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- A film of liquid with kinematic viscosity and density p spreads over a flat horizontal surface due to gravity as shown. Assume that the spread is planar in x - y plane with unit width into page. The height of the film is h(x, t) which varies in a direction and time t. The flow is incompressible and the x-velocity is u(x, y, t) which is governed by the lubrication theory due to the small thickness of the film (h < L). The pressure outside the film is uniform and atmospheric. Inside the film the pressure variation is hydrostatic in y direction. At x = 0, h = h, and h is symmetric in z (i.e., h(L,t) = h(-L, t)). The gravitational acceleration is g. x=-L hexat) ấy x u(x, y, t) = -2y(2h - y). x=L Show that +(udy) = 0 Using lubrication theory show that the velocity profile ↓g Assuming that Oh/dt = constant, use (a) and (b) results to find h(r, t) in term of ho, x, ah/ot, v, L and g.Viscosity of fluid plays a significant role in the analyses of many fluid dynamics problems. The viscosity of water can be determined from the correlation: H = c, where µ = viscosity (N/s · m²) T = temperature (K) C, = 2.414 × 10-5 C; = 247.8 (K) Cz = 140 (K) What is the appropriate unit for c, if the preceding equation is to be homogeneous in units?Problem 2: The potential energy for a particular two-dimensional force field is given by V(x, y) = Axe-ky, where the constants A and k are chosen for dimensional consistency. (a) Choose specific (and reasonable) values for A and k, and use Mathematica, Desmos, or the equivalent to plot contours of constant potential energy for various energy levels, over a region of the plane centered on the origin. (b) Show that an infinitesimal displacement along an equipotential line has the form dr dr = dr x + ŷ. kx (c) Find the expression for the force field, and plot this vector field over the same region of the plane (and with the same choice of constants) as in part (a).
- The continuity equation of fluids describes the property of energy conservation; and Bernoulli's equation indicates conservation of mass. Ture OR FaleHome Work (steady continuity equation at a point for incompressible fluid flow: 1- The x component of velocity in a steady, incompressible flow field in the xy plane is u= (A /x), where A-2m s, and x is measured in meters. Find the simplest y component of velocity for this flow field. 2- The velocity components for an incompressible steady flow field are u= (A x* +z) and v=B (xy + yz). Determine the z component of velocity for steady flow. 3- The x component of velocity for a flow field is given as u = Ax²y2 where A = 0.3 ms and x and y are in meters. Determine the y component of velocity for a steady incompressible flow. Assume incompressible steady two dimension flowThe sliding plate viscometer shown below is used to measure the viscosity of a fluid. The top plate is moving to the right with a constant velocity of 10 m s in response to a force of 3 N. The bottom plate is stationary. What is the viscosity of the fluid? Assume a linear velocity distribution.
- Q4) Set up the differential equations for the two masses [Fig.1] 2cos (3t) Fig. 1 C1 K1 M1 K2 М2 K3The fluid flow in the region above the two planes: y = x and y = -x can be approximated by the velocity field: V(x,y,z,t) = -V,ko +yz- y çî-VoLo x² +y2 x2 + y2 where V=0.004 [m/s] and Lo =1.0 [m] a) Make a sketch of the region in which the fluid flows for 0The fluid particles do not rotate about their own axis in an irrotational fluid flow. Select one: True False1. A flow in the x-y plane is given by the following velocity field: u =3 and v=6m/s for 01. Any equation that comes out of the differential approach is valid at every point in the flow field. On the other hand, the integral approach gives us a point by point value and not an average value. True or False 2. During the analysis of fluid flow, we are more often interested in obtaining the average velocity, and one of the common ways of obtaining the average velocity is to take an average across the flow cross-section. This is called area-average velocity, average along the direction parallel to the principal direction of flow. True or False 3. A crude oil with a density of 875 kilogram per cubic meter is pumped and flowing at 1.5 meter per second through a pipeline 1 meter in diameter. The maximum viscosity of the oil to exhibit a laminar flow behavior is 625 centipoise. True or FalseAn Eulerian velocity vector field is described by V = i + 2zj − 3x2k, where i, j and k are unit vectors in the x-, y- and z-directions, respectively. (i) Is the flow incompressible or compressible? (ii) Find the rate of change of velocity following a fluid particle. (iii) Find the viscous force per unit volume (that is, the viscous term in the Navier-Stokes equation).SEE MORE QUESTIONSRecommended textbooks for youElements 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 EducationControl 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:WILEYElements 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 EducationControl 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