To determine The velocity potential of the fluid flow. The stream function of the fluid flow. Whether the Laplace equation satisfied or not. Explanation Draw the diagram for the fluid flow on the inclined plate making an angle θ with the horizontal. Figure-(1) Write the expression for the fluid velocity parallel to the plate. V x = ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] (I) Here, the density of the fluid is ρ , the viscosity of the fluid is μ , the film thickness is t , the acceleration due to gravity is g , the angle of the inclined plate with the horizontal is θ and the coordinate normal to the plate is y . Write the expression for the fluid velocity in the x- direction. u = ∂ ϕ ∂ x (II) Here, the velocity potential is ϕ . Substitute V x for u in Equation (II). V x = ∂ ϕ ∂ x ∂ ϕ = V x ∂ x (III) Write the expression for the fluid velocity in the y- direction. u = ∂ ψ ∂ y (IV) Here, the stream function of the fluid flow is ψ . Substitute V x for u in Equation (II). V x = ∂ ψ ∂ y ∂ ψ = V x ∂ y (V) Write the expression for the potential gradient. ∇ 2 ϕ = ∂ 2 ϕ ∂ x 2 + ∂ 2 ϕ ∂ y 2 + ∂ 2 ϕ ∂ z 2 (VI) Write the expression for the Laplace equation. ∂ 2 ϕ ∂ x 2 + ∂ 2 ϕ ∂ y 2 + ∂ 2 ϕ ∂ z 2 = 0 (VII) Conclusion: Substitute ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] for V x in Equation (III). ∂ ϕ = ( ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] ) ∂ x (VIII) Integrate the equation (VIII) on both sides. ∫ ∂ ϕ = ∫ ( ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] ) ∂ x ϕ = ( ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] ) ∫ 1 ∂ x ϕ = ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] x (IX) Thus, the velocity potential of the fluid flow is ϕ = ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] x . Substitute ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] for V x in Equation (V). ∂ ψ = ( ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] ) ∂ y (X) Integrate the equation (X) on both sides. ∫ ∂ ψ = ∫ ( ρ t 2 g cos θ 2 μ [ 1 − ( y t ) 2 ] ) ∂ y ∫ ∂ ψ = ρ t 2 g cos θ 2 μ ∫ [ 1 − ( y t ) 2 ] ∂ y ψ = ρ t 2 g cos θ 2 μ [ y − ( y 3 3 t 2 ) ] Thus, the stream function of the fluid flow is ψ = ρ t 2 g cos θ 2 μ [ y − ( y 3 3 t 2 ) ]

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Chapter 6.5, Problem 47P

To determine

The velocity potential of the fluid flow.

The stream function of the fluid flow.

Whether the Laplace equation satisfied or not.

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Chapter 6.5, Problem 46P

Chapter 6.5, Problem 48P

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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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