Homework Answers
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Assumptions |
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Fig 10.1 Laminar flow in a horizontal pipe |
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Intuitively guess the velocity profile |
| Since the flow is steady and laminar, we may intuitively say that the velocities in r direction and θ direction are zero. Due to steady state conditions, the fluid velocity in z direction, vz, is not dependent on time t. Furthermore, due to the axisymmetric geometry fluid flow the velocity vz is independent of θ. Thus, |
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| By applying the equation of continuity in cylindrical coordinates |
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Hence,  |
| Since the fluid is flowing in z direction, we may conclude the following. |
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An incompressible Newtonian fluid flow through a horizontal circular tube is shown in the following figure....
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 a...
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...
A wire of radius ri is pulled coaxially at a constant velocity V through an incompressible...
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Considera steady, incompressible laminar flow of a Newtonian fluid in a pipe ignoring the effects of...
Considera steady, incompressible laminar flow of a Newtonian fluid in a pipe ignoring the effects of gravity. When a constant pressure gradient is applied in the x-direction, demonstrate that the maximum velocity of the fluid is given by 2 times of its average velocity.
Navier-Stokes Equation: An incompressible Newtonian liquid is confined between two concentric cy...
Navier-Stokes Equation:
An incompressible Newtonian liquid is confined between two
concentric cylinders of infinite length—a solid inner cylinder of
radius RA and a hollow outer cylinder of radius RB. The inner
cylinder rotates at angular velocity ω and the outer cylinder is
stationary. The flow is steady, laminar, and two-dimensional in the
r-θ plane. The flow is rotationally symmetric, meaning that nothing
is a function of the coordinate θ. The flow is also circular so
that ur=0 everywhere.
Found Uθ=...
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The laminar flow of a permanent incompressible Newtonian fluid in a long cylindrical pipe with a...
The laminar flow of a permanent incompressible Newtonian fluid in a long cylindrical pipe with a diameter D in vertical position is considered. Gravitational effects are taken into account, flow is carried out with a constant pressure gradient and gravity effect in the z- direction. a. Express the problem on the figure, write the given and accepted. b. Find the velocity profile in the fluid. c. Develop the relations that express the volumetric flow and shear stress in the pipe...
An incompressible viscous fluid is placed between horizontal, infinite parallel plates as shown. The two plates...
An incompressible viscous fluid is placed between horizontal, infinite parallel plates as shown. The two plates move in the same direction but with different velocities, U1 and U2. The pressure gradient in the x direction is zero and the only body force is due to the fluid weight. Using the continuity and the Navier-Stokes equations, find an expression for the velocity profile between the plates. Show ALL work for full credit. U1 V=O b KO wo U2
An incompressible viscous fluid is placed between horizontal, infinite parallel plates as shown. The two plates...
An incompressible viscous fluid is placed between horizontal, infinite parallel plates as shown. The two plates move in the same direction but with different velocities, U1 and U2. The pressure gradient in the x direction is zero and the only body force is due to the fluid weight. Using the continuity and the Navier-Stokes equations, find an expression for the velocity profile between the plates. Show ALL work for full credit. U1 V=O g u K wo U2
Tutorial 2. Incompressible Navier-Stokes equations 18 September, 4-5 pm in FN2 In Lecture Notes 1 the Navier-Stokes equ...
Tutorial 2. Incompressible Navier-Stokes equations 18 September, 4-5 pm in FN2 In Lecture Notes 1 the Navier-Stokes equations (momentum balance) for incompressible flow were derived. They were eventually written in the following form dr In this equation, the viscosity μ and the density ρ are constants. We now consider two simple flow configurations. Config. 1. The steady state flow of a liquid in the space between two very large static parallel plates at distance H of each other in the...
An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates as shown below. The two...
An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates as shown below. The two plates move in opposite directions with constant velocities U 10 m/s and U2 = 5 m/s as shown. The pressure gradient in the x direction is zero and the only external force is gravity (in the y-direction). Use the Navier-Stokes equations to determine where the fluid velocity is zero (in terms of a fraction of b, i.e. 0.75 for y-75% of b) Enter Number...
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...
A wire of radius ri is pulled coaxially at a constant velocity V through an incompressible Newtonian fluid in a horizontal tube of inner radius r2. Using the Navier-Stokes equation, derive the steady-state velocity distribution in the fluid neglecting end effects. 4.
Considera steady, incompressible laminar flow of a Newtonian fluid in a pipe ignoring the effects of gravity. When a constant pressure gradient is applied in the x-direction, demonstrate that the maximum velocity of the fluid is given by 2 times of its average velocity.
Navier-Stokes Equation:
An incompressible Newtonian liquid is confined between two
concentric cylinders of infinite length—a solid inner cylinder of
radius RA and a hollow outer cylinder of radius RB. The inner
cylinder rotates at angular velocity ω and the outer cylinder is
stationary. The flow is steady, laminar, and two-dimensional in the
r-θ plane. The flow is rotationally symmetric, meaning that nothing
is a function of the coordinate θ. The flow is also circular so
that ur=0 everywhere.
Found Uθ=...
10. Immiscible fluids Two immiscible incompressible Newtonian fluids flow together through in thedirection two lates separated by a distance H in the y-direction. Let us make thé top plate /move with ection while fixing the bottom plate. At steady state, however, there be a little slip velocity of the more dense fluid only at the lower boundary The flow constant vetocity V in the x-dir is ynidirectional and faminar. For convenience, we take that x is the flow direction and...
An incompressible viscous fluid is placed between horizontal, infinite parallel plates as shown. The two plates move in the same direction but with different velocities, U1 and U2. The pressure gradient in the x direction is zero and the only body force is due to the fluid weight. Using the continuity and the Navier-Stokes equations, find an expression for the velocity profile between the plates. Show ALL work for full credit. U1 V=O b KO wo U2
An incompressible viscous fluid is placed between horizontal, infinite parallel plates as shown. The two plates move in the same direction but with different velocities, U1 and U2. The pressure gradient in the x direction is zero and the only body force is due to the fluid weight. Using the continuity and the Navier-Stokes equations, find an expression for the velocity profile between the plates. Show ALL work for full credit. U1 V=O g u K wo U2
Tutorial 2. Incompressible Navier-Stokes equations 18 September, 4-5 pm in FN2 In Lecture Notes 1 the Navier-Stokes equations (momentum balance) for incompressible flow were derived. They were eventually written in the following form dr In this equation, the viscosity μ and the density ρ are constants. We now consider two simple flow configurations. Config. 1. The steady state flow of a liquid in the space between two very large static parallel plates at distance H of each other in the...
An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates as shown below. The two plates move in opposite directions with constant velocities U 10 m/s and U2 = 5 m/s as shown. The pressure gradient in the x direction is zero and the only external force is gravity (in the y-direction). Use the Navier-Stokes equations to determine where the fluid velocity is zero (in terms of a fraction of b, i.e. 0.75 for y-75% of b) Enter Number...
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