ENB360 ASSIGNMENT 1

Consider two incompressible Newtonian fluids of different viscosities flowing down an inclined plane and subject to gravity. Assume that the flow of both fluids is steady state and purely along the inclined plane so that the velocity only depends on the direction normal to the plane.

Assume that the flow of both fluids is steady state and purely along the inclined plane so that the velocity depends only on the direction normal to the plane.

The plane is set at an angle ? from the horizontal. The depth of the fluid closest to the inclined plane is h(1), the depth of the topmost fluid only is h(2). The total height of both fluids is H = h(1) + h(2).

We thus have to solve the Navier-Stokes equations for two different fluids.

Parameters

Let

? = 45?, h(1) = 0.005m h(2) = 0.01m, g = 9.8ms-2,

?(1) =?(2) = 6kgm-3.

Boundary conditions

Along the inclined plane, a ‘no-slip’ condition is applied so that the velocity of the fluid at the plane is zero.

At the interface between the two fluids, the velocities of the two fluids must match.

Also at the fluid-fluid interface, the viscous stress tensor for the two fluids must match.

As the top-most fluid has a fluid-air interface, we assume the viscous stress at this interface is zero.

Problems

1. Draw a schematic of the configuration including the 2 layers of fluid flowing down the inclined plan, all the parameters as well as the coordinate system. (2 Points)

2. List all the assumptions in mathematical notation and write the reduced form of the viscous stress tensor. (3 Points)

3. From the text, write explicitly the four boundary conditions required to solve the problem. (2 Points)

4. Based on the assumptions you gave in Problem 2, and the assumption of thin film flow, write the momentum equations to solve in Cartesian coordinates. (3 Points)

5. Using the boundary conditions, solve the equations to determine the velocity profiles of both fluids. (5 Points)

6. (a) Assuming µ(1) constant, equal to 0.01Pa s, plot the velocity profiles as a function of µ(2). (1 Point)

(b) Assuming µ(2) constant, equal to 0.02Pa s, plot the velocity profiles as a function of µ(1). (1 Point) (c) Discuss. (2 Points)

7. Assuming µ(2) constant, equal to 0.02Pa s, calculate the velocity at the top of the external fluid layer. (1 Point)

CRICOS No. 00213J ENB360 1

Consider two incompressible Newtonian fluids of different viscosities flowing down an inclined plane and subject to gravity. Assume that the flow of both fluids is steady state and purely along the inclined plane so that the velocity only depends on the direction normal to the plane.

Assume that the flow of both fluids is steady state and purely along the inclined plane so that the velocity depends only on the direction normal to the plane.

The plane is set at an angle ? from the horizontal. The depth of the fluid closest to the inclined plane is h(1), the depth of the topmost fluid only is h(2). The total height of both fluids is H = h(1) + h(2).

We thus have to solve the Navier-Stokes equations for two different fluids.

Parameters

Let

? = 45?, h(1) = 0.005m h(2) = 0.01m, g = 9.8ms-2,

?(1) =?(2) = 6kgm-3.

Boundary conditions

Along the inclined plane, a ‘no-slip’ condition is applied so that the velocity of the fluid at the plane is zero.

At the interface between the two fluids, the velocities of the two fluids must match.

Also at the fluid-fluid interface, the viscous stress tensor for the two fluids must match.

As the top-most fluid has a fluid-air interface, we assume the viscous stress at this interface is zero.

Problems

1. Draw a schematic of the configuration including the 2 layers of fluid flowing down the inclined plan, all the parameters as well as the coordinate system. (2 Points)

2. List all the assumptions in mathematical notation and write the reduced form of the viscous stress tensor. (3 Points)

3. From the text, write explicitly the four boundary conditions required to solve the problem. (2 Points)

4. Based on the assumptions you gave in Problem 2, and the assumption of thin film flow, write the momentum equations to solve in Cartesian coordinates. (3 Points)

5. Using the boundary conditions, solve the equations to determine the velocity profiles of both fluids. (5 Points)

6. (a) Assuming µ(1) constant, equal to 0.01Pa s, plot the velocity profiles as a function of µ(2). (1 Point)

(b) Assuming µ(2) constant, equal to 0.02Pa s, plot the velocity profiles as a function of µ(1). (1 Point) (c) Discuss. (2 Points)

7. Assuming µ(2) constant, equal to 0.02Pa s, calculate the velocity at the top of the external fluid layer. (1 Point)

CRICOS No. 00213J ENB360 1

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