An incompressible fluid flows between two infinite stationary parallel plates. The velocity prof

An incompressible fluid flows between two infinite...

Key Concepts

Boundary Conditions in Fluid Mechanics

In fluid mechanics, boundary conditions are used to determine unknown constants in a velocity profile by applying known conditions at the boundaries. For flow between stationary parallel plates, the no?slip condition means the fluid velocity must be zero at both plates. These conditions are used to formulate equations that the constants must satisfy, which then determines their magnitude and spatial units (e.g., inverse length squared or inverse length) based on the overall gap width.

Dimensional Analysis

Dimensional analysis ensures that the constants in the velocity profile have the correct units to make the equation physically meaningful. In a parabolic profile expressed as a quadratic function in the spatial variable y, the constants multiplying y², y, and the constant term must combine with y (measured in length units) to yield a nondimensional factor when multiplied by the maximum velocity. This analysis interprets coefficients such that, for example, a constant multiplying y² must have units of 1/length².

Parabolic Velocity Profile in Pressure?Driven Flow

The velocity profile given is parabolic in form, which is characteristic of pressure-driven (plane Poiseuille) flow between two infinite, stationary parallel plates. The parabolic nature of the profile results from the solution of the Navier–Stokes equations for steady, incompressible flow in a channel with no-slip boundary conditions. The profile achieves its maximum value at the channel mid?plane, making it a useful model for understanding shear and velocity distributions in such flows.

Volume Flow Rate Calculation by Integration

The volume flow rate per unit depth in a channel is obtained through integration of the velocity profile across the gap between the plates. By integrating the profile over the spatial variable from one boundary to the other, one calculates the total volumetric delivery per unit depth. This integral also aids in finding the average velocity in the channel, which is then compared to the maximum velocity to provide insights into the flow distribution.

Relationship Between Maximum and Average Velocity

For a parabolic velocity profile, the average velocity is typically a constant fraction of the maximum velocity. After integration across the gap, the ratio of the average velocity to the maximum velocity emerges as a simple numerical value. This relationship is a key characteristic of plane Poiseuille flow and reflects the distribution of velocity in the channel, highlighting that although the maximum speed occurs at mid-channel, the overall mean speed is lower due to the no-slip conditions at the boundaries.

Transcript

00:01 The velocity profile here is given as u is equal to u is equal to u max multiplied by u max multiplied by a y square plus plus b y plus c let this equation be equation number one now we will write boundary condition at y is equal to zero we have velocity u is equal to zero at y is equals to h at y is equal to h we have velocity u is equal to 0.

00:36 At at at y is equals to h by 2 we have u is equals to u max u is equal to u max let this equation be equation number 2 this is equation number 3 and this is equation number 4 now substituting the value of equation 2 in equation 1 we get by by equation 1 .2 we get we get 0 is equal to u max multiplied by 0 plus 0 plus c from here we get this value of constant c is equals to 0 now from equation 3 and 1 from equation 3 comma 1 we have we have we have have 0 is equal to u max multiplied by a h square plus b h plus c or from here we have bh is equals to minus a h square or from here we get b is equals to minus a h from equation 4 .1 we have u max is equal to u max multiplied by a h squared by 4 plus b h by 2 plus c or we can write 1 is equal to a h squared divided by 4 h squared divided power minus a h squared divided by 2 since b is equals to minus a h or from here we get a is equal to a is equal to minus 4 divided by h squared so this is the value of a in terms of h now we will put this in equation 5 so that we get value of b.

03:07 From equation 5, from equation 5 we have b is equal to we get b is equals to minus 4 divided by h squared by h or it will be equals to 4 divided by h.

03:32 Substitute of the value of a, b, c, in equation 1, we get u is equal to u max multiplied by a y square, value of a is minus 4 divided by h square, a y square plus b, the value of b is 4 divided by h, vy plus c c, the value of c is 0.

03:53 Hence, this is the, this is the velocity profile in in terms of h.

04:01 Now we will calculate the volume flow rate q which is equal to integration of 0 to h velocity multiplied by area, area is b, d ,y.

04:12 So let's put the value of u in terms of u max...

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