Starting with two coaxial cylinders with a small gap between the inner and outer cylinders and the outer clylinder is fixed while the inner cylinder rotates with an angular velocity w1. The governing equation for the setup is 0 = d/dr ((1/r) (d/dr)(rvthetha)). Show the velocity profile is vthetha(r) = (w1r12)/(r22-r12)*((r22/r )-r)
Given the governing differential equation:
The boundary conditions are;
Now, at
At,
Solving simultaneously;
On substitution;
or
Starting with two coaxial cylinders with a small gap between the inner and outer cylinders and...
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...
Fluid is Non-Newtonian.
(3) Consider the steady laminar flow between the coaxial cylinders shown below. The inner cylinder rotates with angular velocity 2 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 2-direction hence we may ignore edge effects near the top and bottom surfaces. - R2 Assume that gravity is negligible, v, is zero and that are zero for...
Tangential laminar flow of a Newtonian fluid with constant density and occurring between two vertical coaxial cylinders in which the outer rotating with an angular velocity of ω and the inner cylinder is fixed a. Write the simplified continuity equation and the simplified momentum balance equations using necessary assumptions and determine the velocity. b. Determine the shear stress distributions for this flow. c. Calculate the necessary torque. outside cylinder rotates 2 inside cylinder Figure: Top view of the coaxial cylinders
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the second prob pic
Consider a medical device where blood is circulated in the annular space between two coaxial cylinders (Figure 1). The inner cylinder (radius cylinder (radius R) is rotating with constant anacibeNewtonian fluid (density o. are infinitely long, and that blood behaves as an tncompcessiole viscosity . Ignore the effect of gravity. whereas the outer velocity oAssume that the cylinders 1a. Write a conservation equations appropriate to determine the fluid velocity profile insido the annular gap, along...
Consider two concentric, infinitely long cylinders. The cylinders are oriented such that the center-line is along the z-axis, and the radii exist in the r-direction. The inner cylinder has a radius of ra and the outer cylinder has a radius Tb. The inner cylinder rotates with an angular velocity of w whereas the outer cylinder is stationary. There is no pressure gradient applied nor gravity. The fluid contained between the cylinders is assumed to be Netwonian, incompressible, isotropic and isothermal....
Radial flow between two coaxial cylinders. Consider an incompressible fluid, at constant temperature, flowing radially between two porous cylindrical shells with inner and outer radii xR and R (a) Show that the equation of continuity leads to V C/r where C is a constant (b) Simplify the components of the equation of motion to obtain the following expressions for the modified-pressure distribution: ds dr dz (c) Integrate the expression for dP/dr above to get (d) Write out all the nonzero...
Page 5 Nane: Johnson, Perri NN: 35 1000 omawork 10 10.5 Aoni cous fuid is contained betwen two infinitely vertical, concentric cylinders. The outer cylinder has a radius fixedd rotates with an angular velocity o. The inner cylinder is and has a radius r. The Navier-Stokes equations can be used As utain an exact solution for the velocity distribution in the gap. that the flow in the gap is axisymmetric (neither velocity Fluid nor pressure are functions of angular position...
An incompressible Newtonian fluid is contained between two long concentric cylinders of radii AR (< 1) and R, as shown in the figure. The inner cylinder rotates with an angular velocity Ω (a) Compute the velocity distribution between the cylinders. End effects caused by (b) Compute the torque required to hold the outer cylinder stationary. (8 Pts)
An incompressible Newtonian fluid is contained between two long concentric cylinders of radii AR (
A variable capacitor consists of two thin coaxial metal cylinders of radii a and b, with (b - a) << a, free to move with respect to each other in the axial direction. The length of the cylinders is L, and the potential difference between the two cylinders is V. Initially, the inner cylinder (radius = a) is completely enclosed by the outer cylinder (radius = b). Using energy methods, find the magnitude and direction of the force on the...
4. Consider the situation of radial flow between two concentric cylinders. The outer cylinder has a radius of R and the inner cylinder has a radius KR. Assume flow is only in the radial direction and assume v, = v(r). Use the continuity equation and the relevant momentum balance equations to derive an expression for the pressure difference Pi-Po between the outer and inner cylinders as a function of the volumetric flow rate with L being the length of the...