Consider steady, incompressible, and fully developed laminar flow of a viscous liquid down an in

Consider steady, incompressible, and fully developed laminar...

Key Concepts

Velocity Profile in Inclined Flow

The velocity profile is the distribution of fluid velocity from the solid boundary up to the free surface in the film flow. For a viscous, laminar, free-surface flow down an inclined plane, the momentum equation typically leads to a parabolic profile, reflecting the balance between the gravitational driving force and viscous shear stresses. This profile is essential for analyzing flow characteristics like maximum velocity and shear rates.

Kinematic Viscosity

Kinematic viscosity is defined as the ratio of the fluid's dynamic viscosity to its density. It quantifies the relative influence of viscous forces compared to inertial forces in a flow. In problems involving film flows, kinematic viscosity can be deduced by relating the observed velocity profile (and specifically the maximum velocity) to the film thickness, the gravitational force component along the slope, and the derived theoretical profile.

Boundary Conditions

Boundary conditions are crucial in solving the fluid flow equations. For film flow down an inclined plane, the no-slip condition at the solid wall (zero velocity at the surface) and the stress-free condition at the free surface (zero shear stress) are typically applied. These conditions ensure that the derived velocity profile accurately reflects the physical constraints of the problem.

Gravitational Body Force

In flows down an incline, gravity is the primary driving force acting on the fluid. The component of gravitational acceleration along the incline causes the fluid to flow, and it appears as a body force in the momentum equations. This force is key in deriving the velocity profile for free-surface flows down a slope.

Laminar Flow

Laminar flow describes a fluid motion regime where the fluid particles move along smooth paths in layers, with minimal mixing between layers. This is in contrast to turbulent flow and is characterized by low Reynolds numbers, making the mathematical analysis more straightforward since the flow is orderly and predictable.

Steady, Incompressible, Fully-Developed Flow

This concept refers to a flow that does not change with time (steady), has a constant density (incompressible), and where the velocity profile does not change in the flow direction (fully-developed). This simplifies the governing equations considerably and is a common assumption for many canonical fluid mechanics problems.

Free-Surface Film Flow

This term describes a thin layer of liquid flowing over a solid surface with a free, stress-free surface exposed to the atmosphere. The dynamics of such flows include a balance between gravitational forces and viscous resistance, and the flow behavior is influenced by the film thickness and the angle of inclination.

Transcript

00:01 We can write the velocity profile u is equal to g sine theta divided by nu multiplied by hy minus y square divided by 2.

00:17 Let this equation be equation number 1.

00:20 From here we get the kinematic viscosity.

00:24 New can be written as g sine theta, g sine theta divided by u multiplied by h y minus y squared divided by two now as we know the velocity will be maximum at y is equal to h by two so at y is equal to h by two we have we have u is equal to u max hence our viscosity term mu will be equal to g sine theta divided 2 u max multiplied by h squared by h squared so this will be at y is equal to h the velocity will be maximum at y is equals to h not h by 2 velocity will be max at at y is equals to h so upon putting this value in equation 1 we get new is equal to g so let's put the values of here.

01:29 So g is 9 .81 multiplied by sine 30, xx0.

01:38 Multiplied by the value of h square h is 0 .8 multiplied by 10 to the power of minus 3 meter this square divided by 2 multiplied by u maxes 15 .7 multiplied by 10 to the power of minus 3 15.

01:59 7 multiplied by 10 to the power of minus 3.

02:02 From here we get kinematic viscosity new is equal to 1 multiplied by 10 to the power of minus 5 meter square per second.

02:13 So this is the kinematic viscosity of the liquid given in our problem.

02:24 Now in order to plot the velocity profile, we need to assume some value for y by h and tabulate the values for u by u max.

02:32 So we will write u divided u max u divided by u x is equal to g sine theta divided by new multiplied by divided by u max for u max it is g sine theta multiplied by h squared divided by two new multiplied by h y minus y squared divided by two or upon further solving we can write u.

03:02 Divided by u max is equal to 2 divided by h squared multiplied by h y minus y square divided by 2 so from here we can write u divided by u max is equal to 2 multiplied by y divided by h minus y divided by h whole square let this equation be equation number three now we will form it table so let's form the table so let's put y by h and let's calculate let us calculate the value of u by u max so let us put the values so at y by h equal to zero we have u by u max at y by h equal 0 .2 we have u by u max is got to 0 .36.

04:08 At y by h.

04:09 Is equal to 0 .4, we have u by u max jumped to 0 .6 4.

04:14 At y by h is equal to 0 .6, we have u by u max going to 0 .8 4...

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