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Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate moves with a constant velocity of U = 4m/s. Separation between the plates is 5m m . There is no pressure gradient in the direction of flow. The density of fluid is 800 kg/m3 , and the kinematic viscosity is 1.25 ×10−4 m2 /s . The average shear stress in the fluid is ________ Pa (round off to the nearest integer).
Correct answer is '80'. Can you explain this answer?
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Consider fully developed, steady state incompressible laminar flow of ...
Given : ρ= 800 kg/m3 , v = 1.25 ×10−4 m2 /s
For plane coutte flow
Linearization of Newton’s law of viscosity is used
τ = 80 Pa
For plane coutte flow
Linearization of Newton’s law of viscosity is used
τ = 80 Pa
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Consider fully developed, steady state incompressible laminar flow of ...
X10^-6 m^2/s.
To analyze this flow, we can use the Navier-Stokes equation for incompressible flow, which is given by:
ρ(u∂u/∂x + v∂u/∂y) = μ(∂^2u/∂x^2 + ∂^2u/∂y^2)
where ρ is the density of the fluid, u is the velocity in the x-direction, v is the velocity in the y-direction, μ is the dynamic viscosity of the fluid, and x and y are the spatial coordinates.
Since there is no pressure gradient in the x-direction, we can assume the flow is fully developed and the velocity in the y-direction is zero. Therefore, the equation simplifies to:
ρ(u∂u/∂x) = μ(∂^2u/∂x^2)
Since the flow is laminar and steady state, we can assume that the velocity profile is parabolic. Therefore, we can assume that:
u(y) = U(y/h)^2
where U is the velocity of the top plate, y is the vertical distance from the bottom plate, and h is the separation between the plates.
Using this assumption, we can calculate the velocity gradient in terms of U and h:
∂u/∂x = 2U(y/h)^2/h = 2Uy/h^3
Plugging this into the simplified Navier-Stokes equation, we get:
ρ(2Uy/h^3)(2U/h^3) = μ(∂^2u/∂x^2)
Simplifying further, we get:
4U^2y/h^6 = μ(∂^2u/∂x^2)
Since the left side of the equation is a constant, we can solve for the second derivative of u with respect to x:
(∂^2u/∂x^2) = 4U^2y/(μh^6)
Integrating this equation twice with respect to x, we get:
u = (2U^2y/(μh^6))x^2 + c1x + c2
Using the boundary condition that the velocity at y=0 is zero, we can determine that c2 = 0.
Using the boundary condition that the velocity at y=h is U, we can determine that c1 = -2U^2/h^2.
Therefore, the velocity profile in the x-direction is given by:
u = (2U^2y/(μh^6))x^2 - (2U^2/h^2)x
To find the velocity at a specific point, we can plug in the values for U, μ, h, x, and y into this equation.
To analyze this flow, we can use the Navier-Stokes equation for incompressible flow, which is given by:
ρ(u∂u/∂x + v∂u/∂y) = μ(∂^2u/∂x^2 + ∂^2u/∂y^2)
where ρ is the density of the fluid, u is the velocity in the x-direction, v is the velocity in the y-direction, μ is the dynamic viscosity of the fluid, and x and y are the spatial coordinates.
Since there is no pressure gradient in the x-direction, we can assume the flow is fully developed and the velocity in the y-direction is zero. Therefore, the equation simplifies to:
ρ(u∂u/∂x) = μ(∂^2u/∂x^2)
Since the flow is laminar and steady state, we can assume that the velocity profile is parabolic. Therefore, we can assume that:
u(y) = U(y/h)^2
where U is the velocity of the top plate, y is the vertical distance from the bottom plate, and h is the separation between the plates.
Using this assumption, we can calculate the velocity gradient in terms of U and h:
∂u/∂x = 2U(y/h)^2/h = 2Uy/h^3
Plugging this into the simplified Navier-Stokes equation, we get:
ρ(2Uy/h^3)(2U/h^3) = μ(∂^2u/∂x^2)
Simplifying further, we get:
4U^2y/h^6 = μ(∂^2u/∂x^2)
Since the left side of the equation is a constant, we can solve for the second derivative of u with respect to x:
(∂^2u/∂x^2) = 4U^2y/(μh^6)
Integrating this equation twice with respect to x, we get:
u = (2U^2y/(μh^6))x^2 + c1x + c2
Using the boundary condition that the velocity at y=0 is zero, we can determine that c2 = 0.
Using the boundary condition that the velocity at y=h is U, we can determine that c1 = -2U^2/h^2.
Therefore, the velocity profile in the x-direction is given by:
u = (2U^2y/(μh^6))x^2 - (2U^2/h^2)x
To find the velocity at a specific point, we can plug in the values for U, μ, h, x, and y into this equation.
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Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate moves with a constantvelocity of U = 4m/s. Separation between the plates is 5m m . There is no pressure gradient in the direction of flow. The density of fluid is 800 kg/m3 , and the kinematic viscosity is 1.25 ×10−4 m2 /s . The average shear stress in the fluid is ________ Pa (round off to the nearest integer).Correct answer is '80'. Can you explain this answer?
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Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate moves with a constantvelocity of U = 4m/s. Separation between the plates is 5m m . There is no pressure gradient in the direction of flow. The density of fluid is 800 kg/m3 , and the kinematic viscosity is 1.25 ×10−4 m2 /s . The average shear stress in the fluid is ________ Pa (round off to the nearest integer).Correct answer is '80'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate moves with a constantvelocity of U = 4m/s. Separation between the plates is 5m m . There is no pressure gradient in the direction of flow. The density of fluid is 800 kg/m3 , and the kinematic viscosity is 1.25 ×10−4 m2 /s . The average shear stress in the fluid is ________ Pa (round off to the nearest integer).Correct answer is '80'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate moves with a constantvelocity of U = 4m/s. Separation between the plates is 5m m . There is no pressure gradient in the direction of flow. The density of fluid is 800 kg/m3 , and the kinematic viscosity is 1.25 ×10−4 m2 /s . The average shear stress in the fluid is ________ Pa (round off to the nearest integer).Correct answer is '80'. Can you explain this answer?.
Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate moves with a constantvelocity of U = 4m/s. Separation between the plates is 5m m . There is no pressure gradient in the direction of flow. The density of fluid is 800 kg/m3 , and the kinematic viscosity is 1.25 ×10−4 m2 /s . The average shear stress in the fluid is ________ Pa (round off to the nearest integer).Correct answer is '80'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate moves with a constantvelocity of U = 4m/s. Separation between the plates is 5m m . There is no pressure gradient in the direction of flow. The density of fluid is 800 kg/m3 , and the kinematic viscosity is 1.25 ×10−4 m2 /s . The average shear stress in the fluid is ________ Pa (round off to the nearest integer).Correct answer is '80'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider fully developed, steady state incompressible laminar flow of a viscous fluid between two large parallel horizontal plates. The bottom plate is fixed and the top plate moves with a constantvelocity of U = 4m/s. Separation between the plates is 5m m . There is no pressure gradient in the direction of flow. The density of fluid is 800 kg/m3 , and the kinematic viscosity is 1.25 ×10−4 m2 /s . The average shear stress in the fluid is ________ Pa (round off to the nearest integer).Correct answer is '80'. Can you explain this answer?.
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