Velocity Profile of a Flow

Navier-Stokes equation again, but this time with the initial and boundary conditions for v(y, t) to obtain the following equation for v_1(y, t):ρ (∂v_1/∂t + v(∂v_1/∂y)) = -∂p_1/∂x + μ ∂²v_1/∂y²where p_1 is the pressure at steady state. The boundary conditions for v_1(y, t) are given by:v_1(-a, t) = v_1(a, t) = 0This means that the fluid does not move at
  • #1
Sophist
1
0
Can someone please help me to solve this problem?

Stokes’ fluid is resting in a long channel, whose boundary planes are [tex]y = a[/tex] and [tex]y = -a[/tex] (which do not move). At time [tex]t = 0[/tex] a pressure gradient suddenly begins to act which is constant and equals [tex]G = G e_x[/tex]. If body forces do not exist, and velocity is [tex]v = v (y, t) e_x[/tex], find the equation which [tex]v(y, t) [/tex] satisfies for [tex]t > 0[/tex], as well as boundary and beginning conditions for [tex]v(y, t) [/tex]. When t --> inftinity v(y, t) is expected not to dependant on time t (motion becomes stationary). If we suppose that [tex]v(y,t) = V(y) + v_1 (y, t) [/tex], where [tex]V(y)=lim_{t-->infinity}v(y,t) [/tex] form the eqauation and boundary conditions that [tex]v_1 (y, t) [/tex] satisfies.

Thank you,
 
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  • #2


I am happy to help you solve this problem. Stokes' fluid is a type of fluid that exhibits laminar flow, which means that it flows in smooth layers without any turbulence. In your problem, the fluid is resting in a long channel with boundaries at y = a and y = -a, and a pressure gradient suddenly begins to act at time t = 0. The velocity of the fluid is given by v = v(y, t) e_x, where e_x is the unit vector in the x direction.

To find the equation that v(y, t) satisfies for t > 0, we can use the Navier-Stokes equation, which describes the motion of a fluid. In this case, since there are no body forces acting on the fluid, the Navier-Stokes equation reduces to:

ρ (∂v/∂t + v ⋅ ∇v) = -∇p + μ ∇²v

where ρ is the density of the fluid, p is the pressure, and μ is the viscosity. We can also use the continuity equation, which states that the rate of change of mass in a fluid is equal to the divergence of the velocity field:

∂ρ/∂t + ∇ ⋅ (ρv) = 0

Combining these two equations and using the given conditions, we can obtain the following equation for v(y, t):

ρ (∂v/∂t + v(∂v/∂y)) = -∂p/∂x + μ ∂²v/∂y²

This equation is known as the one-dimensional Navier-Stokes equation for laminar flow.

Next, we need to determine the boundary and initial conditions for v(y, t). At t = 0, the pressure gradient suddenly begins to act, so we can set the initial condition as:

v(y, 0) = 0

This means that initially, the fluid is at rest. As t approaches infinity, the motion becomes stationary, so we can set the boundary condition as:

v(y, ∞) = V(y)

where V(y) is the velocity of the fluid at steady state. This means that as time goes on, the velocity of the fluid will approach V(y).

Now, let's consider the equation and boundary conditions for v_1(y, t). Since v_1(y, t) represents
 

What is the velocity profile of a flow?

The velocity profile of a flow is a graphical representation of how the velocity of a fluid varies across its cross-sectional area. It shows the distribution of velocities, with the highest velocity typically being in the center and decreasing towards the edges.

What factors affect the velocity profile of a flow?

The velocity profile of a flow is affected by several factors including the type of fluid, the shape and size of the channel or pipe, the flow rate, and the presence of obstacles or rough surfaces within the flow.

What is the significance of the velocity profile in fluid dynamics?

The velocity profile is an important concept in fluid dynamics as it provides valuable information about the behavior of a fluid in motion. It can help determine the flow rate, pressure distribution, and energy losses within a flow, and is also used in the design and analysis of various fluid systems.

How is the velocity profile of a flow measured or calculated?

The velocity profile can be measured using various techniques such as flow visualization, laser Doppler anemometry, and hot-wire anemometry. It can also be calculated using mathematical equations and numerical methods based on the principles of fluid mechanics.

What are the different types of velocity profiles?

The velocity profile of a flow can vary depending on the type of flow, such as laminar or turbulent, and the shape of the channel or pipe. Some common types include parabolic, flat, and logarithmic velocity profiles. The type of profile can also change along the length of the flow, depending on the flow conditions.

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