is ∇ × V and the operator is ∇ . Write the expression of cross product of volumetric strain. ∇ × V = | i ^ j ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z u v w | (VI) Write the expression of angular deformation. γ ˙ = ∂ v ∂ x + ∂ u ∂ y (VII) Write the expression of shear stress at the plate surface. τ plate = μ ( d u d y ) y = h (VIII) Conclusion: Differentiate Equation (I) partially with respect to x on both sides. ∂ u ∂ x = ∂ ∂ x { 1 2 μ ( ∂ p ∂ x ) ( y 2 − h 2 ) } = 1 2 μ ( y 2 − h 2 ) ∂ ∂ x ( ∂ p ∂ x ) = 1 2 μ ( y 2 − h 2 ) ∂ 2 p ∂ x 2 Substitute 1 2 μ ( y 2 − h 2 ) ∂ 2 p ∂ x 2 for ∂ u ∂ x in Equation (III). ∇ ⋅ V = 1 2 μ ( y 2 − h 2 ) ∂ 2 p ∂ x 2 = ∂ 2 p ∂ x 2 1 2 μ ( y 2 − h 2 ) Thus, the volumetric strain rate is ∂ 2 p ∂ x 2 1 2 μ ( y 2 − h 2 )

Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version

8th Edition

ISBN: 9781119080701

Author: Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein

Publisher: WILEY

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Chapter 6.1, Problem 6P

To determine

The volumetric strain rate.

The vorticity.

The rate of angular deformation.

The shear stress at the plate surface.

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Chapter 6.1, Problem 5P

Chapter 6.1, Problem 7P

Chapter 6 Solutions

Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version

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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