Problem 7 (10 Pts) For the converging flow, write the continuity equation and show that V, = f(e)/r. Then, write the...
Problem 7 (10 Pts) For the converging flow, write the continuity equation and show that V, = f(0)r. Then, write the r and components of the Navier-Stokes equations, and put the Vr into the r and 0 components of the Navier- Stokes equations. Lastly, write the no-slip boundary conditions using f(e). You do not need to solve the equations Problem 7 (10 Pts) For the converging flow, write the continuity equation and show that V, = f(0)r. Then, write the...
can you solve the last question (e) Q1. Consider a steady, two-dimensional, incompressible flow field has the velocity potential a) b) c) 2 (x-7)(x+y) Determine the velocity components and verify that continuity is satisfied. [4 marks] Verify that the flow is irrotational. [2 marks] Determine the corresponding stream function. [4 marks) Now, consider a steady, two-dimensional, incompressible flow defined by velocity components u = ax + b&v=-ay + cx, where a, b and care constants. Neglect gravity. d) e) Show...
A. In the table below, identify which of the circled terms of the governing equations can be neglected by the given assumption. Write the number of the term in the table. Some assumptions relate to multiple terms, include them all. B. Write the mathematical equations describing the appropriate boundary conditions and identify them in words. C. Applying the appropriate boundary conditions, solve the differential equation remaining after appropriate terms have been neglected to determine the velocity profile in the film: d^2(w)/dx^2 =...
Consider the steady laminar flow between the coaxial cylinders shown below. The inner cylinder rotates with angular velocity Omega and the outer cylinder is stationary. The no-slip condition applies at the inner and outer cylinder surfaces and we are considering the cylinders to be very long in the z-direction, hence we may ignore edge effects near the top and bottom surfaces. a) What are the boundary conditions on the cylinder surfaces at r=R1 , and r= R2 b) Simplify and...
Problem 5. Consider a (i) steady, (ii) incompressible, axisymmetric, (iv) fully- developed, (v) constant viscosity, (vi) laminar flow in a circular pipe. Assume that the pipe is horizontal, so that any gravitational effects can be ignored It is known that an incompressible, constant viscosity fluid can be described by the continuity equation in cylindrical coordinates together with the Naiver-Stokes equations (ak.a., momentum eqns) in cylindrical coor- dinates Ov 00. Or 9-moment um 11ap 2-momentum plus the appropriate boundary conditions. Starting...
Problem 1: Differential Relations for a Fluid Particle (25 points) Two horizontal, infinite, parallel plates are spaced a distance b apart. A viscous liquid is contained between the plates. The bottom plate is fixed, and the upper plate moves parallel to the bottom plate with a velocity U. Assume no-slip boundary conditions. There is no pressure gradient in the direction of flow (a) Demonstrate using the Navier-Stokes equation in the x-direction that the velocity profile is of the form: (15...
4. An incompressible fluid with viscosity u and density p was contained in pipe of length L and radius R. Initially the fluid is in rest. At t=0, a pressure difference of AP is applied across the pipe length which induces the fluid flow in axial direction (V2) Only varies with time (t) and pipe radius (r). There is no effect of gravity. To describe the fluid flow characteristics, after the pressure gradient is applied, answer the following questions: a)...
1. Fluid between parallel plates down an inclined plane (gravity setler). Fluid is flowing between parallel plates, at an angle of β to the vertical. Assume δ<<w, L. th a momentum balance on a differential shell, and using the notation shown below, derive: i. the velocity distribution in the fluid, v,-f(x/b), and sketch result ii. the shear stress distribution in the fluid, -f(x/ö), and sketch result. iii. the volumetric flow rate iv. the maximum velocity, and the position x where...
Consider the steady, laminar flow of two liquids, A and B, with viscosities HA-μ and μΒ 21, respectively, between infinite parallel plates at 2- a, as shown in the diagram below. The plate at 2 a is fixed, while the plate at 2a moves with constant velocity -Vi, where V0. The liquids do not mix, and each forms a layer of depth a. There is an applied pressure gradient acting on both liquids, given by ▽p--Ci (where C > 0...
An important problem in chemical engineering separation equipment involves thin liquid films flowing down vertical walls due to gravity, as shown in this figure yV A. Assume that the wall is long and wide compared to the film thickness, with steady flow that is laminar and fully developed: u= v=0 and w w(x). Using a force balance on a rectangular differential element, derive an expression relating g, p, and τΧΖ . Use τΧΖ-n(-_ +--) for a Newtonian fluid to convert...