How to rewrite this differential equation
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I have the following NS equation and conservation equation:
$$nunabla^2u + v_0frac{partial u}{partial z} - ucdotnabla u - frac{1}{rho}nabla p = frac{partial u}{partial t}$$
$$nablacdot u=0$$
Now, I would like to manipulate the above equations to eliminate $x$ and $y$ components of $u$ together with the pressure $p$, such that I obtain the equation:
$$nunabla^4u_z + v_0frac{partial }{partial z}nabla^2u_z = frac{partial }{partial t}nabla^2u_z$$
Can anyone explain how to get the above final equation?
ordinary-differential-equations pde vectors fluid-dynamics
asked Dec 11 '18 at 9:37
newstudentnewstudent
366
$endgroup$
add a comment |
$begingroup$
I have the following NS equation and conservation equation:
$$nunabla^2u + v_0frac{partial u}{partial z} - ucdotnabla u - frac{1}{rho}nabla p = frac{partial u}{partial t}$$
$$nablacdot u=0$$
Now, I would like to manipulate the above equations to eliminate $x$ and $y$ components of $u$ together with the pressure $p$, such that I obtain the equation:
$$nunabla^4u_z + v_0frac{partial }{partial z}nabla^2u_z = frac{partial }{partial t}nabla^2u_z$$
Can anyone explain how to get the above final equation?
ordinary-differential-equations pde vectors fluid-dynamics
asked Dec 11 '18 at 9:37
newstudentnewstudent
366
$endgroup$
1
$begingroup$
How exactly do you want to eliminate $u$ from the equation?
$endgroup$
– rafa11111
Dec 11 '18 at 11:47
$begingroup$
If $u$ is a scalar field then your second equation means that its gradient is $0$ so the respective term in the first equation vanishes
$endgroup$
– Yuriy S
Dec 11 '18 at 11:56
add a comment |
$begingroup$
I have the following NS equation and conservation equation:
$$nunabla^2u + v_0frac{partial u}{partial z} - ucdotnabla u - frac{1}{rho}nabla p = frac{partial u}{partial t}$$
$$nablacdot u=0$$
Now, I would like to manipulate the above equations to eliminate $x$ and $y$ components of $u$ together with the pressure $p$, such that I obtain the equation:
$$nunabla^4u_z + v_0frac{partial }{partial z}nabla^2u_z = frac{partial }{partial t}nabla^2u_z$$
Can anyone explain how to get the above final equation?
ordinary-differential-equations pde vectors fluid-dynamics
asked Dec 11 '18 at 9:37
newstudentnewstudent
366
$endgroup$
I have the following NS equation and conservation equation:
$$nunabla^2u + v_0frac{partial u}{partial z} - ucdotnabla u - frac{1}{rho}nabla p = frac{partial u}{partial t}$$
$$nablacdot u=0$$
Now, I would like to manipulate the above equations to eliminate $x$ and $y$ components of $u$ together with the pressure $p$, such that I obtain the equation:
$$nunabla^4u_z + v_0frac{partial }{partial z}nabla^2u_z = frac{partial }{partial t}nabla^2u_z$$
Can anyone explain how to get the above final equation?
ordinary-differential-equations pde vectors fluid-dynamics
ordinary-differential-equations pde vectors fluid-dynamics
asked Dec 11 '18 at 9:37
newstudentnewstudent
366
asked Dec 11 '18 at 9:37
newstudentnewstudent
366
edited Dec 11 '18 at 15:26
newstudent
asked Dec 11 '18 at 9:37
newstudentnewstudent
366
asked Dec 11 '18 at 9:37
newstudentnewstudent
366
asked Dec 11 '18 at 9:37
newstudentnewstudent
366
366
1
$begingroup$
How exactly do you want to eliminate $u$ from the equation?
$endgroup$
– rafa11111
Dec 11 '18 at 11:47
$begingroup$
If $u$ is a scalar field then your second equation means that its gradient is $0$ so the respective term in the first equation vanishes
$endgroup$
– Yuriy S
Dec 11 '18 at 11:56
add a comment |
1
$begingroup$
How exactly do you want to eliminate $u$ from the equation?
$endgroup$
– rafa11111
Dec 11 '18 at 11:47
$begingroup$
If $u$ is a scalar field then your second equation means that its gradient is $0$ so the respective term in the first equation vanishes
$endgroup$
– Yuriy S
Dec 11 '18 at 11:56
1
1
$begingroup$
How exactly do you want to eliminate $u$ from the equation?
$endgroup$
– rafa11111
Dec 11 '18 at 11:47
$begingroup$
How exactly do you want to eliminate $u$ from the equation?
$endgroup$
– rafa11111
Dec 11 '18 at 11:47
$begingroup$
If $u$ is a scalar field then your second equation means that its gradient is $0$ so the respective term in the first equation vanishes
$endgroup$
– Yuriy S
Dec 11 '18 at 11:56
$begingroup$
If $u$ is a scalar field then your second equation means that its gradient is $0$ so the respective term in the first equation vanishes
$endgroup$
– Yuriy S
Dec 11 '18 at 11:56
add a comment |
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1
$begingroup$
How exactly do you want to eliminate $u$ from the equation?
$endgroup$
– rafa11111
Dec 11 '18 at 11:47
$begingroup$
If $u$ is a scalar field then your second equation means that its gradient is $0$ so the respective term in the first equation vanishes
$endgroup$
– Yuriy S
Dec 11 '18 at 11:56