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| {{Differential equations}}
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>Should you cherished this information in addition to you would like to acquire more info relating to [http://www.youtube.com/watch?v=90z1mmiwNS8 Dentists in DC] i implore you to check out our own webpage. |
| The '''finite-volume method''' (FVM) is a method for representing and evaluating [[partial differential equation]]s in the form of algebraic equations [LeVeque, 2002; Toro, 1999].
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| Similar to the [[finite difference method]] or [[finite element method]], values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a [[divergence]] term are converted to [[surface integral]]s, using the [[divergence theorem]]. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are [[conservation law|conservative]]. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many [[computational fluid dynamics]] packages.
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| ==1D example==
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| Consider a simple 1D [[advection]] problem defined by the following [[partial differential equation]]
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| :<math>\quad (1) \qquad \qquad \frac{\partial\rho}{\partial t}+\frac{\partial f}{\partial x}=0,\quad t\ge0.</math>
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| Here, <math> \rho=\rho \left( x,t \right) \ </math> represents the state variable and <math> f=f \left( \rho \left( x,t \right) \right) \ </math> represents the [[flux]] or flow of <math> \rho \ </math>. Conventionally, positive <math> f \ </math> represents flow to the right while negative <math> f \ </math> represents flow to the left. If we assume that equation (1) represents a flowing medium of constant area, we can sub-divide the spatial domain, <math> x \ </math>, into ''finite volumes'' or ''cells'' with cell centres indexed as <math> i \ </math>. For a particular cell, <math> i \ </math>, we can define the ''volume average'' value of <math> {\rho }_i \left( t \right) = \rho \left( x, t \right) \ </math> at time <math> {t = t_1 }\ </math> and <math>{ x \in \left[ x_{i-\frac{1}{2}} , x_{i+\frac{1}{2}} \right] }\ </math>, as | |
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| :<math>\quad (2) \qquad \qquad \bar{\rho}_i \left( t_1 \right) = \frac{1}{ x_{i+\frac{1}{2}} - x_{i-\frac{1}{2}}} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \rho \left(x,t_1 \right)\, dx ,</math>
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| and at time <math> {t = t_2}\ </math> as,
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| :<math>\quad (3) \qquad \qquad \bar{\rho}_i \left( t_2 \right) = \frac{1}{x_{i+\frac{1}{2}} - x_{i-\frac{1}{2}}} \int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \rho \left(x,t_2 \right)\, dx ,</math> | |
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| where <math> x_{i-\frac{1}{2}} \ </math> and <math> x_{i+\frac{1}{2}} \ </math> represent locations of the upstream and downstream faces or edges respectively of the <math> i^{th} \ </math> cell.
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| Integrating equation (1) in time, we have:
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| :<math>\quad (4) \qquad \qquad \rho \left( x, t_2 \right) = \rho \left( x, t_1 \right) - \int_{t_1}^{t_2} f_x \left( x,t \right)\, dt,</math> | |
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| where <math>f_x=\frac{\partial f}{\partial x}</math>.
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| To obtain the volume average of <math> \rho\left(x,t\right) </math> at time <math> t=t_{2} \ </math>, we integrate <math> \rho\left(x,t_2 \right) </math> over the cell volume, <math>\left[ x_{i-\frac{1}{2}} , x_{i+\frac{1}{2}} \right] </math> and divide the result by <math>\Delta x_i = x_{i+\frac{1}{2}}-x_{i-\frac{1}{2}} </math>, i.e.
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| :<math> \quad (5) \qquad \qquad \bar{\rho}_{i}\left( t_{2}\right) =\frac{1}{\Delta x_i}\int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}\left\{ \rho\left( x,t_{1}\right) - \int_{t_{1}}^{t_2} f_{x} \left( x,t \right) dt \right\} dx.</math>
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| We assume that <math> f \ </math> is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension <math>f_x \triangleq \nabla f </math>, we can apply the [[divergence theorem]], i.e. <math>\oint_{v}\nabla\cdot fdv=\oint_{S}f\, dS </math>, and substitute for the volume integral of the [[divergence]] with the values of <math>f(x) \ </math> evaluated at the cell surface (edges <math>x_{i-\frac{1}{2}} \ </math> and <math> x_{i+\frac{1}{2}} \ </math>) of the finite volume as follows:
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| :<math>\quad (6) \qquad \qquad \bar{\rho}_i \left( t_2 \right) = \bar{\rho}_i \left( t_1 \right)
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| - \frac{1}{\Delta x_{i}}
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| \left( \int_{t_1}^{t_2} f_{i + \frac{1}{2}} dt
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| - \int_{t_1}^{t_2} f_{i - \frac{1}{2}} dt
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| \right) .</math>
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| where <math>f_{i \pm \frac{1}{2}} =f \left( x_{i \pm \frac{1}{2}}, t \right) </math>.
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| We can therefore derive a ''semi-discrete'' numerical scheme for the above problem with cell centres indexed as <math> i\ </math>, and with cell edge fluxes indexed as <math> i\pm\frac{1}{2} </math>, by differentiating (6) with respect to time to obtain:
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| :<math>\quad (7) \qquad \qquad \frac{d \bar{\rho}_i}{d t} + \frac{1}{\Delta x_i} \left[
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| f_{i + \frac{1}{2}} - f_{i - \frac{1}{2}} \right] =0 ,</math>
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| where values for the edge fluxes, <math> f_{i \pm \frac{1}{2}} </math>, can be reconstructed by interpolation or extrapolation of the cell averages. Equation (7) is ''exact'' for the volume averages; i.e., no approximations have been made during its derivation.
