Concept explainers
Assume the liquid film in Example 5.9 is not isothermal, but instead has the following distribution:
where T0 and Tw are, respectively, the ambient temperature and the wall temperature. The fluid viscosity decreases with increasing temperature and is assumed to be described by
with a > 0. In a manner similar to Example 5.9, derive an expression for the velocity profile.
Example 5.9 ANALYSIS OF FULLY DEVELOPED LAMINAR FLOW DOWN AN INCLINED PLANE SURFACE
A liquid flows down an inclined plane surface in a steady, fully developed laminar film of thickness h. Simplify the continuity and Navier–Stokes equations to model this flow field. Obtain expressions for the liquid velocity profile, the shear stress distribution, the volume flow rate, and the average velocity. Relate the liquid film thickness to the volume flow rate per unit depth of surface normal to the flow. Calculate the volume flow rate in a film of water h = 1 mm thick, flowing on a surface b = 1 m wide, inclined at θ = 15° to the horizontal.
Want to see the full answer?
Check out a sample textbook solutionChapter 5 Solutions
Fox and McDonald's Introduction to Fluid Mechanics
Additional Engineering Textbook Solutions
Thinking Like an Engineer: An Active Learning Approach (3rd Edition)
Statics and Mechanics of Materials (5th Edition)
Engineering Mechanics: Statics & Dynamics (14th Edition)
INTERNATIONAL EDITION---Engineering Mechanics: Statics, 14th edition (SI unit)
Degarmo's Materials And Processes In Manufacturing
Fundamentals of Heat and Mass Transfer
- A Pipe used to transport an oil of specific gravity of 0.90 & 0.03poise at a Velocity of 1.6977m/s with a flow rate of 3cubic meter per second. A Pipe was used to conduct a test using water at 20 degree celsius with a velocity of 5.0931m/s. Find the diameter of the prototype. Viscosity of water at 20 degree celsius is 0.01poisearrow_forwardA rotating viscometer consists of two concentriccylinders—an inner cylinder of radius Ri rotating at angularvelocity (rotation rate) vi, and a stationary outer cylinder ofinside radius Ro. In the tiny gap between the two cylindersis the fluid of viscosity m. The length of the cylinders is L. L is large such that end effects are negligible (we can treat this as a two-dimensional problem). Torque (T) is required to rotate the inner cylinder at constant speed. (a) Showing all of your work and algebra, generate an approximate expression for T as a function of the other variables. (b) Explain why your solution is only an approximation. In particular, do you expect the velocity profile in the gap to remain linear as the gap becomes larger and larger (i.e., if the outer radius Ro were to increase, all else staying the same)?arrow_forwardA rod of diameter 0,08 m and length 0,13 m is inside a concentric cylindrical casing. The rod and its casing are vertically located. A liquid of density 900 kg/m3 and viscosity 0.9 Pa.s is used as a lubrication medium between rod and its casing. The rod is rotated at a speed of 46 rad/s. Due to the narrow clearance 0,03 m between rod and casing gravitational effects and surface curvature effects are ignored . Assume liquid flow in clearance as laminar steady and incompressible and use pi number, π = 3.14 a) What is the magnitude of shearing stress over the rod (in Pa)? b) What is the magnitude of frictional torque on rod (in N m)? c) What is the magnitude of Reynolds Number of flow ? Use clearence as dimensional characteristics in Reynolds Number calculationarrow_forward
- A rotating viscometer consists of two concentric cylinders—a stationary inner cyliner of radius Ri and an outer cylinder of inside radius Ro rotating at angular velocity (rotation rate) ?o. In the tiny gap between the two cylinders is the fluid whose viscosity (? ) is to be measured. The length of the cylinders in Fig is L. L is large such that end effects are negligible (we can treat this as a two-dimensional problem). Torque (T) is required to rotate the inner cylinder at constant speed. Showing all your work and algebra, generate an approximate expression of T as a function of the other variables.arrow_forwardUse the Stokes-Einstein equation to estimate the diffusivity (cm^2/s) for a molecule with a solute radius 2.6 nm, at 44 degrees C,in a fluid with viscosity of 1.01 cP). Enter your answer in decimal form without unitsarrow_forwardA concentric cylinder viscometer may be formed by rotating the inner member of a pair of closely fitting cylinders. The annular gap is small so that a linear velocity profile will exist in the liquid sample. Consider a viscometer with an inner cylinder diameter of 4 in, a height of 8 in, and a clearance gap width of 0.001 in, filled with castor oil at 90 oF. Determine the torque (in lbf×ft) require to turn the inner cylinder at 400 rpm. The dynamic viscosity of the caster oil is 0.0079 lbf×s /ft2.arrow_forward
- A- Womersley number (a) of a human aorta is 20 and for the rabbit aorta is 17, the blood density is approximately the same across the species. The values of viscosity were 0.0035 Ns/m² for the human and 0.0040 Ns/m² for the rabbit. The diameter of the aorta is 2.0 cm for the man, and 0.7 cm for the rabbit, estimate the heart rate beats per minute (bpm) for both speciesarrow_forwardA solid sphere diameter of 6 mm is rising through oil of density 900 kg/m^3, dynamic viscosity of .07 kg/m-s. At a constant velocity of 1 cm/s. Calculate the specific weight of the material.arrow_forwardA cylinder of 250mm in diameter rotates concentrically inside a fixed of 130mm radius. Both cylinders are 300mm long. Determine the viscosity of the liquid which fills the gap between the cylinders if a torque of 0.88N-m is required to maintain an angular velocity of 2π radians/sec. Assume the velocity gradient to be a straight line. Determine the viscosity of the liquid in poise.arrow_forward
- A cylinder of mass m slides down from rest in a vertical tube whose inner surface is covered by a viscous oil of film thickness h. If the diameter and height of the cylinder are D and L, respectively, derive an expression for the velocity of the cylinder as a function of time, t. Discuss what will happen as t → ∞. Can this device serve as a viscometer?arrow_forwardFigure 1 shows a layer of oil (8.95 kN/m3), the thickness of 0.3 mm, between two parallel flat plates. The upper plate is pulled across the bottom plate with a cable connected to a winch. The upper plate moves with a velocity of 0.05 m/s. The area of contact between the upper plate and the oil is 1.5 m2. Determine the kinematic viscosity of the oil if the torque acting on the winch drum should not exceed 24.5 x 10-3 Nm. Take the drum diameter as 50 mm.arrow_forwardTwo concentric cylinders are rotating in opposite directions but the same speed, 25 revolutions per minute. The outer cylinder, 11cm in diameter, is turning counterclockwise and the inner cylinder, 10 cm in diameter, is turning clockwise. The gap between the cylinders is filled with a liquid. The liquid is a Newtonian fluid with a viscosity of 0.1 kg/m.s. Both cylinders are 50 cm long. Assuming linear distribution of velocity in the gap, Sketch the velocity profile inside the liquid in the gap At what radial distance from the center of the cylinders the liquid velocity becomes zero? Find the shear rate Find force acting on each cylinderarrow_forward
- 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