a) The velocity vector in a fluid flow is given as V = 1xi-12x²yj + 3tk. Find the resultant velocity and acceleration of a fluid particle at (1,3,4) at time t=1.
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- Tp = Fq +°P/Q• (1) Here ip/Q is the "position of point P relative to point Q." Similarly the velocities of the two points are related by õp = bq + Up/Q- (2) The quantity õp/Q is the velocity of point P relative to point Q. I want you to use these ideas to solve the following problems. 1. The figure below shows a view from above of a large boat in the middle of the ocean. So that the crew on the ship can get exercise on long journeys, there is a circular walking/running track on the back deck. CA B- -D Suppose that the radius of the track is R = 6 m, and a person is running on the track at a constant speed of v = 3m/s as measured with a stopwatch by a crew-mate on board the ship. Suppose the runner is running counter-clockwise around the track when viewed from above. Write the velocity vector of the runner in terms of basis (ê1, ê2) as perceived by a crew-mate on the ship. (a) What is the velocity vector when the runner is at point A? (b) What is the velocity vector when the runner is…A Fluid Mechanics, Third Edition - Free PDF Reader E3 Thumbnails 138 FLUID KINEMATICS Fluid Mechanies Fundamenteis and Applicationu acceleration); this term can be nonzero even for steady flows. It accounts for the effect of the fluid particle moving (advecting or convecting) to a new location in the flow, where the velocity field is different. For example, nunan A Çengel | John M. Cinbala consider steady flow of water through a garden hose nozzle (Fig. 4-8). We define steady in the Eulerian frame of reference to be when properties at any point in the flow field do not change with respect to time. Since the velocity at the exit of the nozzle is larger than that at the nozzle entrance, fluid particles clearly accelerate, even though the flow is steady. The accel- eration is nonzero because of the advective acceleration terms in Eq. 4-9. FLUID MECHANICS FIGURE 4-8 Flow of water through the nozzle of a garden hose illustrates that fluid par- Note that while the flow is steady from the…The Stokes-Oseen formula for drag force on a sphere at low speed is given asD = 3dV +916V 2d2, where D is drag, V is velocity, is density, d is the sphere diameter, and is the viscosity coe¢ cient.(a) Using the formula given, Önd the dimensions of the viscosity coe¢ cient. (Donít simply look upthe dimensions; use the formula to show them.) Be sure to show your work. Find the primaryunits of viscosity in SI and British units.(b) Verify that the Stokes-Oseen formula is dimensionally homogeneous.
- The terminal velocity of a sphere (maximum drop velocity) depends on sphere diameter (D), sphere density (ρs), fluid density (ρf), fluid viscosity (μ) and acceleration due to gravity (g). By using Buckingham π-Theorem, create a nondimensional form for the terminal velocity. Show all the steps and calcOrange JO A O X 91|4 2:26 ch1_introductio.. Shear stress (t) is the resistance per unit area of the upper plate t = R/A=T/A Water responds to shear stress by continuously yielding in angular deformation in the direction of the shear. IThe rate of angular deformation in the fluid, d(8)/dt ,is proportional to the shear Istress, as shown in Figure 1.1. do dt dx ,and v = dy dx Angular deformation (Shear strain), 0 = dt do Rate of shear strain = dt dx dv (Velocity gradient) dy dy dt dv Therefore, to dv T = constant dy dv T = - dy The proportionally constant, u, is called the absolute viscosity of the flyid Example A flat plate of 50 cm² is being pulled over a fixed flat surface at a constant velocity of 45 cm/sec (Figure 1.1). An oil film of unknown viscosity separates the plate and the fixed surface by a distance of 0.1 cm. The force (T) required to pull the plate is measured to be 31.7 N, and the viscosity E of the fluid is constant. Determine the viscosity (absolute). 22 Example A flat…fluıd mechanıcs Values for a fluid flowing in a circular pipe are given in the table.a) by writing Newton's law of viscosity,b) Determine the flow behavior of the fluids, show on the graph that they obey the Newtonian or non-Newtonian flow behavior. Calculate the dynamic viscosity values on the graph.c) If the viscosity of the liquid is 0.121 g/cm3, calculate the kinematic viscosity and convert it to the English Unit system.
- Consider the following three-dimensional velocity vector: V = 4xy² i + fj- z²y² k a) Find the appropriate form of the function f such that the velocity vector represents a physical incompressible flow. b) What is the stream function for this flow? c) Write down the expression for the velocity potential along the plane z = 0.A velocity field of the two-dimensional, time-dependent fluid flow is given by where t is time. Find the material derivative Du/Dt and hence calculate the acceleration of the fluid at any time t > 0 and any pont x > 0, y > 0. a) Incompressibility a) Is this flow incompressible (i.e. it has zero divergence)? Yes No ди Ət b) Time derivative of flow field Calculate the time derivative of the velocity. Represent your answer in the form i+ || 3 3 u(t, x, y) =r? (x² + y² ) i− {etxtyj X уј 3 a = c) Material derivative and acceleration Calculate the material derivative of the velocity and hence the acceleration a. Represent your answer in the form Du Dt || j i+ jCauchy's ΣF ) equation of motion : pDV/Dt =pg + VT (like pa Newtonian viscous stress relations by the tensor relation : Ti j = - pôij + µ[Əvj/əxi + əvi/axj] where dij is the kroneker delta function (1 for i = T includes pressure and viscous surface forces. into Cauchy's equation, and assume constant viscosity, to get the Navier-Stokes vector eq'ns : pDV/Dt Pg -vp + μ^2 V the acceleration DV/Dt av/at+ (VV)V, which for steady state flow gives DV/Dt =(V.) V. Because (VV) V is a non-linear term on the LHS of the N-S equation Reynolds Number RepVL/μ, a measure of the ratio of inertial to viscous forces. : Patm 10^5 = = N = N = ; pwater 1000; pair 1.2; μwater 10^-3 N s/m^2 ; Hair 2 x 10^-5 N•s/m^2 ; g 9.8 m/s^2 = j; 0 for i j ); N
- 2. Experimental measurements of relationship between velocity gradient and shear stress for different liquids are shown in the table given below. a) Plot the data on Excel and determine the fluid types. (Show your graph!) b) Which model do you use to describe the data? Specify the ranges of values for the coefficients. Velocity gradient Shear stress Fluid A Fluid B Fluid C 0.2 2.60 7.80 61.55 0.5 5.80 17.40 63.67 0.95 10.51 31.53 66.45 1.4 14.52 43.56 69.30 1.9 21.02 56.75 72.89 2.45 28.22 64.91 77.60 3.05 31.50 69.30 80.38 3.7 38.91 71.98 84.50 4.35 45.21 76.86 88.81 5.05 51.71 80.15 93.78Suppose you have crude oil flows through an annulus between two horizontal pipes with the same center, if the velocity distribution, and the average velocity Vavp are expressed by: R R In (R₂/R.) (F)] In- 4μl. R-R ΔΡ SµL R+R In(R./R) Where AP: is the pressure drop through the annulus, u: is the fluid viscosity, L: is the pipe length, R, and R₂: are the inside radius of inner and outer pipes, respectively. Write a program in a script file that calculates the velocity distribution and the average velocity. When the script file is executed, it requests the user to input AP. u. L. R, and R₂ where r has many values between R₁ and R₂. The program displays the inputted values and the calculated average velocity (using fprintf) followed by a table with the values r in the first column and the corresponding values of the velocity distribution in the second column.1. The potential function of fluid flow motion is given by, [(x+c)2+y²7 p = log la-e)2+y². a) Prove that, fluid is incompressible, b) Find the velocity field, c) Find the geometric location of the providing IV? = constant. %3D