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| == General conservation law ==
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| We can also consider the general [[conservation law]] problem, represented by the following [[partial differential equation|PDE]],
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| :<math> \quad (8) \qquad \qquad {{\partial {\mathbf u}} \over {\partial t}} + \nabla \cdot {\mathbf f}\left( {\mathbf u } \right) = {\mathbf 0} . </math>
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| Here, <math> {\mathbf u} \ </math> represents a vector of states and <math>\mathbf f \ </math> represents the corresponding [[flux]] tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, <math>i \ </math>, we take the volume integral over the total volume of the cell, <math>v _{i} \ </math>, which gives,
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| :<math> \quad (9) \qquad \qquad \int _{v_{i}} {{\partial {\mathbf u}} \over {\partial t}}\, dv
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| + \int _{v_{i}} \nabla \cdot {\mathbf f}\left( {\mathbf u } \right)\, dv = {\mathbf 0} .</math>
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| On integrating the first term to get the ''volume average'' and applying the ''divergence theorem'' to the second, this yields
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| :<math>\quad (10) \qquad \qquad
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| v_{i} {{d {\mathbf {\bar u} }_{i} } \over {dt}} + \oint _{S_{i} }
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| {\mathbf f} \left( {\mathbf u } \right) \cdot {\mathbf n }\ dS = {\mathbf 0}, </math>
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| where <math> S_{i} \ </math> represents the total surface area of the cell and <math>{\mathbf n}</math> is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (8), i.e.
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| :<math> \quad (11) \qquad \qquad
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| {{d {\mathbf {\bar u} }_{i} } \over {dt}} + {{1} \over {v_{i}} } \oint _{S_{i} }
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| {\mathbf f} \left( {\mathbf u } \right)\cdot {\mathbf n }\ dS = {\mathbf 0} .</math>
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| Again, values for the edge fluxes can be reconstructed by interpolation or extrapolation of the cell averages. The actual numerical scheme will depend upon problem geometry and mesh construction. [[MUSCL scheme|MUSCL]] reconstruction is often used in [[high resolution scheme]]s where shocks or discontinuities are present in the solution.
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| Finite volume schemes are conservative as cell averages change through the edge fluxes. In other words, ''one cell's loss is another cell's gain''!
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| ==See also==
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| *[[Finite element method]]
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| *[[Flux limiter]]
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| *[[Godunov's scheme]]
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| *[[Godunov's theorem]]
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| *[[High-resolution scheme]]
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| *[[KIVA (Software)]]
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| *[[MIT General Circulation Model]]
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| *[[MUSCL scheme]]
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| *[[Sergei K. Godunov]]
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| *[[Total variation diminishing]]
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| *[[Finite volume method for unsteady flow]]
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| ==References==
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| *'''Eymard, R. Gallouët, T. R. Herbin, R.''' (2000) ''The finite volume method'' Handbook of Numerical Analysis, Vol. VII, 2000, p. 713–1020. Editors: P.G. Ciarlet and J.L. Lions.
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| *'''LeVeque, Randall''' (2002), ''Finite Volume Methods for Hyperbolic Problems'', Cambridge University Press.
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| *'''Toro, E. F.''' (1999), ''Riemann Solvers and Numerical Methods for Fluid Dynamics'', Springer-Verlag.
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| ==Further reading==
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| *'''Patankar, Suhas V.''' (1980), ''Numerical Heat Transfer and Fluid Flow'', Hemisphere.
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| *'''Hirsch, C.''' (1990), ''Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows'', Wiley.
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| *'''Laney, Culbert B.''' (1998), ''Computational Gas Dynamics'', Cambridge University Press.
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| *'''LeVeque, Randall''' (1990), ''Numerical Methods for Conservation Laws'', ETH Lectures in Mathematics Series, Birkhauser-Verlag.
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| *'''Tannehill, John C.''', et al., (1997), ''Computational Fluid mechanics and Heat Transfer'', 2nd Ed., Taylor and Francis.
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| *'''Wesseling, Pieter''' (2001), ''Principles of Computational Fluid Dynamics'', Springer-Verlag.
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| == External links ==
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| * [http://www.cmi.univ-mrs.fr/~herbin/PUBLI/bookevol.pdf The finite volume method] by R. Eymard, T Gallouët and R. Herbin, update of the article published in Handbook of Numerical Analysis, 2000
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| * [http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/imsTOC/Lectures%20and%20Tips/Simulation%20I/FVM_introDocu.html The Finite Volume Method (FVM) – An introduction] {{dead link|date=January 2010}} by Oliver Rübenkönig of [[Albert Ludwigs University of Freiburg]], available under the [[GNU Free Document License|GFDL]]. ({{wayback|url=http://www.imtek.uni-freiburg.de/simulation/mathematica/IMSweb/imsTOC/Lectures%20and%20Tips/Simulation%20I/FVM_introDocu.html}})
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| * [http://www.ctcms.nist.gov/fipy/ FiPy: A Finite Volume PDE Solver Using Python] from NIST.
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| * [http://depts.washington.edu/clawpack/ CLAWPACK]: a software package designed to compute numerical solutions to hyperbolic partial differential equations using a wave propagation approach
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| {{Numerical PDE}}
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| [[Category:Numerical differential equations]]
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| [[Category:Fluid dynamics]]
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| [[Category:Computational fluid dynamics]]
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| [[Category:Numerical analysis]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
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