Concept explainers
Consider a disk of radius R rotating in an incompressible fluid at a speed ω. The equations that describe the boundary layer on the disk are:
Use the characteristic dimensions to normalize the differential equation and obtain the dimensionless groups that characterize the flow.
P7.6
Want to see the full answer?
Check out a sample textbook solutionChapter 7 Solutions
Fox and McDonald's Introduction to Fluid Mechanics
Additional Engineering Textbook Solutions
Applied Fluid Mechanics (7th Edition)
Thinking Like an Engineer: An Active Learning Approach (3rd Edition)
Mechanics of Materials (10th Edition)
Fundamentals Of Thermodynamics
Applied Statics and Strength of Materials (6th Edition)
Degarmo's Materials And Processes In Manufacturing
- Prove that, if a fluid experiences a pure rotation, the divergence of it's velocity is 0 and the curl of it's velocity is twice the angular velocity.arrow_forwardlower end of a string and is moving in a horizontal circle of radius R = 0.15 m at constant speed v = 0.51 m/s. The string has length l and negligible mass and makes an angle 0 with the vertical. The angle 0 is %3D marrow_forward3. Viscous fluid occupies the region 0arrow_forwardIn an oil pool, a small steel ball is released from the surface (y=0) without initial velocity. The strength of the resistance force exerted by the oil against the movement of the ball is directly proportional to the speed of the ball (Fd = k*V , k: constant). Neglect the buoyant force exerted by the oil. (m = 0.2kg, k = 0.843550 kg/s, g = 9.81 m/s^2). a-) What is the limit speed of ball ( Vlim)? b-) What is the time it takes for the speed of the ball to reach 99% of the limit speed after it is released from the surface? c-) What is the depth at which the ball's velocity reaches 99% of the limit velocity after it is released from the surface?arrow_forwardIn deriving the vorticity equation, we have used the identity V × (VP) = 0. Show that this identity is valid for any scalar λ by checking it in Cartesian and cylindrical coordinates.arrow_forwardIn the pipe below, an incompressible fluid flows. If the radius at A₁ is twice the at A2, V₁ V2. 2 V2 4 V2 V2/2 V2/4 A₁ V₁ A₂ 2 V₂arrow_forward1.A) A fluid of density ρ=850 kg/m3 and viscosity µ=0,8 Pa s is completely filling the spacing h= 2 cm between two parallel horizontal plates of length 2,5 m and width 1 m . The upper plate is pulled at a speed of U while lower plate is fixed . In order to generate a laminar steady incompressible flow of fluid in the spacing at a rate of 0,85 kg/s determine total shearing force in the field.ANSWER:10 N 1.B) A fluid of density ρ=850 kg/m3 and viscosity µ=0,8 Pa s is completely filling the spacing h= 2 cm between two parallel horizontal plates of length 2,5 m and width 1 m . The upper plate is pulled at a speed of U while lower plate is fixed . In order to generate a laminar steady incompressible flow of fluid in the spacing at a rate of 0,85 kg/s determine Reynolds Number of flow using the spacing as the characteristic dimension.ANSWER: 2,125 1.C) A fluid of density ρ=850 kg/m3 and viscosity µ=0,8 Pa s is completely filling the spacing h= 2 cm between two parallel horizontal plates of…arrow_forwardRemark The expression (2·10-3) for the work done is much more general than the derivation based on the simple cylinder would indicate. To show this, consider an arbitrary slow expansion of the system from the volume enclosed by the solid boundary to that enelosed by the dotted boundary in Fig. 2.10-1. If the mean pressure is p, the mean force on an element of area dA is p dA in the direction of the normal n. If the displacement of this element of area is by an amount ds in the direction making an angle 0 with the normal, ds 07 V. dA Fig. 2.10 1 Arbitrary expansion of a system of volu me V. why do we expand this piston into that image? what part is expanded? and what is the meaning of the normal angle?arrow_forwardA 2-D channel on the x-y plane has a rectangular inlet surface and a cylindrical outlet surface, as shown in the figure. The depth of the channel in z-direction is W. Air of constant density ρ enters the channel with uniform velocity of u = U, v = V, where U and V are positive constants. The inlet height is ℎ. The outlet is a quarter cylindrical surface with radius R = 2ℎ, and the outlet velocity only has a constant radial component Vr, and no tangential component, that is Vθ = 0. The flow field is in steady state. a) Use mass conservation law and integral analysis to compute Vr as a function of U, V, and ℎ. b)Use momentum conservation law and integral analysis to compute the horizontal force (in x-direction) to anchor the channel in place. (hint: vector integral must be done in rectangular coordinate)arrow_forward3. Consider a fluid motion at a velocity Vol with a space-time varying density such that, n(x, t) = no eα²²xVoxt Find the total time change of n(x, e) as seen by an observer co-moving with the fluid (the convective or material derivative)arrow_forwardWater with density of 1000 kg/m^3 flows through a horizontal pipe (in the x-z plane) bend as shown. The weight of the pipe is 350 N and the pipe cross-sectional area is constant and equals to 0.35 m^2. The magnitude of the inlet velocity is Section (1) 4 m/s. The absolute pressures at the entrance and exit of the bend are 210 kPa and 110 kPa, respectively. Assuming the atmospheric pressure is 100 kPa and neglecting the weight and viscosity of the water , find the following: Control volume The mass flow rate is 180° pipe bend Section (2) The exit velocity is The force (in the z-axis direction) acting on the fluid isarrow_forwardA small, spherical bead of mass 2.80 g is released from rest at t = 0 from a point under the surface of a viscous liquid. The terminal speed is observed to be v, = 2.34 cm/s. (a) Find the value of the constant b in the equation R = -bv. N-s/m (b) Find the time t at which the bead reaches 0.632VT. (c) Find the value of the resistive force when the bead reaches terminal speed. Narrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY