Differential Equations - Acceleration and velocity models

An expression for the laminar velocity profile on a flat plate is u=C1sin(C2y)+C3 Where the argument of the sine function is in radians. Using the three common physicalconditions that the velocity profile should satisfy, determine  d) the skin friction coefficient, Cf as a function of lengthe) displacement thickness (δ*/x) as a function of lengthf) the momentum thickness (θ/x) as a function of lengthg) the drag force (FD) on both sides of the plate

Free stream 8 (x) Free stream • 8,(x) Thermal Velocity boundary layer boundary layer (a) (b) A differential energy analysis of this flow under common assumptions reduces the energy equation to 2 a²T + v. dy Cp where T = T(r, y) is the temperature of the fluid in the boundary layer, u is the velocity parallel to the plate, v is the velocity perpendicular to the plate, a is the thermal diffusivity of the fluid, v is the kinematic viscosity, and c, is the specific heat of the fluid at constant pressure. a) Find the base dimensions, in [M LT0], where 0 is temperature, of a and c, using the given equation. Recall that v has SI units of [m?s-'). Do not simply use the internet, textbooks, or your prior knowledge to determine the dimensions of these variables. b) It is common to "shift" both the fluid temperature in the boundary layer, T, and the far-field fluid temperature, To, by considering these temperatures relative to the surface temperature of the plate, T,. This effectively reduces…

The Laws of Physics are written for a Lagrangian system, a well-defined system which we follow around – we will refer to this as a control system (CSys). For our engineering problems we are more interested in an Eulerian system where we have a fixed control volume, CV, (like a pipe or a room) and matter can flow into or out of the CV. We previously derived the material or substantial derivative  which is the differential transformation for properties which are functions of x,y,z, t. We now introduce the Reynold’s Transport Theorem (RTT) which gives the transformation for a macroscopic finite size CV. At any instant in time the material inside a control volume can be identified as a control System and we could then follow this System as it leaves the control volume and flows along streamlines by a Lagrangian analysis. RTT:DBsys/Dt =   ∂/∂t ʃCV (ρb dVol) + ʃCS ρbV•n dA; uses the RTT to apply the laws for conservation of mass, momentum (Newton's Law), and energy (1st Law of…

Wind provides an increasingly important source of renewable energy in certain locations of the world. In fact, wind was the second largest renewable energy source worldwide (after hydropower!) and provided more than six percent of global electricity in 2020; in the U.S. that number was 8.4%. Suppose that wind turbines can convert up to 30% of the kinetic energy of incoming wind blowing through their blades into electricity (i.e., useful shaft work). In the following, we will analyze the power output of a wind turbine with a diameter of d 80 m placed at an offshore location where the wind blows steadily at an average speed of 20 knots. Explicitly state all relevant approximations or assumptions. = (a) Draw a diagram that includes appropriate elements for assessing the amount of work extracted from the wind turbine. Your diagram should have annotated any known or approximated proper- ties. What is your system and surroundings? What kind of boundary divides these two? Hint: In terms of…

A turbine model has the following characteristics: Effective power: 13, 6 kW; Rotor Diameter: 0.38 meters, Flow: 0.34 ms, Energy Jump: 26.5 J / Kg and 3 R.P.S. rotation It is desired to construct a geometrically similar 1.20 meter diameter turbine that provides a power on the 656 kW axis, we ask: to calculate and justify its answers: a) the specific rotation; b) the unit quantities (rotation, flow and power); c) the rotation, flow and power of the model (bi-unit quantities).

The Laws of Physics are written for a Lagrangian system, a well-defined system which we follow around – we will refer to this as a control system (CSys). For our engineering problems we are more interested in an Eulerian system where we have a fixed control volume, CV, (like a pipe or a room) and matter can flow into or out of the CV. We previously derived the material or substantial derivative  which is the differential transformation for properties which are functions of x,y,z, t. We now introduce the Reynold’s Transport Theorem (RTT) which gives the transformation for a macroscopic finite size CV. At any instant in time the material inside a control volume can be identified as a control System and we could then follow this System as it leaves the control volume and flows along streamlines by a Lagrangian analysis. RTT:DBsys/Dt =   ∂/∂t ʃCV (ρb dVol) + ʃCS ρbV•n dA; uses the RTT to apply the laws for conservation of mass, momentum (Newton's Law), and energy (1st Law of…

1) Is it possible to prescribe the velocity v0, so that the pendulum keeps rotating with alpha = pi/2 Explain clearly why?  Given: L, g, and m. 2) After transient motion, the pendulum settles at steady state rotation atalpha = pi/3. Find the initial velocity v0. Given: L, g, m, sin(pi/3) = (squareroot(3)/2), cos(pi/3) = (1/2). [Hint: Use the conservation of energy and the 2nd Newton’s law.]

Looking for a handwritten solution with fbd.